Let us consider a periodic box filled with a fluid. There are solid fixed inclusions in the box. The Reynolds number is low and the Stokes equations apply. A space varying but time invariant volume force drives the fluid. What conditions must the volume force satisfy so that the flow rate across the box is zero in permanent regime ?
I am not sure about your question...if you are talking of flow rate I assume you are talking about
Int [S] n.(rho v) dS
where S is a surface of the box. Of course, the banal solution is n.v=0 that is a local condition for a not permeable surface.
But you wrote "flow rate across the box" and that seems not totally clear ...
By " the flow rate across the box is zero" I mean: Int [V] v dV = 0
ie, the volume integral of the velocity over the box is zero.
If you write the momentum equation and integrate over the volume, using the Gauss theorem and the periodicity condition any term would vanish...
if you have a body inside the box, the equilibrium prescribes that the surface integral of the pressure balances the integral of the stresses...
Assuming Int [V] v dV = 0 at any time, you have from integration of the momentum (fixed and non-permeable walls), remembering that Grad p = Div (pI)
Int [t0,t] dt ( Int[S] n p - n. 2mu Grad v dS) =0
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