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dominik March 28, 2004 18:07

steady state creeping flow

x-momentum equation for steady-state creeping flow in FV formulation is:

\eta \int_A grad(u) \cdot \hat{n} dA - \int_A p \hat{x} \cdot \hat{n} dA = 0

where u is x-component of velocity, \int_A is surface integral, \cdot is dot product, \hat{x} and \hat{n} denote cartesian unit x vector and normalized outward normal to the surface, respectively. In steady state creeping flow in a tube the two integrals have both to be zero or both be equal. Now, the problem is that the first one is zero then, as expected (no gradients in fully developed duct flow, forget viscosity on walls), but the second one is not - for square grid example:

\int_A p \hat{x} \cdot \hat{n} dA = (p_e-p_w)\delta y

where p_e, p_w denote pressures on east/west faces and \delta y is grid spacing. THIS IS NOT ZERO in steady state (otherwise fluid doesnt flow) but the sum of diffusive fluxes IS zero (no gradients). That way I cant reach steady state. Please help me find the error.

regards, Dominik

Tom March 29, 2004 04:37

Re: steady state creeping flow
The nonzero (constant) pressure gradient is balanced by the shear u_y which is nonzero.

(If you're working with the depth averaged equations then you need to incorporate the viscous stress at the walls.)

dominik March 29, 2004 07:36

Re: steady state creeping flow
Thank you for an answer. but I have shear stress term close to walls: I replace diffusive flux by -eta*du/dy do you mean I should have it too in ANY control volume? or do you mean that the flaaaaat parabolic profile extending to tube center is able to balance pressure gradient even there? thank you dominik

Tom March 29, 2004 09:26

Re: steady state creeping flow
I mean the 2nd; i.e. the shear stress u_y exactly balances the pressure gradient in the flat case so that your first term in your equation is not exactly zero (d/dx is zero for u as you say but d/dy is not)

dominik March 29, 2004 09:29

Re: steady state creeping flow
I see your point. I was underestimating the parabolic profile in the domain due to noslip walls (I thought it was effectively flat). Thank you. Dominik

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