Fluids - boundary conditions for rotating sphere

[davcrai]

Homework Statement

A sphere under uniform rotation R, in a simple shear flow, given at infinity by
u_i = G(x₂ + c)delta_i1
The centre of sphere is fixed at x₂

Boundary conditions are u_i = E_ijkR_jx_k on sphere,
and u_i = G(x₂ + c) at infinity

Homework Equations

d_ij is the kronecker delta
E_ijk is the permutation symbol

The Attempt at a Solution

Just trying to understand the boundary condition on the sphere.

 

[tiny-tim]

hi davcrai!

u_i = ε_ijkR_jx_k is the standard formula for rotational velocity …

usually written v = ω x r, which is the same as v_i = ε_ijkω_jr_k

(in other words, the fluid velocity at the surface of the sphere is the same as the velocity of the surface, as one might expect)

 

[davcrai]

Thanks