Question: 3 consider flow between infinite horizontal parallel plates as shown...
3. Consider flow between infinite horizontal parallel plates as shown below. The bottom plate at y = 0 is fixed, and the top plate at y = 2h moves to the right with a velocity V2. A third, very thin, "floating" plate at y = h separates the fluid above it (at h < y < 2h) from that below it (at 0 < y < h); no external force is applied to this plate. You will solve for V1, the unknown velocity of the floating plate.
•The fluid properties are the same in the top and bottom layers.
• Take the velocity profiles in the bottom and top layers to be u1(y) and u2(y), respectively.
• The velocity in each layer is only in the x-direction.
• Both velocity profiles are fully developed.
• Pressure in the fluid is uniform.
(a) Write shell balances on x-momentum for fluid on either side of the middle plate.
(b) (15 pts) Taking the middle plate velocity as V1, write the boundary conditions that apply to u1(y) and u2(y). Then solve for the velocity profiles u1(y) and u2(y). Feel free to use shortcuts based on online notes, class notes or homework, but explain carefully why the shortcuts are valid.
(10 pts) Use a force balance to determine V1, the velocity of the floating plate. If you didn't get (b), you'll receive ample partial credit for writing down the key equation needed to find V1.