Take the Navier –Stokes equation in its standard form shown in Fluid Mechanics books:

fluid density (local accel. + convective accel.) = - grad p + μ Laplacian u + fluid density x g

Notice that all terms have dimensions of force/fluid unit volume.

Inertial force (per fluid unit volume) is simply the name of the term to the left of the equal sign in the above equation. The justification usually given for this nomenclature is related to the fact that the term is proportional to fluid inertial mass, via fluid density. Then, the left hand side can be written as [ local (or unsteady) inertial force + convective inertial force ] / fluid unit volume.

Notice also that, from the right hand side of the above equation, it is clear that viscous forces (per unit fluid volume), pressure forces (per fluid unit volume) and gravitational forces (per fluid unit volume) are all parts of the inertial force !

Unfortunately, a frequent confusion arises when non inertial frames of reference are used. These allow for forces like centrifugal and Coriolis (only perceived by observers in such reference frames) to appear, in addition to the more common interaction forces. Centrifugal and Coriolis forces are also called inertial forces, for the same reason: they are proportional to the fluid inertial mass. For this reason they are preferably referred to as fictitious forces.

These are not defined technical terms but have their meaning from plain language:

convective inertial force = inertial force as observed in convective motion

unsteady inertial force = inertial force as observed in unsteady (e.g. turbulent) motion.

If you found the words in a context for which my explanation is not plausibe please give an example of this usage.

What is the physical meaning of those?

Hi, 

Thanks for your reply.

In two phase flow, particle can have induced inertial force. 

My question is that whether this induced inertial force is due to the fluids own convective and unsteady force?

Please read it and give some comments.

Thanks

Vivek

Vivek, are you speaking of liquids or of particles suspended in a liquid? What are the two phases? You see, you probably gave too little information for enabling a reasonable answer.

Sorry for little information.

I am talking about the multiphase flow, where liquid is continuous phase and particles are dispersed phase.

Inherently fluid in motion will have unsteady and convective inertial forces. My question is why they are termed as inertial forces? Then other terms in the N-S equation, like diffusive transport term can also be said as the inertial term?

Now we can talk about the particle motion. 

Due to the fluid motion, particle will experience some induced inertial force/acceleration.

Is this inertial force equivalent to the buoyancy force?

Thanks

Vivek

What the particles feel from the surrounding liquid is not only force but also torque. Both can be seen also in situations where there is no buoyancy. Unfortunately I'm not familiar with contemporary work on multiphase flow (I once worked on granular flow) and don't know the established terminology of that field. I would guess that the most efficient way to understand all these things would be to familiarize yourself with one of the existing software tool-boxes for simulating multiphase systems on a computer.

Inertial force arises due to the shear interaction with the background medium for the propagation of light. Viscous force arises from the physical interaction of a fluid medium comprised of the particles of gross matter.

Dear Vivek,

You said in you question that "momentum equation by the term ρ(du/dt) or (ρv)v. "

Can you please tell me how inertia force is equal to (ρv)v ?

The inertial forces follow directly from Newton's laws of motion expressed in an inertial frame of reference. They are only exposed however with respect to a polar point origin. They are radial and transverse with respect to that origin and they follow from the tendency of a body to move in uniform straight line motion in the absence of any externally applied forces.

Take the Navier –Stokes equation in its standard form shown in Fluid Mechanics books:

fluid density (local accel. + convective accel.) = - grad p + μ Laplacian u + fluid density x g

Notice that all terms have dimensions of force/fluid unit volume.

Inertial force (per fluid unit volume) is simply the name of the term to the left of the equal sign in the above equation. The justification usually given for this nomenclature is related to the fact that the term is proportional to fluid inertial mass, via fluid density. Then, the left hand side can be written as [ local (or unsteady) inertial force + convective inertial force ] / fluid unit volume.

Notice also that, from the right hand side of the above equation, it is clear that viscous forces (per unit fluid volume), pressure forces (per fluid unit volume) and gravitational forces (per fluid unit volume) are all parts of the inertial force !

Unfortunately, a frequent confusion arises when non inertial frames of reference are used. These allow for forces like centrifugal and Coriolis (only perceived by observers in such reference frames) to appear, in addition to the more common interaction forces. Centrifugal and Coriolis forces are also called inertial forces, for the same reason: they are proportional to the fluid inertial mass. For this reason they are preferably referred to as fictitious forces.

R. P. Peçanha,

                        The inertial forces follow directly from Newton's laws of motion in an inertial frame of reference. They show up as transverse and radial forces when expressed in polar coordinates. Contrary to what the text books tell you, a rotating frame of reference has got nothing to do with inertial forces, although if a system is actually rotating and an object is dragged such as to co-rotate, this can isolate the already existing inertial forces by contrast with the rotating system.

Frederick David Tombe

It seems you miss the point I made (last paragraph in my message). In Mechanics the adjective “inertial” is commonly used in two ways: (1) in naming the acceleration term of Newton’s second law (proportional to the body inertial mass), which implies an inertial reference frame; (2) in classifying forces (like centrifugal and Coriolis), only perceived by observers in non-inertial reference frames, WHICH ARE ALSO PROPORTIONAL TO THE INERTIAL MASS OF THE BODY UNDER ANALYSIS (actually, this fact also give  those forces the status of body forces).

Take a look in Classical Mechanics books. For instance, Analytical Mechanics by Fowles, Classical Dynamics by Thornton-Marion, An Introduction to Mechanics by Kleppner –Kolenkow, Classical Mechanics by Taylor.     

R. P. Peçanha,

                        Yes, it is the second meaning which I have in mind. We are talking about the same thing. I'm not talking about inertial mass. I am indeed talking about the inertial forces that you are talking about.

                        The textbooks may well claim that the inertial forces only show up in a rotating frame of reference. But you need to do a bit of critical thinking, because all that a rotating frame of reference does is superimpose a circular motion on top of the already existing motion. The actual inertial forces are intrinsic to the inertial path and follow from Newton's three laws of motion. They are expressed in polar coordinates relative to an inertial frame of reference.

                   I have written many articles on this topic. This one here is an example,

http://gsjournal.net/Science-Journals/Research%20Papers-Mathematical%20Physics/Download/6220

            And please have a look at the attached picture. Do we need to be in a rotating frame of reference to observe what will happen?

http://gsjournal.net/Science-Journals/Research%20Papers-Mathematical%20Physics/Download/6220

Frederick David Tombe

Do we agree that a referential frame is a material body where an observer (human being or sensor or robot) stands? Examples: your own body, the Earth, the car you are driving, the plane you are traveling, the sun, the Sirius star (in Canis Majoris) or any of the 212 extragalactic objects forming the International Celestial Reference Frame (ICRF: check Google) used by space agencies. Do we also agree that none of these are 100% inertial frames, but only sufficiently inertial for very different purposes? So, in discussing inertial forces (the second meaning in my previous text !), why bother with specific coordinate systems, like polar coordinates you referred to? Vector characteristics (magnitude etc.) do not depend on the coordinate system. Also, a linearly accelerated frame relative to, say, Earth (taken as a sufficiently inertial in a given problem) is a non-inertial reference frame. In other words, rotation is not essential to give rise to inertial forces; acceleration is ! Actually, Lanczos (The Variational Principles of Mechanics, 1970) call Einstein forces those associated with linear acceleration of the reference frame. Thus, imposing the validity of Newton’s law in a non-inertial reference frame leads mathematically (coordinate transformation) to inertial forces (or fictitious forces as preferred by some) proportional to the body inertial mass.

Regarding the picture, the man attached to the rotating propeller can be a very complex non-inertial observer. The worst case will occur when the plane is flying in a curved trajectory and the propeller accelerates (angular acceleration) under the pilot action. In this case the man on the propeller would perceive Euler forces (after Lanczos, 1970) in addition to centrifugal and Coriolis forces, acting on an external body moving relative to him.

I’ll have a look on the link sent.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               

R. P. Peçanha,

                        It's got nothing to do with rotating frames of reference, or accelerating frames of reference, or coordinate systems. When a body is undergoing its uniform straight line inertial path in the absence of any other forces, there will be a centrifugal force acting on it relative to any arbitrarily chosen point origin. That's a reality which becomes apparent in a gravitational field or if we apply a constraint. A centrifugal force acts on the constraint leading to a reactive centripetal force which curves the path of motion.

                        In the picture, an outward centrifugal force will kill the man who is strapped to the propeller. That's a reality which requires no rotating frame of reference in order to observe it.

Frederick David Tombe

We drifted too much away from the original question posed by Vivek Arora. The question was answered in my first message and, apparently, satisfied him.

The question whether the man in the cartoon would die or not, is in fact not relevant for the discussion (Newton’s Laws). For that purpose, an adequate sensor or robot could take his place easily. Also, as you certainly know, astronauts are tested in huge centrifuges and they can withstand accelerations up to 9 g without loss of consciousness. Death risk is around 15 g for a period of 1 minute (Google High-G training).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            

The cartoon was relevant for the purpose of making one aware that centrifugal force is a physical reality that is not dependent on us observing the effects from a rotating frame of reference.

The man strapped to the propeller (cartoon) would die most certainly. For simplicity let the plane be grounded with the propeller rotating with a high constant angular velocity. The Earth is sufficiently inertial to analyze the man’s motion:

(1) From the point of view of a grounded observer (that is, an inertial one), the man will die because his body parts and blood circulatory system would not stand the centripetal acceleration imposed on him. This situation is similar to that of astronauts training on the high g centrifuge. Just remember that the associated centripetal force is not a new force; centripetal is merely an adjective indicative of direction, for the contact forces exerted by the straps on the man.

(2) From the point of view of the strapped man himself, a non-inertial observer since he is accelerated in relation to the Earth, he would die because his body parts would not stand the centrifugal and Coriolis (his blood is flowing!) accelerations imposed on him.

Conclusion: from the strict point of view of Newton’s law, the cause of his death depends on the referential used by the observer.  However the relevant fact is that most certainly he would die!

R. P. Peçanha,

                        From any point of view, all the parts get flung outwards from the centre of the propeller. The force is called centrifugal force. Some go to great extents to use an alternative formula of words to describe what is going on.

                       But whatever, centrifugal force exists and it is an example of an inertial force.

Not quite. Pieces don’t fly outwards “from the centre of the propeller” (that is, in the radial or centrifugal direction). Rather, they fly tangentially from the position of detachment. As soon as the material link is lost, rotation ceases and inertia commands: the trajectory is linear along the instantaneous tangent. It is away from the centre but not in the radial or centrifugal direction.

My first text regarding the original question by Vivek Arora is missing in this link. So I reproduce it below.

Take the Navier –Stokes equation in its standard form shown in Fluid Mechanics books:

fluid density (local accel. + convective accel.) = - grad p + μ Laplacian u + fluid density x g

Notice that all terms have dimensions of force/fluid unit volume.

Inertial force (per fluid unit volume) is simply the name of the term to the left of the equal sign in the above equation. The justification usually given for this nomenclature is related to the fact that the term is proportional to fluid inertial mass, via fluid density. Then, the left hand side can be written as [ local (or unsteady) inertial force + convective inertial force ] / fluid unit volume.

Notice also that, from the right hand side of the above equation, it is clear that viscous forces (per unit fluid volume), pressure forces (per fluid unit volume) and gravitational forces (per fluid unit volume) are all parts of the inertial force !

Unfortunately, a frequent confusion arises when non inertial frames of reference are used. These allow for forces like centrifugal and Coriolis (only perceived by observers in such reference frames) to appear, in addition to the more common interaction forces. Centrifugal and Coriolis forces are also called inertial forces, for the same reason: they are proportional to the fluid inertial mass. For this reason they are preferably referred to as fictitious forces.