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In this module, we begin the process of deriving a local, spatial form of the equations of motion expressing the balance of linear momentum (Newton's 2nd Law of motion).
We shall ultimately arrrive at the Navier-Stokes equations governing the motion for the incompressible flow of a linearly viscous (Newtonian) fluid.
Newton's Second Law of Motion, applied to a piece of the continuum, states, in words,
The rate of increase of linear momentum within a material volume capital V equals the Resultant external force acting on capital V.
We need to examine both sides of this equation; we'll begin with the forces.
There are two types of forces with which we must deal:
Body forces - Long-range forces which permeate the matter, e.g., gravity & electromagnetic forces. Body forces depend upon the mass of the body.
Surface forces - Forces exerted by material on one side of a surface element on material on the other side. The length scale over which these act is the intermolecular distance.
This notion of surface forces stems from the Cauchy Stress Principle
The two types of forces shall be designated as follows:
Body Forces - f is the body force per unit mass.
Surface Forces - t is the surface force per unit area.
(Note: When a capital V is written in this section, it is understood to be a capital V with a line through it)
We now create a mathematical statement of Newton's 2nd Law as applied to an arbitrary material volume V(t) bounded by surface S(t):
(In red) Capital D over capital D lowercase t of the integral over capital V with a line through it as a function of t of lowercase rho times lowercase bold lowercase v lowercase d capital V equals (In dark purple) the integral over capital V with a line through it as function of lowercase t of lowercase rho times bold lowercase f lowercase d capital V, plus (In light purple)the integral over capital S as a function of lowercase t of bold lowercase t lowercase d capital S.
(In red) Time rate of change of linear momentum in capital V with a line through it as a function of lowercase t.
(In dark purple) Total body force acting on the fluid material in capital V with a line through it as a function of lowercase t.
(In light purple) Total surface force acting on the fluid material in capital V with a line through it as a function of lowercase t.
(Note: When a capital V is written in this section, it is understood to be a capital V with a line through it)
Using the corollary to the RTT, the frist term in the equation may be rewritten to yield
The integral over capital V with a line through it as a function of lowercase t of lowercase rho times (capital D bold lowercase v over capital D lowercase t) lowercase d capital V equals the integral over capital V with a line through it as a function of lowercase of lowercase rho times lowercase bold f lowercase d capital V with a line through it plus the integral over capital S as a function of lowercase t of lowercase bold t lowercase d capital S.
However, we do not yet have a "working" form.
We wish to have a local form of our equation, i.e., one valid at each point in the fluid. In pursuit of this goal, we prove the following:
Theorem (local equilibrium) - The stress forces are in local equilibrium, i.e., they balance themselves at each point.
Proof: Begin by recognizing that
capital V with a line through it as a function of lowercase t equals capital O as a function of lowercase l superscript 3
Capital S as a function of lowercase t equals capital O as a function of lowercase l superscript 2
where lowecase l is a length scale representative of capital V with a line through it as a function of lowercase t.
Our equation of motion,
The integral over capital V with a line through it as a function of lowercase t, of lowercase rho times (capital D lowercase bold v divided by capital D lowercase t) lowercase d capital V with a line through it equals the integral over capital V with a line through it as a function of lowercase t of lowercase rho times lowercase bold f lowercase d capital V with a line through it, plus the integral over capital S as a function of lowercase t of lowercase bold t lowercase d capital S,
may be written as
(Lowercase 1 divided by lowercase l superscript 2) times the integral over capital V with a line through it of lowercase rho times ( ( Capital D lowercase bold v over capital D lowercase t) minus lowercase bold f) lowercase d capital V with a line through it equals (lowercase 1 divided by lowercase l superscript 2) times the integral over capital S as a function of t of lowercase bold t lowercase d capital S.
Taking the limit as lowercase l approaches 0,
The limit of lowercase l as it approaches 0 of (lowercase 1 divided by lowercase l superscript 2) times the integral over capital S as a function of t of lowercase bold t lowercase d capital S equals 0.
This is local equilibrium.
We now apply the principle of local equilibrium to the volume of fluid contained in the tetrahedron shown below.
[D1]
The area and unit outward normal vector have been indicated for each face; Sigma is the area of the slanted face.
Applying local equilibrium,
The limit as sigma approaches zero of one over sigma, open curly brace, sigma, open bracket, lowercase bold t as a function of lowercase bold x and t, time, as well as lowercase bold n, plus lowercase n subscript one times lowercase bold t as a function of lowercase bold x and t, time, for negative lowercase i, plus lowercase n subscript two times lowercase bold t as a function of lowercase bold x and time, lowercase t, for negative j, plus lowercase n subscript three times lowercase bold t as a function of lowercase bold x and time, lowercase t, for negative lowercase k, close bracket, close curly brace, equals zero.
(The sigmas cancel out)
where the stress on each face has been approximated by the value at the point bold lowercase x, the origin of the indicated coordinate system (the value approached in the limit).
Taking the limit, we see, locally, that
Lowercase bold t as a function of lowercase bold x, and t, for lowercase bold n, plus lowercase n subscript one times lowercase bold t as a function of lowercase bold x, and lowercase t, for lowercase bold negative i, plus lowercase n subscript 2 times lowercase bold t as a function of lowercase bold x, and lowercase t, for lowercase bold negative j, plus lowercase n subscript 3 times lowercase bold t as a function of lowercase bold x, and lowercase t, for lowercase bold negative k, equals zero.
Now the stress vector lowercase bold t must be an odd function of the outward normal vector lowercase bold n, therefore,
Lowercase bold t as a function of lowercase x and lowercase t, for lowercase bold n, equals lowercase n subscript one times lowercase bold t as a function of lowercase bold x and lowercase t, semi-colon lowercase i, plus lowercase n subscript two times lowercase bold t as a function of lowercase bold x and lowercase t, semi-colon lowercase j, plus lowercase n subscript three times lowercase bold t as a function of lowercase bold x and lowercase t, semi-colon lowercase k.
That is, in tensor notation,
Lowercase t subscript lowercase i equals capital T subscript lowercase i lowercase j times lowercase n subscript lowercase j.
where capital T subscript lowercase i lowercase j equals capital T subscript lowercase i lowercase j as a function of lowercase bold x, and lowercase t.
The quantity
Capital T subscript lowercase i lowercase j equals capital T subscript lowercase i lowercase j as a function of lowercase bold x, and lowercase t.
is known as the stress tensor; it represents the i-component of force per unit area exerted across a surface element normal to the j-direction at location lowercase bold x at time lowercase t.
In this module we begin the process of deriving the local, spatial equations of motion (Newton's Second Law).
We discuss the nature of surface and body forces and the principle of local equilibrium applying to the former.
Application of the principle of local equilibrium to a tetrahedron leads us to the existence of the stress tensor.
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