Tensors for Physics

10.4 Further Applications in Electrodynamics 177

The interaction between two magnetic dipoles has the same functional form, the
factor 4 πε^10 , however is replaced by 4 μπ^0. The dipole–quadrupole and the quadrupole–
quadrupole interactions involve the third and the fourth multipole potentials, which
are proportional tor−^4 andr−^5 , respectively,

Wdip−quad=−

1

4 πε 0

1

6

p(μ^1 )XμνλQ(νλ^2 )=−

1

4 πε 0

5

2

r−^7 pμ(^1 )rμrνrλ Q(νλ^2 ).

(10.54)

Wquad−quad=

1

4 πε 0

1

36

Q(μν^1 )XμνλκQ(λκ^2 )=

1

4 πε 0

35

12

r−^9 Q(μν^1 )rμrνrλrκQ(λκ^2 ).

(10.55)

These multipole-multipole interactions are of importance for the anisotropy of the
long range interaction between two molecules, labelled by 1 and 2. The multipole
moments depend on the orientation of the molecules. For electrically neutral linear
molecules, without permanent dipole moments, the quadrupole-quadrupole interac-
tion is the lowest order contribution. The induced dipole-dipole interaction, referred
to asvan der Waals interaction, as well as the repulsive interaction, associated with
the shape of the molecules, in general, possess anisotropic parts. The van der Waals

interaction is proportional toα(μν^1 )Xνλα(λκ^2 )Xκμ, whereαμν(^1 )andα(λκ^2 )are the molecular
polarizability tensors of particles 1 and 2.

10.5 Applications in Hydrodynamics

10.5.1 Stationary and Creeping Flow Equations

In a stationary situation, the local momentum conservation equation (7.52), for an
isotropic fluid with the spatially constant shear viscosityη,cf.(7.55), is

ρvν∇νvμ+∇μp=ηΔvμ. (10.56)

Hereρis the constant mass density,pis the hydrostatic pressure which may depend
on the positionr. The right hand side of (10.56) is obtained from (7.52) with the
Newton ansatz (7.55) for the friction pressure tensor

pνμ=− 2 η∇νvμ =−η(∇νvμ+∇νvμ), ∇λvλ= 0.

Due the incompressibility condition∇μvμ=0, application of∇μto (10.56) implies

ρ∇μvν∇νvμ+Δp= 0. (10.57)

The solutions of these equations, for specific hydrodynamic applications, have to
obey the relevantboundary conditions.