EATS 2470.03 - Introduction to the mechanics of fluids and solids.
FINAL EXAM (3 hr, SUNDAY 9 April, 2006; 2.30-5.30 in VH 3006)
Test covers Dimensional Analysis, Cartesian Tensor algebra and calculus, Stress and strain in
continua, elastic solids and fluid dynamics.
Part A: Answer all questions. (8 questions, 5 marks per question)
Part B: Answer as many questions as you can (8 questions, 10 marks per question)
WRITE YOUR ANSWERS IN THE EXAM BOOKLETS - indicate clearly which question
is being answered where. EXPLAIN your methods and calculations with words wherever
appropriate.
All tensor questions relate to right handed Cartesian coordinates
Calculators plus two pages (or 4 sides) of notes (8.5x11 inch paper) are permitted.
One form of the "ε-δ" relationship is εijkεpqk = δipδjq - δiqδjp, in case you did not include this in
your notes.
In the absence of any body forces, Navier's equations for displacements ui in an elastic solid
can be written,
x x
+ u
x x
= ( + ) u
t
u
k k
i
2
i k
k
2
2
i
2
¶ ¶
¶
¶ ¶
¶
¶
r ¶ m l m
where λ and μ are the Lamé constants
Hooke's law for a linear isotropic elastic solid is σij = λεkkδij + 2μεij where σij is the stress
tensor and εij the strain tensor. The inverse can be written as εij = [(1+ν)/E]σij - [ν/E]δijσkk
where E is Young's modulus and ν is Poisson's ratio.
In the absence of any body forces, the Navier - Stokes equations for velocity and pressure in a
viscous fluid can be written,
x x
+ u
x
)= - p
x
+u u
t u
(
k k
i
2
k i
i
k
i
¶ ¶
¶
¶
¶
¶
¶
¶
r ¶ m
while the continuity equation for an incompressible fluid is,
=0
x
u
j
j
¶
¶
---------------------------------------
1
Part A:
A1. Consider a projectile launched, on Earth, with a velocity U at an angle of 45 degrees
above a horizontal surface. Suppose it travels a distance L before striking the ground.
WITHOUT solving the equations of motion, use dimensional analysis to show that,
ignoring aerodynamic drag, the same projectile launched with speed 2U would travel a
distance of 4L.
A2. Determine the value of each element of the vector ak = δijεijk +δijδijbk, where δij and εijk
have their standard meanings as the substitution and alternating tensors and bk = (1,0,1).
A3. Write axb in tensor form. Suppose you have two vectors a and b and find that akbj = ajbk
for all j,k. What can you deduce about the vectors a and b? Explain your reasoning.
A4. Use tensor notation to prove the vector identity,
(a x b).(c x d) = (a.c)(b.d) - (a.d)(b.c)
[You may assume the "ε-δ" relationship]
A5. Using tensor notation show that, curl (curl u) = grad (div u) - 2u
A6. What, briefly, is "the stress tensor" and what are its units? Explain why it is symmetric
(a 2D explanation will do).
A7. Briefly explain how Poisson’s ratio is defined for an elastic solid, and show that
Poisson's ratio = 0.5 for an incompressible isotropic material.
A8. Briefly explain the concept of Reynolds number and the transition from laminar to
turbulent flow in a pipe at high Reynolds number
Part B:
B1. a) Say which of the following are legitimate expressions in Cartesian tensor notation,
and which are not, and explain your reasoning.
i) aiijbijj = 9 ii) aijbij = ck iii) aijbjkcki = d
b) Compute or otherwise evaluate
i) εijkεijk ii) εijk εikj iii) εijk δi1δj2δk3
c) From the Hooke’s law relationships given on page 1, obtain an expression for μ in
terms of E and ν.
2
B2. a) Explain what is meant by a "direction cosine" (αij) for a transformation from xj
Cartesian coordinates to xi Cartesian coordinates. Show that αipαjp = δij
b) Find the direction cosine tensor for the transformation resulting from a rotation of π/4
about the x3 axis.
c) Show that the trace (aii) of any Cartesian tensor aij is a scalar invariant, i.e. it has the
same value after transformation, so that ajj = aii.
d) Illustrate this result by using the transformation from part b) above.
B3. Use Cartesian tensor methods to show that curl(φu) = φ curl(u) + (grad φ) x u
where φ is a scalar function of position and u is a vector function of position.
Suppose φ = |x|2 and u = xi. Obtain tensor forms for v = grad φ and w = curl(φu) ; as
functions of position and evaluate them at (0,0,1).
B4. Explain the link between the principal stresses associated with a given stress tensor
and the eigenvalues and eigenvectors of the corresponding matrix.
Find the principal stresses and the direction corresponding to the largest of the
principal stresses if the stress tensor (τij) is
| 2 0 0 | where the units are 106 Nm-2
| 0 2 √3 |
| 0 √3 4 |
B5. Consider an experiment where an 0.1m x 0.1m x 0.1m cube of elastic material is
subjected to a uniformly distributed compressive force along one axis of 100 N.
Compute the corresponding normal stress. If the material compresses by 1% in the
direction in which the force is applied, and expands by 0.02% in each of the other two
directions, compute the Young’s modulus and Poisson’s ratio for the material.
B6. Starting from Navier's equations, and ignoring effects of gravity and any inhomogeneity
of the Earth, show that the speed of a plane Secondary or Shear wave is Cs = (μ/ρ)1/2.
Given that the speed of a Primary wave is Cp = ((2μ+λ)/ρ)1/2, and taking the local
properties of the ground to be μ = 21GPa, λ = 14 GPa and ρ = 2.5 kg m-3, compute Cp
and Cs. If a seismograph records the first arrival of an S wave 120s after the first arrival
of the P wave corresponding to the same seismic event, determine how far away from
the seismograph the event is and when it occurred (relative to the arrival of the P wave).
3
B7. Consider a thin sheet of fluid flowing steadily under gravity on an inclined plane at an
angle α = 10 degrees to the horizontal. Suppose an equilibrium has been reached and the
flow is not accelerating or decelerating. Take Cartesian coordinates with x1 along the
sloping surface and x3 normal to it, as in the sketch below.
Set up the x1 component of the Navier-Stokes equations, assuming u2 = u3 = 0. Suppose
that the fluid’s kinematic viscosity is ν = 1 x 10-6 m-2s-1, and the depth of the fluid in the
equilibrium state is 1 mm (normal to the sloping surface). Assuming no shear stress on
the free surface what is the fluid velocity at the free surface?
g
free surface
Horizontal α
B8. Explain what is meant by the terms, "rate of change following the fluid", "local rate of
change" and "advective rate of change", in the context of the equation,
x
T
+u
t
= T
Dt
DT
k
k ¶
¶
¶
¶
You are outside at York and observe that the local air temperature increases steadily
from 5 C at 6am to 21 C by 10am, while a friend who drove from near Guelph (100km
due West) reports that the temperature there was 10C at 6am. If there is a steady
40km/hr wind from the west during this period, and the temperature variations in space
and time are assumed linear, how much of the local observed temperature change (i.e.
∂T/∂t) should you attribute to advection (u ∂T/∂x), and how much to heating of the air
by the sun.
4

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