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Volume and Surface Forces
Generally speaking, fluids are acted upon by two distinct types of force. The first type is long range in naturethat is, such that it decreases relatively slowly with increasing distance between interacting elementsand is
capable of completely penetrating into the interior of a fluid. Gravity
is an obvious example of a longrange force. One consequence of the relatively slow variation of
longrange forces with position is that they
act equally on all of the fluid contained within a sufficiently small volume element. In this situation, the net force
acting on the element becomes directly proportional to its volume. For this reason, longrange forces are often called volume forces. In the following, we shall write the total volume force acting at time
on the fluid contained within a
small volume element of magnitude
, centered on a fixed point whose position vector is
,
as

(1.1) 
The second type of force is short range in nature, and
is most conveniently modeled as momentum transport within the fluid. Such transport is generally due to a combination of
the mutual forces exerted by contiguous molecules, and momentum fluxes caused by relative molecular motion.
Suppose that
is the net flux density of
directed fluid momentum due to
shortrange forces
at position
and time
. In other words, suppose that, at position
and
time
, as a direct consequence of shortrange forces,
momentum is flowing at the rate of
newtonseconds per meter squared per second in the direction of vector
. Consider an infinitesimal plane surface element,
, located at point
. Here,
is the
area of the element, and
its unit normal. (See Section A.7.) The fluid which lies
on that side of the element toward which
points is said to lie on its positive side, and vice versa.
The net flux of
momentum across the element (in the direction of
) is
newtons, which implies (from Newton's second law of motion) that the fluid on the positive side of the surface element
experiences a force
in the
direction due to shortrange
interaction with the fluid on the negative side.
According to Newton's third law of motion, the fluid on the negative side of the surface experiences a force
in the
direction due to interaction with the fluid on the positive side.
Shortrange forces
are often called surface forces, because they are directly proportional to the area of the surface element across which
they act.
Let
and
be the net flux density of
 and
momentum, respectively, at position
and
time
. By a straightforward extension of previous argument, the net surface force exerted by the fluid on the positive side of some
planar surface element,
, on the fluid
on its negative side is
In tensor notation (see Appendix B), the previous equation can be written

(1.3) 
where
,
,
, et cetera.
(Note that, because the subscript
is repeated in the previous equation, it is assumed to be summed from
to
. Hence,
is shorthand for
.)
Here, the
are termed the local stresses in the fluid at position
and
time
, and have units of force per unit area. Moreover, the
are
the components of a secondorder tensor (see Appendix B) known as the stress tensor. [This follows because the
are the components of a firstorder tensor (as all forces are proper vectors), and the
are the components of an arbitrary firstorder tensor (as surface elements are
also proper vectorssee Section A.7and Equation (1.3) holds for
surface elements whose normals point in any direction), so application of the quotient rule (see Section B.3) to Equation (1.3) reveals that the
transform under rotation of the coordinate axes as the components of a secondorder tensor.] We can interpret
as the
component of the force per unit area exerted, at position
and time
, across a plane surface element normal to the
direction. The three diagonal components of
are
termed normal stresses, because each of them gives the normal component of the force per unit area acting
across a plane surface element parallel to one of the Cartesian coordinate planes. The six nondiagonal components
are termed shear stresses, because they drive shearing motion in which parallel
layers of fluid slide relative to one another.
Next: General Properties of Stress
Up: Mathematical Models of Fluid
Previous: What is a Fluid?
Richard Fitzpatrick
20160122