PROBLEM SET 4: DUE NOVEMBER 10, 2014
FLUID MECHANICS I
1. Consider the uniform slow motion with speed U of a viscous fluid past a spherical
bubble of radius a, filled with air. Do this by modifying the Stokes flow analysis for a rigid
sphere as follows. Apply conditions relevant to the situation where fluid within the bubble
is unable to support shear stress. Determine the new stream function and estimate the
drag on this bubble D. What is D/(4πU a)?
2. For a cylindrical jet emerging from a hole in a plane wall, we have a problem
analogous to the 2D jet considered in class. Consider only the boundary-layer limit. (a)
Show that
(1)
ν ∂ ∂uz
∂ 2 1 ∂
u +
(rur uz ) −
(r
) = 0,
∂z z r ∂r
r ∂r ∂r
and derive the corresponding jet-momentum conservation equation.
(b) Defining a stream function ψ that allows for the calculation of (ur , uz ), where ψ(0, z) =
0, determine the form of ψ.
3. Consider two-dimensional Stokes flow past a circular cylinder of radius a. Show that
the problem reduces to the biharmonic equation for the two-dimensional stream function
ψ, determine the appropriate boundary conditions. Determine the form of the solution for
ψ and discuss the implications for the existence of solutions to the 2-D problem.
4. A square-duct wind tunnel of length L = 1 m is being designed to operate at room
temperature and atmospheric conditions (see Figure 1). A uniform airflow at U = 1 m/s
enters through an opening of D = 20 cm. Due to the viscosity of air, it is necessary to
design a variable cross-sectional area if a constant velocity is to be maintained in the middle
part of the cross-section throughout the wind tunnel.
a) Determine the duct size, D(x), as a function of x.
b) How will the result be affected if U = 20 m/s? At a given value of x, will D(x) be larger
or smaller (or the same) than the value obtained in part a)? Explain.
c) How will the result be affected if the wind tunnel is to be operated at 10 atm (and U = 1
m/s)? At a given value of x, will D(x) be larger or smaller (or the same) than the value
obtained in part a)? Explain.
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2
FLUID MECHANICS I
Figure 1. Geometry for problem 4.
5. Derive the formula for the temporal Taylor microscale λt by expanding the definition
of the temporal correlation function into a two-term Taylor series and determining where
the correlation falls to zero.
6. Starting from Equations (12.38), (12.39) and (12.40) of Kundu, set r = re1 , show
that F = u2 f (r) and G(r) = u2 g(r).
(a) Compute ∂Rij /∂rj for incompressible flow.
(b) For homogeneous-isotropic turbulence use the result of part a) to show that the longitudinal, f (r), and transverse, g(r), correlation functions are related by g(r) = f (r) +
(r/2)(df (r)/dr).
(c) Use part b) and the integral length scale and Taylor microscale definitions to find relationships between them.
In homogeneous turbulence, Rij (rb − ra ) = ui (x + ra ) uj (x + rb ).
(d) Determine (∂ui (x)/∂xk )(∂uj (x)/∂xm ).
(e) If the flow is incompressible, determine the relationship between (∂u1 (x)/∂x1 )2 and
d2 f /dr2 .