Tensors for Physics

10.5 Applications in Hydrodynamics 179

of the functionspandvare

∇μp=−AXμνVν,

and

∇μvν=a′r̂μVν+b′r̂μXνλVλ−bXμνλ.

The divergence of this expression for the velocity field is

∇μvμ=a′r̂μVμ+b′r̂μXμλVλ=(a′+ 2 r−^3 b′)r̂μVμ= 0.

Thusa′+ 2 r−^3 b′=0 has to hold true. Due to

Δvμ=ΔaVμ+(b′′− 4 r−^1 b′)XμνVν,

cf. (10.13), the creeping flow equation (10.59) implies

a′′+ 2 r−^1 a′= 0 , −A=ηR−^1 (b′′− 4 r−^1 b′).

The boundary conditions impose

r→∞:r−^2 A→ 0 , a(∞)= 1 , r−^3 b→ 0 , (10.61)

and

a( 1 )= 0 , b( 1 )= 0. (10.62)

The solution of the differential equations (10.59), which obey the boundary

conditions, are determined by

A=

3

2

ηR−^1 , a= 1 −r−^1 , b=

1

4

( 1 −r^2 ). (10.63)

To verify that (10.60), with (10.63) are solutions of the creeping flow equation, make
use of (10.13). The presence of the fixed sphere in the streaming fluid has far reaching
consequences. There are parts of the distortion of the velocity field which decrease
with increasing distance from the sphere, both upstream and downstream, only like
r−^1. This is an effect typical for hydrodynamics.
The part of the velocity differenceVμ−vμproportional tor−^1 isr−^1 (δμν+
1
4 r

(^3) Xμν)Vν. The tensorr− (^1) (δμν+^1
4 r
(^3) Xμν)=^3
4 r
− (^1) (δμν+r̂μr̂ν)is proportional to
theOseen tensor
( 8 πηr)−^1 (δμν+r̂μ̂rν),
which determines the hydrodynamic interaction between colloidal particles [29–33].