ME 6601: Introduction to Fluid Mechanics
Module 8 Equations
Table of Contents
Slide 5
 Equation 1 from Slide 5 – Balance of Linear Momentum – mathematical
statement
Slide 6
 Equation 1 from Slide 6 – Balance of Linear Momentum – mathematical
statement (continued)
Slide 7
 Equation 1 from Slide 7 – Local Equilibrium
Slide 8
 Equation 1 from Slide 8 – Local Equilibrium – proof (continued)
 Equation 2 from Slide 8 - Local Equilibrium – proof (continued)
Slide 9
 Diagram 1 from Slide 9 - Stress Tensor
Slide 10
 Equation 1 from Slide 10 - Stress Tensor (continued)
 Modification to Equation 1 from Slide 10 - Stress Tensor (continued)
 Equation 2 from Slide 10 - Stress Tensor (continued)
Slide 11
 Equation 1 from Slide 11 - Stress Tensor (continued)
Slide 12
 Equation 1 from Slide 12 - Stress Tensor (continued)
Equation 1 from Slide 5 – Balance of Linear Momentum –
mathematical statement:
Capital D over capital D lowercase t of the integral, over capital V with a
line through it as a function of lowercase t, of lowercase rho times lowercase
v 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 f lowercase d capital V with a line through it, plus the
integral, over capital S as a function of lowercase t, of lowercase t lowercase
d capital S.
Equation 1 from Slide 6 – Balance of Linear Momentum –
mathematical statement (continued):
The integral, over capital V with a line through it as a function of lowercase
t, of lowercase rho, capital D lowercase v over 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 f lowercase d capital V with a line through it, plus the integral,
over capital S as a function of lowercase t, of lowercase t lowercase d capital
S.
Equation 1 from Slide 7 – Local Equilibrium:
Capital V with a line through it as a function of lowercase t equals capital O
as a function of lowercase l superscript three.
Capital S as a function of t equals capital O as a function of lowercase l
superscript two.
Equation 1 from Slide 8 – Local Equilibrium – proof (continued):
One over lowercase l squared times the integral, over capital V with a line
through it as a function of lowercase t, of rho, times, open parentheses,
capital D lowercase v over capital D lowercase t, minus lowercase f, closed
parentheses, lowercase d capital V with a line through it, equals one over
lowercase l squared times the integral, over capital S as a function of
lowercase t, of lowercase bold t lowercase d capital S.
Equation 2 from Slide 8 - Local Equilibrium – proof (continued):
The limit as lowercase l approaches zero of one over lowercase l squared
times the integral, over capital S as function of lowercase t, of lowercase
bold t lowercase d capital S equals zero.
Diagram 1 from Slide 9 - Stress Tensor:
This diagram shows a three dimensional graph, with three axes. The x axis is
labeled x subscript one, the y axis is labeled x subscript two, and the z axis is
labeled x subscript three. A green dot that exists in the area bounded by the x
subscript one axis, x subscript two axis, and a line that connects the two axes
is labeled lowercase n subscript three, sigma. A green arrow pointing from a
location next to the green dot in the same region points to negative k, which
is also in green. In the location bounded by the axes x subscript one and x
subscript three, along with the line that connects the two axes, there is a
magenta arrow that points to a magenta negative lowercase j. Next to this
arrow, also in magenta, is another dot. This dot is labeled lowercase n
subscript two, sigma. In a third region that is bounded by x subscript two, x
subscript three, and the line that connects the two axis is a blue arrow that
points from this region to negative lowercase i, also in blue. A blue dot in
the same region is labeled by a lowercase n subscript one, sigma. In the
fourth region, the one bounded by the three lines that connect the axes, an
arrow pointing from a location on this region points to an equation,
lowercase n equals lowercase bold n equals lowercase n subscript one times
lowercase bold i, plus lowercase n subscript two times lowercase bold j, plus
lowercase n subscript three times lowercase bold k. This equation and the
arrow pointing to it are labeled in red. In this same region, there is a red dot
that is labeled by the term ‘area sigma’, also in red.
Equation 1 from Slide 10 - Stress Tensor (continued):
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
lowercase t, time, as well as lowercase n, plus lowercase n subscript one
times lowercase bold t as a function of lowercase bold x and lowercase 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 lowercase 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.
Modification to Equation 1 from Slide 10 - Stress Tensor (continued):
Eliminate all of the sigmas from the equation.
Equation 2 from Slide 10 - Stress Tensor (continued):
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, for lowercase i, plus lowercase n
subscript two times lowercase bold t as a function of lowercase bold x and
lowercase t, for lowercase j, plus lowercase n subscript three times
lowercase bold t as a function of lowercase bold x and lowercase t, for
lowercase k.
Equation 1 from Slide 11 - Stress Tensor (continued):
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 time, lowercase t.
Equation 1 from Slide 12 - Stress Tensor (continued):
Capital T subscript lowercase i lowercase j equals capital T subscript
lowercase i subscript lowercase j, as a function of lowercase bold x and
lowercase t.