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The Stress tensor
Introduction
In his
Principia
, Newton defined the concept of a force in terms of the laws of motion. These laws in effect define
the protocol for measuring the force exerted on a particle ( defined as a discrete object) in terms of the rate of
change of momentum of that particle. The extension of these ideas to a continuum is attributed to Euler (1775)
Euler's First Law can be stated as
"The time rate of change of linear momentum of a body relative to the fixed stars (i.e., an inertial frame of refer
ence) is equal to the sum of forces acting on the body."
There are essentially two types of forces that can act on a fluid or continuum: body forces and surface forces.
Body Forces
Body forces are distributed throughout the continuum and are proportional to the mass. They arise as a conse
quence of the continuum being placed in a force field (gravitational, magnetic, electrostatic, or more generally
electromagnetic). We will denote these forces with the symbol
b,
which characterizes the vector field.
Body forces can be conservative or nonconservative. A conservative body force can be expressed as a gradient of
a scalar potential, i.e.,
(1)
b
= “f
where
f
denotes the scalar potential. Forces that are directed centrally from a source are conservative; examples
are gravity, electrostatic, and magnetic. The case of gravity (the body force we will primarily be concerned with in
these notes) we can write
(2)
g
= “f
where
f =
g
ÿ
x
, and
»
g
»
ª
G
is the gravitational constant.
Surface Forces
Surface forces are shortrange forces, molecular in origin, and depend on the interactions of molecules and/or
atoms in the body . In a fluid body each molecule interacts with every other molecule of the fluid, but because this
interaction (e.g. van der Waals) is short range (the penetration depth of the forces, typically no more than tens of
nanometers), molecules only interact strongly with their nearest neighbors. Shortrange forces thus decrease
rapidly with increase of distance between interacting molecules, and are significant only when that distance is of
the order of the molecular separation.
Consider now a fluid element that is acted on by shortrange forces arising from interactions with another element
(either a solid or fluid). Since the shortrange forces can act only on a thin layer adjacent to the boundary of the
fluid element, the net force acting on the element due to the shortrange forces is thus determined by the surface
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 Spring '10
 Franics

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