# meaning of pressure

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 September 1, 2004, 23:09 meaning of pressure #1 Junseok Kim Guest   Posts: n/a What is meaning of pressure term in the incompressible viscous Navier-Stokes equation?

 September 2, 2004, 12:22 Re: meaning of pressure #2 noName Guest   Posts: n/a An isotropic forcing potential?

 September 2, 2004, 16:54 Re: meaning of pressure #3 Jonas Holdeman Guest   Posts: n/a The pressure in incompressible flow can be divided into two parts, hydrostatic and hydrodynamic. The hydrostatic pressure is that which balances conservative forces such as gravitational. By the Helmholtz decomposition, the incompressible Navier-Stokes (momentum) equation can be separated into two parts, a solenoidal part which becomes a pressureless governing equation for the velocity (and depends on any nonconservative forces), and an irrotational part for the pressure (gradient) as a function of the velocity. The two equations are integro-differential equations, which is one reason you usually don't see them in this form. When the two equations are combined, the result is a partial differential equation (actually an differential algebraic equation (DAE)in time). The dependence of the pressure on the velocity remains in that there is no explicit mechanism for independently advancing the pressure in time. For computation, the equations may be multiplied by solenoidal test or weight functions for the velocity and irrotational test functions for the pressure to form and equivalent formulation. In this way, the test functions serve as projection operators and the integro-differential form is not evident. This "weak" form is the basis for the finite element method, and is the form used by mathematicians to "prove" the existence and uniqueness of solutions of the NSE. Actually, that existence has not been proved in 3D, and there is a million dollar prize offered to anyone who can prove it. Normally, the sum of solutions of two equations is not equal to the solution of the sum of the equations. However in this case it is true because the two are orthogonal; the velocity is the curl of a stream function or vector potential and the pressure gradient is the gradient of a scalar potential. In this view, the incompressible pressure is that quantity which is consistent with the velocity in the NSE, and is an approximation to the physical velocity in compressible constant-density flow. acgnipper likes this.

 September 3, 2004, 02:47 Re: meaning of pressure #4 Junseok Kim Guest   Posts: n/a What does mean by the sentence in your paper, Governing Equation for Incompressible Flow; Revisiting the Navier-Stokes Equation. What kind of adjustments you have in mind? The problem with the classical derivation of the incompressible NSE is irrefutable, and appropriate adjustments should be made in the physics curriculum. Thanks

 September 3, 2004, 07:51 Re: meaning of pressure #5 Tom Guest   Posts: n/a Pressure is simply the macroscopic manifestation of the inter-molecular force which resists compression. For an inviscid fluid it can be interpreted as the Lagrange muliplier for mass conservation in the varaitional principle - hence it's usual interpretation of enforcing incompressibility. ( In Lagrangian mechanics this is actually how the pressure enters the problem with the Lagrange multiplier being interpreted as the pressure after application of the Euler-Lagrange equations to the variational principle). Tom.

 September 4, 2004, 06:09 Re: meaning of pressure #7 Tom Guest   Posts: n/a I think you've misinterpreted my above comment since I didn't say anything about the Stokes equations - I was talking about the Euler equations of an ideal fluid (in which case when you perform the variation all you do is restrict yourself to the condition that the mass should not be varied). Incompressibility then arises through the condition that variations in the volume must vanish. The Stokes equations are a bit strange since they are derived on the assumption of strong viscosity and end up being time reversible. The restrictions on the variations appear because, as opposed to the usual case of starting with a variational principle and then deriving the equations, you start knowing the equations and reconstruct the variational principle. Another difference, because of the above point, is that in the case of Stokes flow the equations will be in Eulerian form whereas in the ideal fluid case the equations appear in Lagrangian form. Tom.

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