Dear Bahadır Ozan Aktaş ,

There is no difference between the density of the fluid in the flow and in a static condition. Density is the property of the fluids. If the fluid is compressible, it can change its density which depends on the pressure, temperature, etc. Both for simple compressible fluids, the density depends on the two other properties.

There are several assumptions for Bernoulli's equation as following:

1) the flow is steady

2) adiabatic and no shaft work

3) it is valid for a streamline

4) the fluid is inviscid

5) the flow is incompressible

We might modify Bernoulli's equation for other situations such as viscous and compressible flow with heat transfer and shaft work.

Thank you.

The influence of the flow dynamic can be generally discussed after introducing the mass equation

d rho/dt = -(v.grad rho + rho*div v)

in static condition (at the rest), the density can not change in time but it is possible to have a spatial gradient of density.

Hi sinaga

In the field of fluid mechanics we have compressible(Variable density) and incompressible(constant density) fluids. Most fluids usually are considered as incomressible unless we have some phenomena same as shock wave or etc... For example in water-hammer, water able to be incompressible because of incredibly suddenly movement . Water behind a dam is incompressible but when it subjects to earthquake it will able to be compressible. So density of a fluids same as water , dose not change due to normal movement.

Dear Ramin Alipour

as a general physics property of a fluid we could always think about compressibility as an intrinsic property. Any medium has some degree of compressibility.

What is different is to discern between compressible and incompressible flow models. That depends on the dynamics of the flow problem.

Dear Mr. Bahadır Ozan Aktaş, your simple and natural question addresses important topics:

1. What is fluid and what physical quantities can be used to describe its state?

2. What is a fluid flow and by what physical quantities it can be characterized?

3. What describes the Bernoulli equation?

The first remark:

The density of a real fluid cannot be constant due to variations of temperature, pressure, concentration of dissolved or suspended matter.

According to modern concepts, a fluid is an easily deformable moving medium.

The properties of a stationary fluid at rest in an equilibrium state are described by thermodynamic potentials (the Gibbs potential was selected as primary) and their derivatives. Density, pressure, entropy, temperature, the concentration of dissolved substances, and suspended particles are selected as the base value. All quantities are considered continuously distributed in space and described by continuous functions.

The special role of density in fluid mechanics is determined by the choice of the form of the basic equation of state in the form of its dependence on pressure, temperature, the concentration of dissolved substances and suspended particles.

More details about the definition of the thermodynamic properties of a fluid can be found in papers published by the International Association for the Properties of Water and Steam (IAPWS) (see, for example, Feistel R. Thermodynamic properties of seawater, ice and humid air: TEOS-10, before and beyond // Ocean Sciences. 2018. V. 14. P. 471–502.)

In Flows, the energy changes sufficiently fast under the direct influence of atomic-molecular processes, it is transferred with the flow velocity, the group velocity of waves, and slowly under the action of dissipative processes at the same time.

Accordingly, the density can change quickly, with a moderate speed and rather slowly, so it is impossible to measure. But one can calculate the density by solving a complete system of fundamental equations with initial and boundary conditions for the fluid flow.

The modern definition of fluid flows has the form: an intrinsic inherent or forced self-consistent transfer of momentum, matter and total energy, causing changes in the pattern of physical fields in space and providing the interaction of the flow with a solid (Chashechkin Yu. D. Differential fluid mechanics – “old-new” generation of solvable models of flows // Waves and vortices in complex media: Proceedings of 10-th International conference-school of young scientists 03-05 December 2019, Moscow. M.: OOO Premium-Print. 368 P. 18-21.).

To describe flows in the approximation of a continuous medium, in the mechanics of fluids, the balanced equations of transport of matter (the continuity equation), momentum (Navier-Stokes equation), and energy are postulated - analogues of conservation laws for equilibrium systems. Intrinsic inherent are flows induced by diffusion on topography. Forced flows are created by the action of gradients of pressure (force), temperature (convective), concentration (gravitational), the combined action of the part of all types of disturbances (forced multicomponent or thermo-concentration convection), and the motion of solids and gas volumes.

Accordingly, the complete solutions of the system describe changes in the fields of all physical quantities, including density.

The Bernoulli equations are not a universal formula for describing the flows of real liquids, but the properties of one of the models of flows of an incompressible ideal fluid with its all restrictions indicated by Nazaruddin Sinaga

You can learn more about the properties of flows on my page in ResearchGate, where examples of optical visualization of density gradient fields in fluid flows are presented.

Perhaps this paper may help:

Hi Bahadir and all.

I stick with Filippo comment. Every material is compressible to some extent. A neutron star has a density around 1017 kg/m3. The density of liquid water is around 103 kg/m3. The mean density of the Earth planet is close to 5.5 X 103 kg/m3. So, even in a static column of liquid water, very small (and difficult to measure!) increases in density occur from top to botton. The same applies to flowing fluids. Now, down to earth, liquids are practically incompressible. As for flowing gases, the density depends on their velocity. Engineers like to work with ratios of similar quantities called dimensionless numbers. One such ratio is de Mach Number (M), defined as the ratio of the fluid velocity to the sound velocity in that fluid at a given pressure and temperature. If M < 0.3 approximately, the fluid can be considered incompressible in engineering calculations.

Dear R. P. Peçanha

I would just add as an engineer that the Mach number is indeed a measure of compressibility in a flow problem but only in terms of the response of the density to a pressure variation (that is the sound velocity). For M->0 it is assumed that the pressure waves spreads istantaneously. However, even at M=0 the density can change depending for example on temperature gradient. A classic model for taking into account that case is the Bousinnesq approximation. Furthermore, combustion is often modelled with low-Mach models as well as flows in seawater (liquid) depend on temperature and salinity that affect the density even if the Mach number is zero.

there is not state equation for incompressible flows, the pressure in a incompressible flow is not the thermodynamic pressure, when we work in fluid static the presurre is the thermodynamic one

you must read about the difference between the mechanic pressure and the thermodynamic pressure in fluid flows

Dear Bahadır Ozan Aktaş

The topic is very long, but briefly;

the density is one of an important physical property of a fluid for the treatment of flow. its can change which depends on the pressure, temperature, Mach number etc. That's why there is no difference between the density of the fluid in a dynamic state or in a static state.

Cordialy.@

Density is a physical property of the fluid. In incompressible it remain constant, in compressble density vary in nrelation to temperature and pressure. When fluid is moving at all it is static. In fluid there is static pressure and dynamic pressure

Hi Filippo

I presumed fluid in thermal equilibrium. The trouble with the presence of a grad T in a static fluid (M = 0) or in a fluid under creeping flow (M => 0) is that small density differences will appear between layers with different temperatures and vertical natural convection currents will establish a bottom-up-bottom flow. The sonar “ping” principle for detecting submarines (sound waves reflection) shows that pressure waves in liquids are in fact far from instantaneous.

Hello R. P. Peçanha

yes, physical sound velocity is always finite but the flow model M=0 is often used in aerodynamics. Liquid can be at rest in stratification state, as happens in the sea under some circumstances.

Basically as the pressure changes the fluid depending on the fliw velocity U, the density changes. In common flows the variation of density is adiabatic. In that case the inverse of the compressibility of the fluid is the product (rho c^2) of the density rho and the square of the speed of sound. For an ideal gas (rho c^2)= gamma p , with gamma the Poisson ratio (order unity, 1.4 for air) and p the static pressure.. For a liquid rho c^2 is the so called internal pressure, a measure for molecular attraction (cohesion). For water for example rho c^2= 2E9 Pa=20000 bar. Using the equation of Bernoulli to estimate the variation in pressure in a Stationary (time independent) flow Do=0.5 rho U^2 one finds a change in density Drho/rho=0.5 (U/c)^2=0.5 M^2. This order of magnitude estimate is also valid in unsteady vortices as encountred in turbulent flows. When considering acoustic waves due to abrupt changes in flow from a velocity U to a velocity U+u’, the corresponding pressure changes are of the order p’=rho c u’ . The corresponding relative density changes scale as rho’/rho=u’/c. Hence while steady flow involve a variation of density quadratic with the Mach number U/c, unsteady flow density variations are typically scaling linearly with the Mach number of the velocity variation u’/c. An example of this is the water hammer phenomenon.

Hi Bahadır Ozan Aktaş ,

In addition to what he said both

Mohsen Jahanmiri and Avraham Hirschberg.

To simplify the equations of fluid mechanics, we often consider that liquids (water, oil, mercury, blood etc.) are incompressible. An incompressible fluid is characterized by a low Mach number, that is to say less than 0.3. The Mach number is a dimensionless number, noted Ma, which expresses the ratio of the local speed of a fluid to the speed of sound in the same fluid. The incompressible fluid assumption can also be used for gas flows, but provided that the Mach number is low.

Good luck.@

A Mach number of 0.3 for the limit of validity of an incompressible fluid approximation is derived from the compressible Bernoulli equation for air: To/T=1+0.2 Ma^2

Thus is only valid for an ideal gas

with constant specific heat. For a liquid like water, Ma=0.3 would imply velocities of 500 m/s. This would imply pressure fluctuations of the order of 1bar in vortices and most probably cavitation ( unless there are no walls and the water is extremely pure). The incompressible approximation breaks down due to the formation of vapor bubbles.

Dear All,

Thank you very much for your detailed explanations, recommendations and contributions. I know that Fluid Mechanics is a very broad subject. My question is originated from the simple macroscopic situation as shown in attached file “file2.pdf”.

As shown in the first attached file “file1.pdf”, as far as I know in a standart Venturimeter we have a U-tupe (Manometer) with liquid (etc. mercury) and a fluid (gas). Liquid in U-tupe is static and gas is fluid. As a consequence of Bernoulli Principle for a fluid at low Mach number the renponse from Case 1 to Case 2 is comprehensible and matches our expectations.

Suppose that we have a Venturimeter-like machinery as shown in second attached file “file2.pdf”. We want to determine the response of the ideal gas (compressible) in U-tube to the narrowing of the cross-sectional area at point 2 from Case 1 to Case 2. Again via Bernoulli and Pascal Principles one can easily show that the pressure of gas is decreased and volume is increased from Case 1 to Case2. If the the cross-section is uniform of U-tupe the increment in volume of the gas is proportional to the difference between the heights of two sides. This means |Δh2|-|Δh1|>0.

The essensial question is this: How can we show that this difference (|Δh2|-|Δh1|) is greater then zero? If this proof is trivial, I would be very pleased if you could explain it simply for non-experts in this field like me.

Thank you very much in advance.

Bests.

PS: I want to understand the relation between this phenomena and possible consequences of embolism in human circulatory system, oxygen and carbon dioxide exchange in alveolus, nutrient diffusion from fine vein to tissue, the pressure of water supplied for the users from water works in city network etc. For this puspose, please try to explain your answer of the essential question above as simple as possible. Thank you very much again.

Hi Bahadir and all.

Let’s stick to the Venturi meter (Inventor: Giovanni Battista Venturi, 1746 – 1822). The first thing to remember is that Venturi meter is aimed to quantify flow rates in pipelines. If you divide the volume flow rate by the pipeline cross-section area, you have the mean fluid velocity in the pipeline.

In your case a gas flows and mercury (Hg) is basiclly a static liquid in the U tube manometer. In case 1 (supposing the figure is to scale!) the pressure drop will be very very small, and its measument will lack precision in practice. In fact, in Case 1 you don’t have a Venturi meter.

Assuming a constant flow rate in the pipeline, in Case 2, due to the sudden decrease in cross-section, the pressure drop will be much biger and its measurement precision will improve a lot. Mind you, Case 2 is hardly a standard Venturi (look on textbooks on Fluid Mechanics or Wikipedia).

Since you are using Bernoulli equation/principle to study the problem, you are assuming that the pipeline fluid is incompressible! This is a hypothesis used in deriving the equation. Then, if you use Bernoulli equation for flowing gases (your case), the gas flow rate calculated via Bernoulli will have an associated error. In your particular case, I believe the error is going to be very small.

However, there is another problem with Bernoulli.The equation also holds only for ideal fluids (not to be confused with ideal gases!). An ideal fluid has viscosity equal to zero! However, in reality, no such fluid exists. Again, under certain conditions, some fluids are almost ideal. This is the case of flow across small distances, as happen in a Venturi!

Now, the answer to your question. I assume a constant flow rate in your horizontal pipeline.

Case 1.

According to Bernoulli, the pressure-drop between section 1 and section 2 is zero. In fact this prediction is wrong, mainly because of the fluid viscosity (however small, all real fluids have viscosity).So there is a very small pressure drop between section 1 and section 2, but it is very difficult to measure it with ordinary manometers. The pressure drops because fluid mechanical energy is partially dissipated as heat (exchanged with pipe and exterior surroundings) and partially increasing the fluid internal energy (a concept from Thermodynamics!) with an associated rise in the fluid temperature from section 1 to section 2. The dissipation mechanism has to do with friction between layers of fluid with different velocities in the pipeline.

Case 2.

In this case, Bernoulli equation predicts a pressure-drop between sections 1 and 2, easily measurable. No doubt, the errors associated to the gas compressibility and viscous effects described above, will also affect calculations based on Bernoulli equation! Since the flow rate is constant, the gas velocity increases from sections 1 to 2 (velocity = volume flow rate/cross-section area). Then, according to Bernoulli equation (however imprecise!), the pressure must drop. By feeding the pressure-drop to Bernoulli equation, we get the gas flow rate.

There are empirical corrections (multiplicative factors) for the unaccounted viscous effects seen above, called “discharge coefficients”(C) whose values depend on the Reynolds number. For standard Venturies the values of C fall in the approximate range 0.83 – 0.98. Thus, Bernoulli equation always overpredicts flow rates.

Dear R. P. Peçanha @R. P. Peçanha. Thank you very much for your simple clarifications.

But I'm confused about this part of your explanation: "Since the flow rate is constant, the gas velocity increases from sections 1 to 2 (velocity = volume flow rate/cross-section area). Then, according to Bernoulli equation (however imprecise!), the pressure must drop. By feeding the pressure-drop to Bernoulli equation, we get the gas flow rate."

In the second case of relevant attached file "file2.pdf", the gas in U-tupe is static and liquid is fluid (it flows). Hence, I wonder that, If the density is uniform throughout the flow, How can we show that the volume variation of gas (well, in here this is an expansion) in U-tube due to the static pressure difference in section 1 and section 2 of liquid?

Thank you.

@R. P. Peçanha, In this part "Since the flow rate is constant, the gas velocity increases from sections 1 to 2 (velocity = volume flow rate/cross-section area). Then, according to Bernoulli equation (however imprecise!), the pressure must drop. By feeding the pressure-drop to Bernoulli equation, we get the gas flow rate." I guess the fluid and static matter in U-tube are confused. In this situation we can neglect the compression of liquid (i.e. mercury). Everything is OK. Am I wrong? However, in the machinery that I am talking about, these have a reverse situation then we should take into account expansion of gas in U-tube. But, how?

Thank you.

Dear Bahadır Ozan Aktaş

The Bernoulli’s law derived for the steady, adiabatic and incompressible flow states that

(\rho v^2)/2+\psi \rho +p =const

for a given streamline (see for example Landau & Lifshitz Fluid Mechanics).

Considering the fact that the U-tube section is separated from the main flow you can include the loses of energy due to compression of gas in the same manner as the other dissipative effects.

This should be quite natural when you switch to Hagen-Poiseuille equation; this change is highly recommended.

I have included also a paper discussing some very practical aspects of your problem.

Good question

Following

Hi Bahadir and all.

First of all we should refer to the gas density (mass/volume), which is the relevant physical property, rather than gas volume.

Take Case 2. The gas in the left arm of the U-tube will have a slightly higher density than that of the right arm. However, in practice, the density of both gases (left and right arms) will be much, much smaller than the density of mercury (something like three orders of magnitude less). So, in the “hydrostatic balance” on the U-tube, simply disregard both gas columns!

If, for some reason (I cannot imagine a practical one!), you want to know the exact gas density on the larger and smaller pipeline, the simplest way is to install a Bourdon gage (type of manometer) in each pipeline. The pressure-drop (required by the Bernoulli equation to get the gas flow rate) across the Venturi will be the difference between the two analogic readings. The gas densities can be calculated summing the local atmospheric pressure to the individual pressure readings (to get the absolute pressures) and using the ideal gas law (good up to approximately 10 atm): gas density = (pressure x gas molar mass) /(universal gas constant x absolute temperature).

Can provide any equations for this case

For incompressible fluid, the density is the same

But, for compressible fluid Derivation of density of any fluid > Derivation of density for static fluid