ME 6601: Introduction to Fluid Mechanics
Module 6 Equations
Table Of Contents
Slide 2
 Diagram 1 from Slide 2 – Coordinate Definition
Slide 3
 Equation 1 from Slide 3 – Jacobian
 Equation 2 from Slide 3 – Jacobian
Slide 4
 Equation 1 from Slide 4 – Reynolds Transport Theorem
Slide 5
 Diagram 1 from Slide 5 – Reynolds Transport Theorem – proof
 Equation 1 from Slide 5 – Reynolds Transport Theorem – proof
 Modification to Equation 1 from Slide 5 – Reynolds Transport Theorem –
proof
Slide 6
 Equation 1 from Slide 6 – Reynolds Transport Theorem – proof (continued)
Slide 7
 Equation 1 from Slide 7 – Reynolds Transport Theorem – alternate form
 Equation 2 from Slide 7 – Reynolds Transport Theorem – alternate form
Slide 8
 Equation 1 from Slide 8 – Reynolds Transport Theorem – alternate form
(continued)
Slide 9
 Equation 1 from Slide 9 – Reynolds Transport Theorem – alternate form
(continued)
 Equation 2 from Slide 9 – Reynolds Transport Theorem – alternate form
(continued)
Slide 10
 Equation 1 from Slide 10 – Reynolds Transport Theorem – physical
interpretation
Diagram 1 from Slide 2 – Coordinate Definition:
This picture shows a three dimensional graph, with x, y, and z axes,
respectively. The origin is labeled zero where all three axes converge. A
vector labeled lowercase x subscript zero extends from the origin to a blue
oval representing a fluid particle. A second vector extends from the origin to
a second fluid particle, and this vector is labeled lowercase x. A curved
arrow points from the first particle to the second, labeled phi superscript
lowercase i.
Equation 1 from Slide 3 – Jacobian:
Capital J equals the partial derivative of the coordinate triple lowercase x
superscript one, lowercase x superscript two, lowercase x superscript three,
with respect to the coordinate triple lowercase x superscript one subscript
zero, lowercase x superscript two subscript zero, lowercase x superscript
three subscript zero, which equals the determinate of the partial derivative of
lowercase x subscript lowercase i with respect to lowercase x subscript
lowercase j subscript zero. Capital J, the Jacobian, is between zero and
infinity.
Equation 2 from Slide 3 – Jacobian:
Capital D capital J over capital D lowercase t equals capital J gradient dot
lowercase v.
Equation 1 from Slide 4 – Reynolds Transport Theorem:
Capital D over capital D lowercase t of the integral over capital V with a line
through it as a function of time lowercase t, of capital F as a function of
lowercase x and 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, open parentheses, capital D capital F over capital D
lowercase t plus F del dot lowercase v, closed parentheses, lowercase d,
capital V with a line through it.
Diagram 1 from Slide 5 – Reynolds Transport Theorem – proof:
This diagram shows a pair of ovals, which contain smaller ovals inside of
them. The first oval, labeled lowercase x subscript zero, contains a smaller
oval labeled capital V with a line through it, subscript zero. Underneath the
bigger oval it expresses that lowercase t equals zero. An arrow points from
the first big oval to the second. This arrow is labeled phi superscript
lowercase i. The second big oval is labeled lowercase x. It contains a smaller
oval, which is labeled capital V with a line through it, as a function of
lowercase t. Below the second big oval, there is a lowercase t.
Equation 1 from Slide 5 – Reynolds Transport Theorem – proof:
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 capital F lowercase d capital V
with a line through it, equals capital D over capital D lowercase t of the
integral, over capital V subscript zero, of capital F as function of capital J
lowercase d capital V with a line through it, subscript zero.
Modification to Equation 1 from Slide 5 – Reynolds Transport
Theorem – proof:
The last equation is also equal to the integral over capital V with a line
through it, subscript zero, of capital D over capital D lowercase t, of capital
F capital J, lowercase d capital V with a line through it, subscript zero.
Equation 1 from Slide 6 – Reynolds Transport Theorem – proof
(continued):
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 capital F, lowercase d, capital
V with a line through it, equals the integral over capital V with a line
through it, subscript zero, of, open parentheses, capital D capital F over
capital D lowercase t, times capital J, plus capital F times capital D capital J
over capital D lowercase t, close parentheses, lowercase d capital V with a
line through it, subscript zero, equals the integral over capital V with a line
through it subscript zero, of, open parentheses, capital D capital F over
capital D lowercase t, plus capital F del dot lowercase v, close parentheses,
capital J lowercase d capital V-not, equals the integral, over capital V with a
line through it as a function of lowercase t, of capital D capital F over capital
D lowercase t, plus capital F del dot lowercase v, lowercase d capital V with
a line through it, equal to the integral, over capital V with a line through it,
as a function of lowercase t, of, open parentheses, the partial derivative of
capital F with respect to t plus del dot lowercase v capital F, close
parentheses, lowercase d capital V with a line through it.
Equation 1 from Slide 7 – Reynolds Transport Theorem – alternate
form:
Capital D capital F over capital D lowercase t equals the partial derivative of
capital F with respect to lowercase t, plus lowercase bold v dot del capital F.
Equation 2 from Slide 7 – Reynolds Transport Theorem – alternate
form:
Capital D over capital D lowercase t of the integral, over capital V with a
line through it as a function of t, of capital F as a function of lowercase bold
x and 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 t, of the partial derivative of
capital F with respect to lowercase t, lowercase d capital V with a line
through it, plus the integral, over capital S as a function of lowercase t, of
capital F times lowercase v dot lowercase n lowercase d capital S.
Equation 1 from Slide 8 – Reynolds Transport Theorem – alternate
form (continued):
The integral, over capital V with a line through it as a function of lowercase
t, of del dot lowercase v capital F lowercase d capital V with a line through
it, equals the integral, over capital S as a function of lowercase t, of capital F
lowercase v dot lowercase n lowercase d capital S.:
The integral, over capital V with a line through it as a function of lowercase
t, of del dot lowercase v capital F lowercase d capital V with a line through
it, equals the integral, over capital S as a function of lowercase t, of capital F
lowercase v dot lowercase n lowercase d capital S.
Equation 1 from Slide 9 – Reynolds Transport Theorem – alternate
form (continued):
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 capital F as a function of
lowercase x subscript lowercase i and 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 t, of the partial derivative of capital F with respect to
lowercase t, lowercase d capital V with a line through it, plus the integral,
over capital S as a function of t, of capital F lowercase v subscript j,
lowercase n subscript lowercase j, lowercase d capital S.
Equation 2 from Slide 9 – Reynolds Transport Theorem – alternate
form (continued):
Capital D over capital D lowercase t of the integral, over capital V with a
line through it as a function of t, of capital F as a function of lowercase x and
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 t, of, open parentheses,
the partial derivative of capital F with respect to t plus v dot del capital F
plus F del dot lowercase v, close parentheses, lowercase d capital V with a
line through it.
Equation 1 from Slide 10 – Reynolds Transport Theorem – physical
interpretation:
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 capital F as a function of
lowercase x and 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 the partial derivative of capital F with respect to lowercase t,
lowercase d capital V with a line through it, plus the integral over capital S
as a function of t, of capital F, lowercase bold v dot lowercase n lowercase d
capital S.