Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infinitely long and wide, parallel flat plates (Fig. P9–104). 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 2z-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 P₀. 

(a) List all the appropriate boundary conditions on both velocity and pressure. 

(b) Solve for the velocity field. 

(c) Solve for the pressure field. 

(d) Let fluid 1 be water and let fluid 2 be unused engine oil, both at 80°C. Also let h₁ = 5.0 mm, h₂ = 8.0 mm, and V = 10.0 m/s. Plot u as a function of z across the entire channel. Discuss the results.

FIGURE P9–104

Transcribed Image Text:

Moving wall m₂ hi N Interface X V Fluid 2 Fluid 1 P2, M2 PI, MI