Saturday, December 12, 2009 Professor Philip S. Marcus
ME 106
Solution to Final Examination
1a) The relevant input parameters are: , Router , inner , H , Rinner , outer , and . Specically, we do not include g , and the air pressure. We choose units of cfw
ME 106, Fluid Mechanics
Assignment 7 Solutions
1. For the following function can we use potential ow theory, and why? (Assume the
uid is inviscid.)
V = (2xt 2yt
i
j)
V = (2xt 2yt
i
j)
u v
+
= 2t 2t = 0
x y
Thus, the ow is incompressible
v u
=00=0
x y
Thus
Saturday, December 12, 2009 Professor Philip S. Marcus
ME 106
Final Examination
1) 20 points You have built a small Couette-Taylor apparatus. In this apparatus, an incompressible ow with constant density and kinematic viscosity is conned between two conce
1. (a) What is the steady state shear stress in Pascals (i.e. N/m2 ) required to move the
top plate shown at 3 m/s? The uid is glycerin at 20o C and the height L is 5 mm?
(See inside cover of text for glycerin properties/viscosity.)
(b) Physically speakin
O. Sava s ME 106 Fluid Mechanics Spring 2011 SCP Special/Computational Problems
Jan 11, 2011
1. Consider a rigid sphere of radius a submerged in a liquid of density . The total force on the sphere due to the hydrostatic pressure distribution is F=
S
p n
Solution of Practice Final
Final Exam, Fall 2014
Problem 2: The hydrostatic equation is 1 P + g~k = 0. This can be achieved by setting all velocities
to zero in Navier-Stokes equation. The x and y direction of these equation gives
P
= 0.
y
P
= 0,
x
So, th
Differential Analysis
Our goal is now to become more sophisticated in our description of fluid dynamics. To start,
lets consider the deformation rate (aka strain rate) of an infinitesimal fluid element. I claim
that deformation rate is described by the ve
Solution of Homework 7
Problem 1:
Problem 2:
1
Problem 3: Conservation of mass equation
+ (V~ ) = 0.
t
(u) (v) (w)
+
+
+
= 0.
t
x
y
z
Substituting u = x/t, v = w = 0 and considering that = (t) is not a function of x, y, z gives
x d(t)
+
=
+ = 0.
t x
t
Solution of Homework 10
Problem 1: n = 5 for number of variables P, , Q, D, . Also [P ] = M L1 T 2 , [] = M L3 , [D] = L,
[Q] = L3 T 1 and [] = T 1 . The primary dimensions that appear are M , L and T , so m = 3. Hence
the number of groups are n m = 2. Fo
ME 106 Fluid Mechanics: Practice Midterm 2
Fall 2015
1. Given the unsteady flow field u = t2 and v = 1 t,
(a) Determine the equation y(x) describing the streamline passing through point x = 0 and y = 0 at
time t = 2.
(b) Determine the equations x(t) and y
Solution of Homework 9
Problem 1: Since the flow is steady, incompressible, inviscid we can use Bernoulli equation along the
streamline. The streamline that passes over the half-circle is continued from the line on the ground.
Assuming the potential theor
Problem 2: Momentum flow-rate uniform = u
uA =
u2 R2 .
Momentum flowrate non-uniform is
Z 1
Z R
r 2 r r u2 R2
2
2
1
d
= c
uu(2rdr) = uc 2R
R
R
R
3
0
0
Obtain relationship between u and uc
2
2
R
Z
uR = 2R uc
0
r 2 r r
1
d
R
R
R
u =
uc
2
thus,
4 2 2 4
u
ME 106 Fluid Mechanics: Midterm 3
Fall 2014
Name & Discussion Section:
1. Can pressure or gravity forces generate vorticity? If they can, describe how. If they cannot, describe why.
This can be answered two ways:
(a) No: (expected answer) Consider a free
Solution of Homework 8
Problem 1: Potential theory can be used when the flow is potential flow, that is (1) flow is irrotatioal
and (2) flow is incompressible. Checking incompressibility for V~ = (2xt)~i (2yt)~j,
u v
+
= 2t 2t = 0
V~ =
x y
Thus, the flow
Solution of Problem Set 3
Problem 1: The cross sections of points (1), (2) and (3) are chosen as in figure.
The hydrostatic relation between points (2) and (3) gives
P2 = P3 w gH,
where w is density of water and P3 = Patm . Conservation of mass between po
1.
Problem 4. Figure 1 shows the bottle before it is squeezed. The test tube is floating since
its weight W is in balance with buoyancy force FB . The buoyancy force is equal to the
weight of displaced fluid, which here is the portion of air inside the te
Solutions Manual Fluid Mechanics, Fifth Edition
260
4.5 The velocity field near a stagnation point (see Example 1.10) may be written in
the form
U o y
U x
u= o
v=
U o and L are constants
L
L
(a) Show that the acceleration vector is purely radial. (b) For
Solution of Homework 6
Problem 1: Method 1: Using inertial reference frame. Set the observer to be on the ground. Let the
control volume move with the cart. The x-momentum equation is
F =
(MC.V. VC.V. ) M Vjet .
t
Let the mass inside control volume be MC.
Problem Set 7
Due in class Friday April 11
1. Examine the steady flow in Figure 1 below.
Water with constant density exits a hose at point 1 with velocity V1 at an angle 1 with respect to
horizontal. Gravity g makes the stream tube of the exiting water cu
ME106 Discussion 4/8, 4/9
I apologize in advance for the unusual format of these solutions. I wanted to make sure the discussion
had them, but I also dont have a whole bunch of time to write up something nicely in LaTeX. Hopefully
they are sufficient.
Sol
Professor Philip S. Marcus
ME106
Problem Set 8
Due in class Friday April 18, 2014
1) We are now chicken experts and also experts in driving trucks across bridges. We
are about to be rocketry experts. We are prepared therefore to combine the two
together.