2. Consider a situation where there are two parallel plates with a gap between them. In the gap, there are two layers of liquid, as shown in the figure below. The bottom plate is fixed and does not move, while the top plate moves with a velocity 3 cm/s. The distance between the 2 plates is 2h, with the origin of the system located at the bottom plate. There is no pressure gradient. The viscosity of the two fluids is given by Hд and μB. You would like to solve for the velocity profile between the two plates, and in both fluids. upper plate, moving → with velocity 3 cm/s fluid B fluid A y = 2h y = 0 bottom plate, fixed a.) This is a shear driven flow problem. Please write the equation for Couette (or shear-driven) flow. Solve this differential equation to obtain the general solution b.) You will need to apply the equation from part a to each layer of fluid individually. Thus, you will need 4 boundary conditions for the system. Please write the boundary conditions. You may write them in words, but I would also like for you to write a mathematical expression for each. 2 of the boundary conditions are no-slip conditions, and 2 are continuity conditions, which state that the velocity and the shear stress of the two fluids are equal at the point where they meet. c.) Plug in the boundary conditions from part b to obtain an equation for the velocity of fluid A as a function of vertical distance (i.e. UA (y) = ?) and an equation for the velocity of fluid B as a function of vertical distance in the channel ((i.e. UB (y) = ?).
2. Consider a situation where there are two parallel plates with a gap between them. In the gap, there are two layers of liquid, as shown in the figure below. The bottom plate is fixed and does not move, while the top plate moves with a velocity 3 cm/s. The distance between the 2 plates is 2h, with the origin of the system located at the bottom plate. There is no pressure gradient. The viscosity of the two fluids is given by Hд and μB. You would like to solve for the velocity profile between the two plates, and in both fluids. upper plate, moving → with velocity 3 cm/s fluid B fluid A y = 2h y = 0 bottom plate, fixed a.) This is a shear driven flow problem. Please write the equation for Couette (or shear-driven) flow. Solve this differential equation to obtain the general solution b.) You will need to apply the equation from part a to each layer of fluid individually. Thus, you will need 4 boundary conditions for the system. Please write the boundary conditions. You may write them in words, but I would also like for you to write a mathematical expression for each. 2 of the boundary conditions are no-slip conditions, and 2 are continuity conditions, which state that the velocity and the shear stress of the two fluids are equal at the point where they meet. c.) Plug in the boundary conditions from part b to obtain an equation for the velocity of fluid A as a function of vertical distance (i.e. UA (y) = ?) and an equation for the velocity of fluid B as a function of vertical distance in the channel ((i.e. UB (y) = ?).
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Transcribed Image Text:2. Consider a situation where there are two parallel plates with a gap between them. In the
gap, there are two layers of liquid, as shown in the figure below. The bottom plate is fixed
and does not move, while the top plate moves with a velocity 3 cm/s. The distance between
the 2 plates is 2h, with the origin of the system located at the bottom plate. There is no
pressure gradient. The viscosity of the two fluids is given by Hд and μB. You would like to
solve for the velocity profile between the two plates, and in both fluids.
upper plate, moving → with velocity 3 cm/s
fluid B
fluid A
y = 2h
y = 0
bottom plate, fixed
a.) This is a shear driven flow problem. Please write the equation for Couette (or
shear-driven) flow. Solve this differential equation to obtain the general solution
b.) You will need to apply the equation from part a to each layer of fluid individually. Thus,
you will need 4 boundary conditions for the system. Please write the boundary
conditions. You may write them in words, but I would also like for you to write a
mathematical expression for each. 2 of the boundary conditions are no-slip conditions,
and 2 are continuity conditions, which state that the velocity and the shear stress of the
two fluids are equal at the point where they meet.
c.) Plug in the boundary conditions from part b to obtain an equation for the velocity of fluid
A as a function of vertical distance (i.e. UA (y) = ?) and an equation for the velocity of fluid
B as a function of vertical distance in the channel ((i.e. UB (y) = ?).
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