Midterm Exam II Solution, ME 342 Fluid Mechanics (Spring 2013) 1. Consider a steady, incompressible, y=+h b viscous flow of viscosity due to y a pressure gradient x inside a channel of two fixed u=u(y) plates at a distance of 2h, as shown in the figure. Neglect a gravity effect. The channel width y=h b is W and it is very long compared to the channel length L (i.e., W>>L) so that it is still valid to assume that the channel flow is a two-dimensional plane flow ( ) and an axial flow (v = w = 0). In contrast to conventional no-slip boundary conditions, consider a slip boundary condition as shown in the figure, where us is defined as a slip velocity at the wall and b as a slip length (distance where the no-slip boundary condition will virtually be satisfactory by linear extrapolation). The relationship between the slip velocity and slip length is defined as ( ) . (a) Solve the momentum (Navier-Stokes) equation to find out u(y) and a maximum velocity umax (i.e., u at y=0) with the considered slip boundary condition. Use the definition of a slip velocity, ( ) , so that your solutions for the velocities should include the slip length b in the equations. (20 pt.)

u

2u 2u u u p v g x 2 2 x y x y x

Solving Navier-Stokes Eq 10 points

2u p 0 2 y x

1 p y 2 u( y) C1 y C2 L 2

Applying BCs (for no slip condition) at y h , u 0 or BCs 6 points at ,

C1 0 and C2

For slip condition

uslip b p h L

p h 2 L 2

2 points

2 points

Midterm Exam II Solution, ME 342 Fluid Mechanics (Spring 2013) (b) Compute the volume flow rate (Q) and the average velocity (Vav). The parallel plates can be considered as a square channel so that the cross-sectional area of the channel can be estimated to be 2hW. Your solutions should include the slip length b in the equations. (20 pt.) Q udA ,

A 2hW

4 points

Q

2Wh3 p 2h2W p b 3 L L...

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