The velocity for the fluid in the upper layer is given by uu=C1+C2y and that in the lower layer is given by uL=C3+C4y.
The value of uu at y = 2h is U
The expression for C1 in terms of C2 and the other variables is C_1=U - C_2(*2h)
The expression for the velocity of the fluid in the upper layer is U_1= C_2 * (y-2h) + U
The value of uL at y = 0 is 0
The value of C3 is 0
The expression for the velocity of the fluid in the lower layer is U_2 = C_4*y
At the interface between the two layers (y = h), the expression for the velocity in the upper layer is: U_1i= -hC_2 + U
At the interface between the two layers (y = h), the expression for the velocity in the lower layer is: U_2i = C_4*h
There is no slip between the two fluids at the interface and so the velocities are equal. The expression for C2 in terms of C4 is: C_2 = -C_4 + (U/h)
At the interface the shear stress in the top layer equals that in the lower layer. The shear stress is given by: Tau=μ*(du/dy) The expression for the shear stress in the upper layer in terms of the viscosity μ1 is: T_1= μ1*C_4
The expression for the shear stress in the lower layer in terms of the viscosity μ2 is: T_2= μ2*C_4
The shear stress at the interface is equal for both fluid layers. Equating the shear stresses, what is the expression for C2 ? For some reason Wiley Plus does not recognize (μ1*C_4)/ μ2 as the correct answer.
There are two expression for C2. One from the fact that the velocities are equal at the interface and the other that the shear stresses are equal at the interface. Equate the two expressions. What is the expression for C4?
Using the expression for C4, what is the velocity at the interface?
