Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between...

 

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Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infinitely long and wide, parallel flat plates as shown below. The flow is steady, incompressible, parallel, and laminar. The top plate moves at velocity V to the right, and the bottom plate is stationary. Gravity acts in the -z-direction (downward in the figure). There is no forced pressure gradient pushing the fluids through the channel-the flow is set up solely by viscous effects created by the moving upper plate. You may ignore surface tension effects and assume that the interface is horizontal. The pressure at the bottom of the flow (z = 0) is equal to Po. (a) List all the appropriate boundary conditions on both velocity and pressure. (Hint: There are six required boundary conditions.) (b) Solve for the velocity field. (Hint: Split up the solution into two portions, one for each fluid. Generate expressions for u as a function of z and u, as a function of z.) (10 pt) Moving wall V Interface h2 Fluid 2 P2. 42 h2 Fluid 1 P1. 41 Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infinitely long and wide, parallel flat plates as shown below. The flow is steady, incompressible, parallel, and laminar. The top plate moves at velocity V to the right, and the bottom plate is stationary. Gravity acts in the -z-direction (downward in the figure). There is no forced pressure gradient pushing the fluids through the channel-the flow is set up solely by viscous effects created by the moving upper plate. You may ignore surface tension effects and assume that the interface is horizontal. The pressure at the bottom of the flow (z = 0) is equal to Po. (a) List all the appropriate boundary conditions on both velocity and pressure. (Hint: There are six required boundary conditions.) (b) Solve for the velocity field. (Hint: Split up the solution into two portions, one for each fluid. Generate expressions for u as a function of z and u, as a function of z.) (10 pt) Moving wall V Interface h2 Fluid 2 P2. 42 h2 Fluid 1 P1. 41