MECH 3300 Assignment # 1

MECH 3300
Assignment # 1
Due: January 24 th , 2011, 9:30
1. Show that the following expression ( ∙ ∇) produces the following differential form
for the convective acceleration:
+
+
2. When a valve is opened, fluid flows in the expansion duct (see figure) according to
the approximation:
= 1 −
tanh
2
Find:
a) Find an expression for the acceleration
of the flow;
b) the fluid acceleration at (x, t) = (L, L/U);
c) the time for which the fluid acceleration
at x = L is zero;
d) Why does the fluid acceleration
becomes negative after the time found in c).
3. A piston compresses gas in a cylinder by moving at constant speed V. Let the gas
density and length at t = 0 be ρ 0 and L 0 , respectively. Let the gas velocity vary
linearly from u = V at the piston face to u = 0 at x = L. If the gas density varies only
with time, find an expression for ρ(t).
4. For a laminar flow between parallel plates, the flow is 2D (v ≠ 0) if the walls are
porous. A special case solution is:
= ( − )(h − ) where A and B are constant.
Find a) a general formula for velocity v if v = 0 at y = 0. b) What is the value of the
constant B if v = v w at y = +h
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5. A frictionless, incompressible steady flow field is given by
= 2 −
in arbitrary units. Let the density be ρ 0 = constant and neglect gravity. Find an
expression for the pressure gradient in the x direction.
6. Consider a steady, 2D, incompressible flow of a newtonian fluid in which the velocity
field is known: u = -2xy, v = y 2 -x 2 , w = 0.
a) Does this flow satisfy conservation of mass;
b) Find the pressure field, p(x,y) if the pressure at the point (x = 0, y = 0) is
equal to p a .
7. From the Navier-Stokes equations for incompressible flow in polar coordinates, find
the most general case of purely circulating motion v θ (r), v r = v z = 0, for flow with no
slip between two fixed concentric cylinders as in the figure.
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