Tensors for Physics

142 8 Integration of Fields

8.4.2 Momentum Balance, Force on a Solid Body

The local conservation equation for the linear momentum densityρvμ, in the absence
of external forces, cf. (7.52), reads

ρ

dvμ

dt

+∇νpνμ= 0. (8.88)

The pressure tensorpνμcharacterizes the part of the momentum transport, which is
not of convective type. The gradient∇νpνμ=kμis an internal force density.
The total momentum associated with a fluid in a volumeVis

PμV≡

∫

V

ρvμd^3 r. (8.89)

Integration of the local conservation equation (8.88) over this volume and application
of the Gauss theorem leads to

d

dt

PμV=−

∫

∂V

nflνpνμd^2 s, (8.90)

wherenflis the outer normal of the volume containing the fluid. The term on the
right hand side of this equation is the forceFflacting on the fluid. Due toactio equal
reactio, the forceFsexerted by the fluid on a solid wall or on a solid body immersed
in the fluid has the same magnitude, but with opposite sign:Fs=−Ffl. When the
normal vectornflis replaced byns=−nfl, which points from the solid into the fluid,
the expression forFshas the same form as that one in (8.90),

Fμs=−

∫

∂V

nνpνμd^2 s. (8.91)

Consider now a plane wall with the surface areaAand the normal vectorn≡ns.
Assuming that the pressure tensor of the fluid is constant at the wall, the forceFw
acting on the plane wall is
Fμw=−nνpνμA. (8.92)

In thermal equilibrium, the pressure tensor of a fluid is just the isotropic tensorPδμν,
with the hydrostatic pressureP,cf.(7.53). Then (8.92) reduces toF=−nPA.The
minus sign means that the fluid pushes against the wall, provided thatP>0, as
valid in thermal equilibrium. In equilibrium, there is only a normal force, i.e. a force
perpendicular to the wall.
In general, the force (8.92) has both normal and tangential components

Fμnorm=−nμ(nνpνλnλ)A, Fμtang=−nν(pνμ−δμνnκpκλnλ)A. (8.93)