Tensors for Physics

178 10 Multipole Potentials

In thecreeping flow approximation, applicable for slow velocities, the nonlinear

terms involvingvν∇νvμare disregarded in (10.56) and (10.57). Applications thereof

are the calculation of the Stokes force acting on a solid body moving in a viscous fluid

or the torque acting on a rotating body. The special case of a sphere is considered

next.

10.5.2 Stokes Force on a Sphere

The friction forceFacting on sphere with radius R, which undergoes a slow

translational motion, with velocityVin a viscous fluid, is

Fμ=− 6 πηRVμ. (10.58)

The expression for the force, first derived by Stokes and referred to asStokes force,

is remarkable since is not proportional to the cross sectionπR^2 of the sphere, but

rather linear inR. This is an effect typical for hydrodynamics. The minus sign in

(10.58) means that the force is damping the motion.

The derivation of the Stokes force starts from the creeping motion version of

(10.56). For convenience, the dimensionless position vectorr∗ =R−^1 ris used.

When no danger of confusion exists,r∗is written asrand it is understood that

∇stands for the derivative with respect to the dimensionless position vector. The

differential equations then read

∇μp=ηR−^1 Δvμ,Δp= 0. (10.59)

The equation is linear in the velocity. The situation of a sphere moving with the
velocity−Vand the fluid at rest, far away from the sphere, is equivalent to the
sphere at rest, with its center corresponding tor=0, and the fluid flowing with
the constant velocityV, far away, i.e. forr→∞. Furthermorep→const. for
r→∞should hold true. In the followingpis associated with the deviation of the
pressure from its constant value atr→∞. In addition to these boundary conditions,
the behavior of the velocity at the surface of the sphere has to be prescribed for the
solution of (10.59). The Stokes force (10.58) pertains to thestickorno slipboundary
conditionvμ=0 at the surface of the sphere, corresponding tor=1. Of course,
for the sphere to stay fixed in the streaming fluid, it has to be held by a force which
has to balance the Stokes force.
Solutions of (10.59) must be linear inV. Symmetry and parity considerations
suggest the ansatz

p=A(r)XνVν,vμ=a(r)Vμ+b(r)XμνVν, (10.60)

where theX···are multipole potential functions and the coefficientsA,a,bhave yet

to be determined. The equationΔp=0 impliesA=const. The spatial derivatives