1. Suggest how one could measure the viscosity of a viscous fluid by using a simple result
from slow motion analysis. What precautions would have to be made to insure that the
measurement was accurate?
2. Please base on the assumptions and implications to discuss the differences among ideal
flow, exact solution of flow, and slow motion of flow. Also try to write out the
equations for each flow from Navier-Stokes equation.
3. Clearly state why the assumption of incompressible potential flow is equivalent to the
assumption of ideal flow. Show and indicate any addition assumptions that have to be
made. What is the deficiency of the potential flow solutions when compared to real
systems?
4. A flat plate of essentially infinite width and breadth oscillates sinusoidally in its own
plane beneath a viscous fluid. The fluid is at rest far above the plate. Making as many
simplifying assumptions as you can, setup the governing differential equation and
boundary conditions for finding the velocity field u in the fluid.
5. Investigate the polar coordinate velocity potential, given below, in general terms; i.e.,
outline the steps in detail that you would undertake to provide a graphic picture of the
flow field. It is not necessary to actually derive the flow field or provide a picture of the
flow.
 (r ,  )  U  r cos  [1  (a 2 / r 2 )]  ( / 2 )
Where a is a known constant that you can set to any value and Γ is a known value of
circulation.
1. Please base on the assumptions and implications to discuss the differences among ideal
flow, exact solution of flow, and slow motion of flow. Also try to write out the
equations for each flow from Navier-Stokes equation.
2. Clearly state why the assumption of incompressible potential flow is equivalent to the
assumption of ideal flow. Show and indicate any addition assumptions that have to be
made. What is the deficiency of the potential flow solutions when compared to real
systems?
3. Investigate the polar coordinate velocity potential, given below, in general terms; i.e.,
outline the steps in detail that you would undertake to provide a graphic picture of the
flow field. It is not necessary to actually derive the flow field or provide a picture of the
flow.
 (r ,  )  U  r cos  [1  (a 2 / r 2 )]  ( / 2 )
Where a is a known constant that you can set to any value and Γ is a known value of
circulation.
4. A rod (radius, Ri) rotates with a constant radial speed ωi inside a cylindrical tube (radius,
Ro). An incompressible viscous fluid in pumped axially in the annular region at a flow
velocity of Vo (see the figure below). It is desired to find the velocity distribution of the
fluid in the annular region. Set up the differential equations and boundary conditions.
Neglect end effects. You don’t need to solve this differential equations, but you do need to
indicate all assumptions and conditions you need to make this problem solvable.
Ri rotating at ωi
Vv
Ro
5. The internal energy change derived from equation of change is as following:
 DU / Dt   (  q)  p(  U )  ( : (U ))   ( J s  Fs )
s
This equation can be expressed in thermodynamic terms of pressure, volume, and
temperature, as following:
cv ( DT / Dt )  (  q)  ( : (U ))  T (p / T ) v (  U )   ( J s  Fs )
s
Please prove it.
6 Solve the flowing problems:
a. What is the flow given by   3 x  ln r
b. A stream function   2 x 3  3 y 2
What is the magnitude and direction of the velocity at (4,3)? Draw the streamlines
corresponding to
ψ=0 andψ=2.
1. A rod of radius Ri move upward with a constant velocity V through a cylindrical
container of radius Ro, that contains an incompressible viscous fluid. The ends of the
cylinder are closed and the rod slides through the ends through a gasket. The fluid
circulates in the cylinder, moving upward along the moving central core and moving
back downward along the fixed wall. It is designed o fine the velocity distribution in
the annular region away from the end disturbances. You do not have to integrate and
solve for the velocity. However, make sure you indicate all the assumptions, equations,
and conditions you need for your solution.
2. A flat plate of essentially infinite width and breadth oscillates sinusoidally in its own
plane beneath a viscous fluid. The fluid is at rest far above the plate. Making as
many simplifying assumptions as you can, setup the governing differential equation and
boundary conditions for finding the velocity field u in the fluid.
4. Clearly state why the assumption of incompressible potential flow is equivalent to the
assumption of ideal flow. Show and indicate any addition assumptions that have to be
made. What is the deficiency of the potential flow solutions when compared to real
systems?
5. The velocity in a two-dimensional field for an incompressible fluid is
U x  x 2 y  2 x  ( y 3 / 3)
U y  ( x 3 / 3)  2 y  xy 2
Is the equation of continuity satisfied?
Is the flow rotational?
Determine the magnitude of the velocity at the point (2,2)?
What is the vorticity at the same point?
1. Prove the following relationships:
(i) (  ( A  B))  ( B  (  A))  ( A  (  B))
(ii)   A  B  B  A  A(  B)  A  (B)  B(  A)
(iii)  : uv  (  u v)
(iv)   vw  v  w  w(  v)
(v) ( A  A)  2( A  (  A))  2( A  ) A
2. Write out  - equation in spherical coordinates for Eq. (4-12). You can use
either one of two methods which are mentioned in the class to get the answer.
But please state clearly how to get each term of Eq. (4-12).
3. Calculate and describe particle paths and streamlines for the flows
(i)
v= (xt, -yt, 0),
(ii)
v= (xt, -y, 0)
(iii)
v= (a cosωt, a sinωt, 0)
4. (20 pts) Explain the following questions
(i)
What are the differences between Eulerian and Lagrangian
descriptions? Give a real example in the real world.
(ii)
What are the differences between streamline and path? What kind of
conditions will these two have the same flow pattern?
1. In the Gibbs tensor notation, the relation between the stress tensor and the velocity gradient
terms is
 ji  [(U i / x j )  (U j / xi )]  [(2 / 3)   ](U l / xl ) ij
What is this a generalization of? Discuss whatever you can about the nature of the second
coefficient of viscosity, κ. Write out this equation for τyz and forτyy.
2. In the general property balance, the term (▽‧ψU) appeared. Where did it come from? What is
its meaning? The derivation was made from the viewpoint of an element moving at U. What
happens to the above term if the view of a stationary element is taken? What happens to the flux
term in the general property balance when the view of a stationary element is taken?
3. What is the equation in the below?
(Please explain more detail.)
What law of the mathematic expression is this equation?
 ( DU / Dt )  (  P)    s Fs
s
1.Outline, in the most general terms, the necessary to obtain the Navier-Stokes equation. (10%)
Equations are not needed! What are the assumptions made? (10%) What is the range of validity
of the assumptions? (5%) What is the meaning of each term in the equation? (10%)
(U / t )  (U  (  U )) 

1
1
1
1
(U  U )   p 
(  U )  [(  U )  (  (  U ))]    s Fs
2

3


4. In the study of fluid flow problems we wish to define the velocity distribution as well as the
states of the fluid over the whole space for all time. There are 6 unknowns which are
functions of the Cartesian coordinates and time. (4 unknowns in vectors space and time)
Name these unknowns. (10%) In order to determine these unknowns we have to find 6
relations (4 in terms of vectors). What are the equations to relate these unknowns. (10%)
1. (25%) Please proof
( A  A)  2( A  (  A))  2( A  ) A
2. (25%) Write out r - equation in spherical coordinates for Eq. (4-12). You can use either
one of two methods which are mentioned in the class to get the answer. But please state
clearly how to get each term of Eq. (4-12).
3. (25%) Calculate and describe particle paths and streamlines for the flow
v  (ax,ay, b(t ))
What could be modified by the case b(t) = constant?
Please draw the simple figures for paths and streamlines.
4. (25%) Find streamlines and particle paths for the two-dimensional flows
(i) v = (asint, acost, 0)
(ii)v = (x, y-Vt, 0)
(iii)vr=rcos(/2), v=rsin(/2), vz=0, 0<  < 2
Please draw the simple figures for streamlines.
1. The following streamlines are given
  Ar 2 sin  cos
Explain how you would obtain a picture of the flow given by this stream function. Can
you suggest what the picture of the flow is without selecting and calculating individual
streamlines? Show your work.
2. Obtain the flow picture for
  Ar 2 cos 2
3. The power law is often used to describe non-Newtonian flow of materials over a limited
range of shear rates. It is given by
  K (dU z / dr ) n
where K and n are assumed to be constant. How would you proceed to solve for the
flow of this material in a pipe by the same general methods used in textbook page 90?
Please setup the basic equations and B.C.s and solve it. Write down all assumption.
4. Solve the rotation of single cylinder in an infinite real fluid. What is the frictionless flow
that is exactly equivalent to this real fluid flow? Show and explain the equivalence by
comparing the solutions. What is the value of the constant in the ideal flow solutions?
1. Suggest how one could measure the viscosity of a viscous fluid by using a simple result
from slow motion analysis. What precautions would have to be made to insure that the
measurement was accurate? (at least 5 terms)
2. The equation of motion in spherical coordinates is given on the attached figure. Obtain
the basic differential equations that describe the flow for a plate-end-cone viscometer
shown below. IF the flow between the plate-end-cone is very very slow, what
assumption can be made to simplify the equation and what if any terms can be dropped
from the differential equations you obtained? With the very very slow flow assumption,
given the final differential equation and boundary conditions under the assumption of
steady state. If the flow were such that the very very slow assumption was not valid,
what would be the dimensionless groups that would to be used to empirically describe
the problem? DO NOT FORGET THAT YOU MUST ALSO CONDISER THE
EXISTANCE OF BOUNDARY CONDITIONS.
3. Set up the differential equations and boundary conditions for the flow of a fluid in an

Cone rotating at an
Angular velocity of 
1

Liquid held between
cone and plate by
surface tension
O
stationary
flat plate
annual space with the inner cylinder in rotation.
direction and flow in that direction.
R
There is a pressure gradient in the axial
Flow in (inside hole radius is small compressed to radius
of the plate, thus neglect any entrance effects)
Top plate is
Stationary

lower plate is rotating
4. Set up the equations and boundary conditions for the flow: