Fluid Dynamics I

Course Work

(Due before 4pm, 11th November)

1. Consider an incompressible viscous fluid motion through a tube of elliptic cross section, given by the equation

y2

a2 +

z2

b2 = 1.

The pressure difference between the tube ends is ?p. Assuming that the tube length L is large, find the velocity distribution in the tube.

Suggestion: Use the Cartesian coordinates with x-axis directed along the centre- line of the tube. You may assume without proof that the y- and z-components of the velocity are zeros. Seek the solution for the x-component in the form

u = C1 + C2y 2 + C3z

2,

where C1, C2 and C3 are constants to be found.

2. A layer of viscous fluid of thickness h is sliding down a flat slope in the gravitational field g. The angle between the slope and horizon is ?; see Figure E1.1. Find the velocity distribution across the layer.

x

y

h

?

g

Figure E1.1: Fluid layer on the downslope.

Hint : Use Cartesian coordinates with the x-axis measured down the slope, and notice that the tangential stress

?yx = µ

(

?v

?x +

?u

?y

)

is zero at the upper edge of the fluid layer.

1

3. Consider a single circular cylinder of radius R surrounded by viscous fluid of density ? and dynamic viscosity µ. The cylinder rotates around its axis with angular velocity ?. Assuming that the fluid remains at rest far from the cylinder (r ? ?), prove that the velocity field and pressure are given by the “potential vortex” solution

Vr = 0, V? = ?

2?r , Vz = 0, p = p? ?

??2

8?2r2 . (e3.1)

How does the circulation ? depend on the cylinder radius R and angular velocity