Chapter 9 Viscous flow

18 CHAPTER 9. VISCOUS FLOW
Instead of having a discrete set of point forces, if we have a force distribution,
in which the force per unit volume is given by f i (x), the velocity and pressure
fields can be calculated by integrating the appropriate product of the Oseen
tensor and the force distribution over the volume,
∫
u i (x) = dx ′ J ij (x − x ′ )f j (x ′ )
∫
p(x) = dx ′ K i (x − x ′ )F i (x ′ ) (9.71)
9.2.2 Green’s function for force dipoles:
There are often situations where the suspended particles in a fluid a ‘neutrally
buoyant’, so that they exert no net force on the fluid. Examples are for a
solid sphere in a uniform shear flow in equations and , and for a rotating
particle in equations and . The velocity disturbance at a large distance
from the center of the particle decays proportional to r −2 , in contrast to the
decay proportional to r −1 in equation for the flow around a particle which
exerts a net force on the fluid. The far-field velocity can be expressed as a
function of the ‘force dipole moment’, which is defined as the integral over
the sphere of the tensor product of the surface force and the position vector,
∫
Force dipole moment = dSx i f j
∫
= dSx i τ jk n k (9.72)
Note that this definition is analogous to the ‘dipole moment’ for the temperature
field due to a point dipole. The force dipole moment is a second
order tensor, and it is convenient to separate it into a symmetric and an
antisymmetric part. The symmetric part is given by,
S ij = 1 ∫
dS(x i τ jk n k + x j τ ik n k ) (9.73)
2
while the antisymmetric part is,
A ij = 1 2
∫
dS(τ ik n k x j − τ jk n k x i ) (9.74)
The velocity and pressure fields around a point particle which exerts no net
force on the fluid can be expressed in terms of the symmetric and antisymmetric
dipole moments S ij and A ij .