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March 28, 2004, 18:07 
steady state creeping flow

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xmomentum equation for steadystate 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 xcomponent 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_ep_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 

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March 29, 2004, 04:37 
Re: steady state creeping flow

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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.) 

March 29, 2004, 07:36 
Re: steady state creeping flow

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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


March 29, 2004, 09:26 
Re: steady state creeping flow

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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)


March 29, 2004, 09:29 
Re: steady state creeping flow

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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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