Tensors for Physics

Appendix: Exercises... 405

polar vector is the ansatzvκ=c(r)εκλνΩλXν, with a coefficientc. For the present
problem, the creeping flow equation reduces toΔvκ =0. Thusc =const. is a
solution, which has still to be specified by the boundary conditions.
The fluid is assumed to be at rest, far away from the sphere. This is obeyed by
the ansatz withc=const. A no-slip boundary condition at the surface of the sphere
meansvκ=RεκλνΩλrν,atr=1. Due toXν=r−^3 rν, the solution for the flow
velocity is

vκ=RεκλνΩλXν. (A.12)

According to (8.98), for the present problem, the torque is given by

Tμ=−εμαβR^2

∮

rα̂rτpτβd^2 ̂r=ηεμαβR^3

∮

r̂αr̂τ(∇τvβ+∇βvτ)d^2 ̂r.

The first term in the integrand involves

r̂τ∇τvβ=R−^1

∂

∂r

vβ=− 2 R−^1 vβ.

The expressionr̂τ∇βvτ, occurring in the second term of the integrand, is equal to

−r̂τετλνΩλXνβ=r−^3 r̂τετλβΩλ=−R−^1 vβ.

Thus one obtains

Tμ=− 3 ηεμαβεβλτΩλR^3

∮

r̂α̂rτd^2 ̂r.

The surface integral

∮

r̂αr̂τd^2 ̂r=( 4 π/ 3 )δατleads to

Tμ=− 4 πηR^3 εμαβεβλαΩλ=− 8 πηR^3 Ωμ,

which is the expression for the friction torque mentioned above. The minus sign and
η>0 imply that the rotational motion is damped. Time reversal invariance is broken,
as typical for irreversible processes.

Exercises Chapter 11

11.1 Contraction Rules for Delta-Tensors(p.185)
Verify(11.6)for=2.
The contraction rule for the-thΔ-tensor is