Download Sample problems

Name:__________________________
ME3250 Fluid Dynamics I, Fall 2012
Mid Term Exam II Practice Problem
Problem 1. The velocity of a 3-D flow field is given as: u = y+2z,
v=2x+z, and w=3x+3y.
z
a) Find whether the flow is incompressible.
y
b) Compute the field of fluid element acceleration.
c) Find the angular velocity of a fluid particle in each direction.
x
Determine whether the flow is irrotational.
d) If the dynamic viscosity of the fluid is =2, compute the viscous stress tensor.
e) Compute the viscous force (vector) acted on a circular horizontal surface facing up (i.e.
towards z-direction) with a radius of R=1.
Name:__________________________
Problem 2 Water of constant density =1000kg/m3 flows through a horizontal 2-D
channel shown in Fig.2.2. The flow is in steady state. The entrance of the pipe has a
height of H1=2m and the outlet has H2=4m. The velocity is uniform (u1 = 1m/s) at inlet.
The flow at outlet is laminar and fully developed. Ignore gravity in this problem.
(a) Write down the Navier-Stokes equation for the fully developed flow at the outlet in xdirection; simplify it to a solvable form based on the given flow conditions.
(b) Solve the simplified equation and obtain the solution with three unknown constants,
one of which is the pressure gradient at the outlet.
(c) Write down two boundary conditions and use them to eliminate two of the constants.
(d) Use mass conservation and finite control volume analysis to determine the last
constant, i.e. the pressure gradient.
(e) Assuming that the inlet and outlet are both under atmospheric pressure, use
momentum conservation and finite control volume analysis to determine the net
horizontal force (with the sign) anchoring the pipe.
y
x
Name:__________________________
Problem 3 The velocity of a steady state flow is given as: u = yz, v=xz, and w=xy.
a) Find whether the flow is incompressible.
b) Find whether the flow is irrotational. Find the angular velocity (vector) if so. Explain
otherwise.
c) Find the acceleration of the fluid particle at x=1, y=2 and z=3.
d) Find whether this a potential flow. Find a potential function if so, explain otherwise.
e) If the pressure at x=0, y=1, and z=0 is p=1atm, and the density of the fluid is =2,
compute the gauge pressure at x=1, y=1 and z=1. Ignore gravity.
Name:__________________________
Problem 4 Water with density  and dynamic viscosity  flows down an inclined pipe
with radius R. The flow is in steady state and fully developed. The angle between the
pipe and the ground is 30. There is no axial (z-direction) pressure gradient.
a) Write down the Navier-Stokes equation in the axial direction.
b) Simplify the equation based on the given flow conditions
c) Write down the boundary conditions that are needed to solve the problem.
d) Solve the simplified equation through integration
e) Determine the constants using the B.C.s.
Name:__________________________
Problem 5 An equation for viscous fluid dynamics involves four variables:  (density), u
(velocity), y (height), and  (dynamic viscosity).
a) Starting from the definition of viscous stress, derive the dimension of .
b) Write down the dimensions of each quantity and determine how many base dimensions are
involved.
c) Using the first r variables as repeating variables, where r is the number of base dimensions,
obtain a dimensionless variable using the Buckingham Pi theorem.