I would really appreciate any help with this problem. I'm pretty stuck.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

An incompressible viscous fluid occupies the region between two parallel plates. The lower plate is stationary and the upper plate moves with a velocity u. Find the velocity of a steady, pressure-driven flow between the plates. Sketch the velocity profice and consider separately the cases where the pressure gradient is positive or negative. Compute the force per unit area on the upper and lower plates.

2. Relevant equations

most likely navier stokes for incompressible newtonian fluids found at http://en.wikipedia.org/wiki/Navier-Stokes_equations

and continuity equation for fluid dynamics, found at http://en.wikipedia.org/wiki/Continuity_equation

3. The attempt at a solution

If the upper plate is taken to be moving in the x-direction, then (velocity) Vy = Vz =0.

With the continuity equation, d(rho)/dt becomes zero as the fluid is incompressible so density never changes. WHere U is fluid velocity, Continuity becomes del(dot)(rho*u) = (del(rho))(dot)u + rho(del(dot)u). (del(rho))(dot)u is zero due to incompressibility as well, so this gives dUx/dx + dUz/dz = 0.

Vz is 0 at the bottom and top plate and thus everywhere else. so Vz=0, dUz/dz=0, dUx/dx=0 by symmetry

so Ux is a function of z. Ux=Ux(z)

WIth the navier stokes equation becomes

0=-dP/dx+μ((d^2)Ux/dz^2)

integrating both sides gives

dUx/dz=1/μ *dP/dx *z + C

Ux(z) = 1/2μ * dP/dx Z^2 + C(1)z + C(2)

Applying the boundary conditions: Ux(0)=0 -> C(2)=0

Ux(h) = V0 -> C(1)=V0/h-1/2μ*dP/dx*h

After this I'm not sure about the pressure gradients part or the forcer per unit area on the plates. I'm not even completely sure I did the flow velocity (Ux(z))) part correctly. Any help would be really appreciated.

I also hope you can understand what I wrote, it's quite hard to type these formulas.

Thanks

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# Homework Help: Fluid Dynamics Forces

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