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Old   April 5, 2013, 10:07
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Naveen Raj N
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pressure is varying downwards in a problem. can anyone help me to find out the format of pressure profile which i have to input for boundary condition.
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Old   April 9, 2013, 11:23
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Jonas T. Holdeman, Jr.
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Are you asking the right question?

I assume you are talking about incompressible flow. If the fluid/flow is incompressible, this means the density (or volume of a quantity of fluid) does not depend on, or is independent of, the pressure. The Navier-Stokes equation contains two dependent variables, v and p, and is generally accepted as the governing equation for incompressible flow. But:
(1) If p is the pressure, the NSE cannot be the governing equation for incompressible (pressure-independent) flow.
(2) If the NSE is the governing equation for incompressible (pressure-independent) flow, then p cannot be the pressure.

In fact, the NSE is a composite equation, the sum of a governing equation for the velocity that does not contain the pressure, and a pressure equation which is a functional of the velocity, which follows from the Helmholtz decomposition theorem. The resulting pressure equation is a once-integrated form of the familiar pressure-Poisson equation and the pressure need be specified at only one point.

The (pressure-independent) velocity equation is more problematic. There is a simple way of solving the weak/variational form of this equation, but most use subtractive projection with the more difficult differential form. This form looks precisely like the standard NSE, but the symbol p is not the pressure (case 2 above). Unlike the true pressure, this quantity p must satisfy the LBB conditions and some other unspecified boundary conditions. Solving this system seems more like an art than a science.

I am not a practitioner of this art form, but since no one else has responded I decided to give you my opinion. That is, that you probably need not be concerned that the pressure is varying when you seek boundary conditions on p. Just do what you would do otherwise.
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