ASSIGNMENT 5 EATS 2470
EATS 2470 - 2011
INTRODUCTION TO CONTINUUM MECHANICS
ASSIGNMENT 5 : All questions marked with are to be handed in by
12.30 pm, March 29. This is the final assignment.
Question 1
In Lab 3 you studied the bending of a beam supported at one end and loaded by W (N) at the
other.
The beam displacement result is y = Wx2(3l-x)/(6EI) where E is Youngs modulus and I is
the area moment of inertia (I= wh2/12) where w and h are the width and thickness of the
beam.
i) Derive the result for I
ii) Derive the result for y.
Question 2
Starting from the Euler Equations (the Navier-Stokes equations with zero viscosity) derive
Bernoulli’s equation for an incompressible, inviscid fluid.
For a Boeing 747 look up wing area (A) and cruise speed (U) at
http://en.wikipedia.org/wiki/Boeing_747. Assuming an air density of 1 kgm-3, calculate the
lift coefficient [Lift/(0.5ρU2A)] necessary to support the aircraft if the mass is 300,000 kg.
What does this mean in terms of relative air speeds above and below the wings.
Question 3
What is the Reynolds number in fluid flow? Explain its role in determining which are the
dominant terms in the Navier Stokes equations.
Question 4
Look up the form of the Navier-Stokes equations in cylindrical polar coordinates. For an
incompressible fluid solve the equations to obtain an expression for the velocity profile U(r)
for steady, laminar, flow, in a long circular pipe of radius r with a pressure gradient dp/dz =
-G where z is distance along the pipe. Assume axial symmetry and no variation of U with z.
In order to maintain laminar flow we need the Reynolds Number = 2Umr/ν < 2000 where Um
is the mean velocity in the pipe. What is the maximum Um for water flow in a pipe of
diameter 1mm if the flow is to remain laminar.

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