pressure drop is the driving force to push fluid to flow through a pipe, that means as far as there is no pressure difference between two points along the pipe length there is no flow. Generally pressure drop takes place even radially, but the value is too smaller than that in length. therefore we just consider the pressure drop in pipe length.

when you have a fully developed flow it doesn't mean that there is no pressure drop, it just means there is no velocity difference along the axis where we have fully developed flow.

Dear Mr. Ghalambaz,

Considering that we are talking about filled pipe flow, i.e. the fluid flow fills the pipe cross-section area, so no open surface (contact with air) exists, then the inertial movement can only develop if there is a pressure gradient applied between two points of the pipe. As you mentioned fully developed flow, then the viscous forces acting contrary to the inertial movement, established the velocity profile which is now permanent and uniform. A point needs remarking: things can go differently when on laminar (Re < 2000) and turbulent (Re > 4000), and so the velocity profile change depending on flow conditions.

We could be treating here a Hagen-Poiseuille flow, for example. Then, answering your question, the pressure found as solution of this problem isn't function of the radius because if you assume that, by absurd, then (dP/dr =/ 0), what would mean that there would have axial flow, meaning fluid particles migrating from the center to the top/bottom. On turbulent flow, you might have disturbances on the pressure gradient, or even rotational fields, but I don't think that this is your case, am I right?

Best regards,

pressure drop is the driving force to push fluid to flow through a pipe, that means as far as there is no pressure difference between two points along the pipe length there is no flow. Generally pressure drop takes place even radially, but the value is too smaller than that in length. therefore we just consider the pressure drop in pipe length.

when you have a fully developed flow it doesn't mean that there is no pressure drop, it just means there is no velocity difference along the axis where we have fully developed flow.

Dear Ali, why the "the pressure drop takes place even radially"?

Dear Guilherme Fiorot , it seems convening. Thank you.

Dear Mohammad, if you have a look on the "Transport Phenomena" written by R. Byron Bird, you would find that in cylindrical coordinate we will have pressure drop in all three-axis (including radial and angular directions). you could refer to appendix B, equation B.5-4 at the page of 847 to see the relation between the velocity and pressure drop of Newtonian fluid in cylindrical coordinate.

explaining the root and reason of these parameters here in this space is not possible. please take a tour on that book to have a better understanding.

thanks

sorry these equations are not limited to Newtonian fluids, they explain the equation of motion for all elastic materials.

Dear Ali, of curse, generally in a cylindrical coordinate system pressure drop in any direction is completely meaningful. However, here, The question is about an specified fully developed region in a pipe.

If you consider the momentum equation by considering that there are no velocities gradient in axial direction, you can see that the pressure gradient in radial direction is not zero.

For simplicity, consider fully developed laminar flow in a horizontal 2D pipe, x the axial direction, y the of gravity. Fully developed flows means \partial( . )/\partial x =0, (not for pressure of course). Applying to NS, the momentum equation in the y direction became hydrostatic

\partial{p}/\partial{y}= \rho g_y

so there is a linear, hydrostatic, pressure in the y direction.

Same arguments apply to any kind of cross section geometry.

I think also for turbulent mean flow, but I'm not certain of that.

Dear Luis and Reza,

Thank you for your valuable comments.

Dear Luis, Can I conclude that if I consider a vertical pipe then the radial pressure gradient is zero?

Certainly Mohammad, you should conclude that.

Thanks Luis.

The pressure will not remain constant since there has to be a pressure gradient in order for the flow to take place...this is true for laminar and turbulent flow.............u can run a simple simulation in any commercial cfd software to see the pressure contour both radially and in the direction of the flow

Dear Deepak,

I am agree with you about the pressure drop with the flow axis (in the opposite direction of the flow ), but some of the members believe that in the redial direction there is not flow and pressure gradient (neglecting the static pressure gradient because of the gravity). If you are not agree, please provide more details, and then we can discuss about that. Thank you.

Looking at the flow in a pipe problem it is clear that the length of the pipe is significantly large compared to the diameter of the pipe. the fluid properties vary in the flow direction example pressure. the velocity is then only function of the radial direction.....

In a turbulent channel or pipe flow the pressure changes in the wall-normal and streamwise directions.

In particular, the sum of the pressure and of the wall-normal velocity fluctuation squared equals the pressure at the wall. Since the wall-normal velocity fluctuation varies in the wall-normal direction also the pressure does. The pressure at the wall varies linearly with the stream-wise direction (pressure drop). More details are given in Turbulent flow, Pope 2002, pg. 266.

Dear Francesco Picano,

What is your idea about the laminar flow in a pipe? For the case of constant thermo-physcial properties, Is the pressure in radial direction constant?

Dear Mohammad,

yes it is constant (see the book of Kundu & Cohen pg. 278).

Dear Francesco Picano,

Thank you

In fully developed laminar flow, each fluid particle moves at a constant

axial velocity along a streamline and the velocity profile u(r) remains

unchanged in the flow direction. There is no motion in the radial direction,

and thus the velocity component in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady and fully developed.

The volume element involves only pressure and viscous effects and thus the pressure and shear forces must balance each other.

Dear Dravid,

"The volume element involves only pressure and viscous effects and thus the pressure and shear forces must balance each other."

Are you agree with zero pressure gradient in radial direction?

Yes i agree to you because viscosity vanishes by gravitational potential force works when fluid flows.

Dear Mohammad,

The pressure is constant in the radial direction for fully developed flow. It only varies in the streamwise or axial direction since this is what drives the flow. This is something that you can read up on in a fluid mechanics text book - E.g. Frank White - Fluid Mechanics or Ronald Panton - Incompressible Flow.

Dear Kiran and Rahul,

Thank you for your response.

Dear Konstantinidis,

If we neglect the gravitational effects, is the pressure in the radial direction of a large diameter pipe also constant? If not why?

Dear Konstantinidis,

Thank you for conclusion.

what is the dimension of the pipe? if it is a "microchannel" with sub-millimeter dimensions, you can ignore the effect of gravity. Thus, the pressure drop is only along the direction of the flow, and is the driving force for the motion of the liquid towards the outlet.

The biogas pressure is fully developed and constant in the outlet pipeline.

no, although you consider gravity acting in the directionof the radious. In that case piezometric pressure is constant with radius

By definition, for steady, laminar, rectilinear flow in a pipe, the only nonzero fluid velocity component is the axial component v_z(r), which is a function of the radial coordinate only. With this assumption all the inertial terms in the linear momentum equations are identically zero!. And the velocity profile also satisfies the continuity equation. Since the body force ( due to gravity) can be written as a gradient of a scalar potential it can be combined with the pressure terms as a gradient of a new scalar say P. One can then show that this P ( sometimes called the modified pressure) is at most a function of z , the axial coordinate. The actual pressure on the other hand may depend on the other coordinates depending on the direction of gravity relative to the flow direction. These components represent the hydrostatic components of the pressure, but do not contribute to the flow in the theta (azimuthal) or radial directions, because of the rectilinear flow assumption.

Of course, rectilinear flow is an assumption, and may not be realized in practice. For example, if the flow rate is very large ( i.e., large Reynolds number> 2000) then the flow becomes turbulent, and we say the rectilinear flow is unstable to small perturbations....

Dear Brian Higgins,

Thank you very much for your answer.

In deed, one day, one of my students has asked me this question. He said, in an interview, someone have asked him if Is the pressure in a fully developed section of a pipe constant?? I was confused to answer. Because in the text books we have assume the velocity as V_z(r), I was wounder what are the assumptions behind this conclusion and how may specially affect the pressure distributions. Now, I realize that we have assumed the flow is a rectilinear flow. Then, considering the other assumptions, steady, laminar ... the flow is such that V=V_z(r). Therefore, neglecting the external body forces (here the gravity force), the pressure can be assumed constant in a section of a tube. Is my conclusion correct?

Thank you again for your time and answer.

Mohammad, Perhaps you have a typo, but the last sentence in your response is not correct. The pressure gradient is a constant, but not the pressure. The pressure varies linearly with z for rectilinear flow. If the pressure were every where constant in the tube there would be no flow ( assuming there are no other driving forces for flow).

Mohammad, If I understood you correctly, you are referring to pressure on a plane perpendicular to the direction of flow. If that is true, then yes you can neglect pressure variation in the vertical direction (assumed direction of gravity field) when gravity effect is assumed zero. This is only true for a straight pipe with no secondary flow.

For a fully-developed flow in a straight pipe, laminar or turbulent; pressure does not vary with radius. I can share my results if you are interested.

Dear Prof. Brian Higgins,

I mean the pressure in "a plane perpendicular to the direction of flow" is constant. Thank very much for you your attention.

Dear Dr. Taherian

Thank you for your response.

Dear Sudhir Bangaru,

Are you sure about the turbulent flow?

To be precise mean pressure in fully developed turbulent flow varies only in stream wise direction. But the fluctuating pressure does vary radially(although very small for most of the engineering applications). I would like to share this paper which contains the DNS data of turbulent flow in a pipe. Figure 11 describes the variation of fluctuating pressure in radial direction. ftp://128.111.100.26/pub/org/lakemix/matlibrary/users/dturney/2009_08_24_WritingPaperWithSally/related_papers/Eggels1994_DNS_TurbulentPipeFlow.pdf

Yes, the pressure is the function of the radius but if the radius is much smaller then length of the pipe, the pressure can be assumed as a constant in section of the pipe.

If there is flow, there has to be pressure gradient along the length. That means, the pressure difference exists between any two points along the length. As far radial velocity variations, I believe at least due to viscosity the central section will have maximum velocity.

A pressure difference in the flow direction is required to overcome the friction. There will not be any pressure variation in the radial direction for a fully developed flow in a straight pipe as the streamlines will be straight and parallel. Radial pressure difference only exists if the streamlines curve, which happens if the flow separates; doesn't happen in the case of a constant diameter straight pipe for steady flow.

The flow is said to be developed when the velocity profile does not vary along the pipe, it is not influenced by entrance effects. Pressure difference is needed to flow the fluid. It is as a driving force. In fully developed flow, the pressure gradient (Δp/Δx) is constant, Δx = length of pipe element, x is flow direction. In Transport Phenomena book that I have read, in annular pipe case, for flow in z direction, p=p(z), so it can be considered that pressure isn't function of radius. But, for flow in falling film that two ends is open to air, p=p(x), x = radial direction. So, it depends on the case.

Considering a section of the tube, if you project the Euler equation along the perpendicular "n" to the flow direction you have that d/dn (z+p/gamma)=v^2/r, being r the curvature of the pipe, gamma the weigth of the fluid, z the elevation and v the velocity. Then, for a rectilinear flow (r=infinite) the piezometric head (z+p/gamma) is constant but not the pressure which varies with z.

Dear Dr. Ghalambaz,

Its a mere coincidence that both of us posted a similar kind of question. Mine is to determine the pressure drop for a given axial distance. For the fully developed flow, pressure is constant along the radius of the annulus, but function of axial distance. For a given axial location, the pressure drop can be assumed as constant, but need to find the value of the constant.

Regards,

Sankar

@Sankar

Yes, i agree with your definition:

For the fully developed flow, pressure is constant along the radius of the annulus, but function of axial distance.

Best regards

Interesting