8.4 Further Applications of Volume Integrals 143
Fig. 8.17The tangential
force density due to
non-diagonal elements of the
pressure tensor. The force
exerted by the fluid on the
solid is in the directions
shown whenpyx<0and
pxy< 0
When thex–z-plane of a coordinate system is on the wall with they-axis antiparallel
ton, the normal component of the force isFy=−pyyA. The tangential components
areFx=−pyxAandFz=−pyzA.
For a cube placed in the fluid, with its sides parallel to the coordinate axes, the
tangential part of the vectornνpνμoccurring in (8.92), has directions indicated in
Fig.8.17,forthex–y-plane. Notice, whenpyx=pxy, i.e. when the pressure tensor
has a non-zero antisymmetric part, the cube experiences a torque, caused by the fluid.
By analogy to (8.91), the total force of the fluid, exerted on a stiff solid body is
Fμs=−∮
∂Vnνpνμd^2 s. (8.94)Here∂Vis the closed surface of the solid body,nis its outer normal.
A remark on Fig.8.17is in order. The force exerted by the fluid on the solid cube,
evaluated with (8.94), has tangential components in the directions indicated by the
arrows, provided thatpyxandpxyare negative. This happens, indeed, for a plane
Couette flow with the geometry chosen as in Fig.7.6.
8.4.3 The Archimedes Principle
The principle of Archimedes states: an impenetrable solid body immersed in a liquid
experiences a lift force, against the direction of gravity. The magnitude of this buoy-
ancy force is equal to the weight of the liquid in a volume, which is as large as that
one occupied by the solid. Why is that so? Why does it apply to solid bodies of any
shape, as long as the liquid does not penetrate into the solid?
Consider the local momentum conservation equation (8.88). In the presence of an
external force, an extra force density has to be taken into account on the right hand
side of the balance equation. In the case of the gravity on earth, this force density