Chapter 9 Viscous flow

30 CHAPTER 9. VISCOUS FLOW
(a) Write down the Stokes equations in terms of two Laplace equations,
one for the velocity and one for the pressure fields.
(b) Solve the Laplace equations for a point source which is independent
of the θ co-ordinate.
(c) Obtain the higher harmonics by taking gradients of the fundamental
solution.
(d) What are the possible solutions for the velocity and pressure fields
based on vector symmetries
(e) Obtain the constants in the solutions from the boundary conditions.
5. Determine the fluid flow field and the stress acting on a particle of
radius a placed in an extensional strain field u i = G ij x j (G ij = G ji )
at low Reynolds number where the fluid and particle inertia can be
neglected. The particle is placed at the origin of the coordinate system,
and the fluid velocity field has the undisturbed value u i = G ij x j for
r → ∞. Find the fluid velocity and pressure fields around the particle.
Find the integral: ∫
dAT il n l x j (9.115)
A
over the surface of the sphere, where n l is the outward unit normal.
6. A sphere is rotating with an angular velocity Ω k in a fluid that is at
rest at infinity, and the Reynolds number based on the angular velocity
and radius of the sphere, Re ≡ ρΩa 2 /µ is small. The velocity at the
surface of the particle is:
u i | r=a = ǫ ikl Ω k x l (9.116)
and the velocity and pressure fields decay far from the sphere.
(a) Find the most general form for the velocity and pressure fields at
zero Reynolds number. Note that the velocity and pressure are
real vectors, whereas Ω k is a pseudo vector.
(b) Use the incompressibility condition and the condition on the velocity
at the surface of the sphere to determine the constants in
your expression for the fluid velocity.