ME 6601: Introduction to Fluid Mechanics
Module 5 Equations
Table of Contents
Slide 1
 Equation 1 from Slide 1 – Kinematics IV – Fluid-material lines
Slide 2
 Equation 1 from Slide 2 – Kinematics IV – Fluid-material lines
 Equation 2 from Slide 2 – Kinematics IV – Fluid-material lines
Slide 4
 Equation 1 from Slide 4 – Streaklines – Solving a Streakline Problem
 Equation 2 from Slide 4 – Streaklines – Solving a Streakline Problem
Slide 5
 Equation 1 from Slide 5 – Streaklines – Solving a Streakline Problem
(continued)
Slide 6
 Diagram 1 from Slide 6 – Streaklines example
Slide 7
 Equation 1 from Slide 7 – Streaklines example (continued)
 Equation 2 from Slide 7 – Streaklines example (continued)
Slide 8
 Equation 1 from Slide 8 – Streaklines example (continued)
 Equation 2 from Slide 8 – Streaklines example (continued)
 Equation modification to Equation 2 from Slide 8 – Streaklines example
(continued)
Slide 9
 Equation 1 from Slide 9 – Streaklines example (continued)
 Equation 2 from Slide 9 – Streaklines example (continued)
Slide 10
 Equation 1 from Slide 10 – Streaklines example (continued)
 Modification to Equation 2 from Slide 11 – Streaklines example (continued)
Slide 11
 Graph 1 from Slide 11 – Streaklines example (continued)
 Graph 2 from Slide 11 – Streaklines example (continued)
Equation 1 from Slide 1 – Kinematics IV – Fluid-material lines:
Capital Y subscript zero equals lowercase f subscript capital Y, as a function
of capital X subscript zero.
Capital Z subscript zero equals lowercase f subscript capital Z, as a function
of capital X subscript zero.
Equation 1 from Slide 2 – Kinematics IV – Fluid-material lines:
Capital bold X subscript capital F equals capital bold faced X as a function
of capital X subscript zero, lowercase f subscript capital Y as a function of
X-not, lowercase f subscript capital Z as a function of X-not, and time.
Equation 2 from Slide 2 – Kinematics IV – Fluid-material lines:
Capital Y subscript capital F equals capital Y subscript capital F as a
function of capital X subscript F and time lowercase t.
Capital Z subscript capital F equals capital Z subscript capital F as a function
of capital X subscript capital F and time lowercase t.
Equation 1 from Slide 4 – Streaklines – Solving a Streakline Problem:
Capital bold-faced X as a function of capital X subscript zero and lowercase
t subscript 1 equals lowercase x superscript star.
Equation 2 from Slide 4 – Streaklines – Solving a Streakline Problem:
Lowercase x superscript star subscript zero equals lowercase x superscript
star subscript zero as a function of lowercase x superscript star and time
lowercase t.
Equation 1 from Slide 5 – Streaklines – Solving a Streakline Problem
(continued):
Capital bold-faced X subscript capital F equals capital bold X as a function
of capital X subscript zero superscript star and time.
Diagram 1 from Slide 6 – Streaklines example:
This diagram displays an example of a plane stagnation-point flow. It
resembles the top half of a two-dimensional graph, with a point along the
positive x-axis labeled x subscript 1, and a point along the negative x-axis
labeled negative x subscript one, with both points equidistant from the y
axis. The first quadrant contains several curves that are bend in towards the
origin. The farther away the line is from the origin, the less of a curve it has.
Each curve has an arrow on it that points down along the curve. The second
quadrant is a mirror image of the first, with the exception of a small dot that
is located directly above the point negative x subscript 1, between the second
and third curves. From this point, a line extends directly to the right, labeled
U, directly above the point x subscript one. An arrow along this line points
to the positive side of the graph.
Equation 1 from Slide 7 – Streaklines example (continued):
Lowercase x superscript star equals negative lowercase x subscript one plus
capital U times lowercase t.
Lowercase y superscript star equals y subscript one.
Equation 2 from Slide 7 – Streaklines example (continued):
Capital X equals capital X subscript zero times lowercase e to the power of
capital K times lowercase t.
Capital Y equals capital Y subscript 0 times lowercase e to the power of
negative capital K times lowercase t.
Equation 1 from Slide 8 – Streaklines example (continued):
Negative lowercase x subscript one plus capital U times t subscript one
equals capital X subscript zero times lowercase e to the power of capital K
times lowercase t subscript one.
Capital Y subscript one equals capital Y subscript zero times lowercase e to
the power of negative capital K times lowercase t subscript one.
Equation 2 from Slide 8 – Streaklines example (continued):
Capital X subscript zero equals lowercase y subscript one over capital Y
subscript zero times, open parenthesis, negative lowercase x subscript one
plus capital U divided by capital K times the natural log of capital Y
subscript zero divided by lowercase y subscript one, end parenthesis.
Equation modification to Equation 2 from Slide 8 – Streaklines
example (continued):
Substitute capital X subscript capital S, to denote a streakline, for the lefthand value of capital X subscript zero, and multiply the entire right-hand
side of the equation by lowercase e to the power of (capital K lowercase t).
Equation 1 from Slide 9 – Streaklines example (continued):
Capital Y subscript capital S equals capital Y subscript zero times lowercase
e to the power of (negative capital K lowercase t).
Equation 2 from Slide 9 – Streaklines example (continued):
Capital X subscript capital S equals lowercase y subscript one over capital Y
subscript capital S, times, open parentheses, negative lowercase x subscript
one plus capital U over capital K times the natural log of capital Y subscript
capital S times lowercase e to the power of capital K times t, both over
lowercase y subscript one.
Equation 1 from Slide 10 – Streaklines example (continued):
Lowercase x subscript one equals negative lowercase x subscript one plus
capital U times lowercase t.
Modification to Equation 2 from Slide 11 – Streaklines example
(continued):
Instead of lowercase e being to the power of (capital K times lowercase t),
lowercase e is now to the power of (two times capital K times lowercase x
subscript one divided by capital U).
Graph 1 from Slide 11 – Streaklines example (continued):
This graph is labeled Streakline, open parentheses, lowercase x subscript one
equals three, lowercase y subscript one equals one, capital K equals one,
capital U equals four.
This graph is a two-dimensional graph that displays positive and negative x
coordinates and only positive y coordinates. The x coordinates go from
negative four to positive four, in increments of one, and the y coordinates go
from zero to one point two in increments of point two. The representation of
the streakline is a red curve that starts out at approximately negative four,
zero point four, and resembles a line with a slight curve upward until the
point two, zero point six, when the graph starts curving upward
exponentially.
Graph 2 from Slide 11 – Streaklines example (continued):
This two dimensional graph is labeled Streakline, open parentheses,
lowercase x subscript one equals three, lowercase y subscript one equals
one, capital K equals one, capital U equals one point five.
This graph displays negative and positive x coordinates, ranging from
negative five to five in increment of one. It displays only positive y
coordinates, from zero to one point two, in increments of point two. The
representation of the streakline starts out at approximately the point negative
five, zero point one. The curve resembles a line with a slight curve upwards
until it hits the point four, zero point three. The streakline then makes a
sharp curve and starts heading in the negative direction, resembling a line
with a slope of approximately negative one.