fluidMechanics

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ME 240A Midterm 2 Fall 2016 Instructor: Marko Princevac

Closed books, closed notes. Each problem brings a maximum of 25 points.

For all problems assume laminar, incompressible flow.

1. For the configuration shown below: find the flow field (5 points), flow rate (5 points), shear

stress exerted on the upper plate (5 points) and shear stress exerted on the slope (5 points). Make a

sketch of the velocity field (5 points). The fluid has dynamic viscosity  and density , plate is moving with velocity Uo, and slope angle is a.

2. Consider a film of liquid draining at volume flow rate Q down the outside of a vertical rod of

radius a, as shown in figure below. Some distance down the rod, a fully developed region is

reached where fluid shear balances gravity and the film thickness remains constant. Find an

expression for vz(r) and a relation between Q and film radius b. (10 points for vz(r), 10 points for Q(b,a))



Uo



g



b

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3. a) The uniform stream of velocity U approaches a channel (of width b). After

flowing a distance le (“entrance length”) flow inside the channel will become “fully developed”.

Show that the entrance length le is proportional to the Reynolds number Re (based on the channel

width b). Fluid’s dynamic viscosity is  and density is . (10 points)

b) The uniform stream of velocity U and temperature T approaches stationary flat

plate whose temperature is TW (TW > T). Fluid’s kinematic viscosity is , thermal diffusivity is

, and Prandtl number Pr = 10. Estimate distances xv and xT (measured from the leading edge) where velocity boundary

layer reaches thickness v ~ L and thermal boundary layer reaches thickness T ~ L, respectively. What is the ratio xT / xv ? (15 points)

U

Fully developed

flow

Entrance region

Potential core

Boundary layer

Boundary layer

le

b

x

U T T

U

TW



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4. Sphere (diameter D, density s) is fully immersed and keeps on sinking in the water (density

w, dynamic viscosity ). The drag force on the sphere is given by FD=3UD. a) Find the terminal velocity of the sphere UT. (10 points)

b) If initial velocity of the sphere is UI (UI = 0), find the time and distance traveled before

the terminal velocity is achieved. Comment on results. (15 points)

Navier-Stokes Equation in Cylindrical Coordinates

 

 

2 2 2

2 2 2

2 2

2 2

2

2 2

1

1 1

1 1 2

1 2

r z

r r r r r

r r

z

V v v v r r z

r r r r r z

r momentum

vv vp V v v g v

t r r r r

momentum

v v v vvp V v g v

t r r r r

z momentum

v

t



 

     





  



   

      

  

         

    



              

   



              

   





   2

1 z z z

p V v g v

z 



       



U

g

D

s



w