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Louisiana Tech UniversityRuston, LA 71272
Vectors
1. What is the projection of the vector (1, 3, 2) onto the plane described by ?4 2 3 1x y z
n v
Louisiana Tech UniversityRuston, LA 71272
Cross Product
1 2 3
1 2 3
a a a
b b b
i j k
a b
Used for:
Moments
Vorticity
In Cartesian Coordiantes:
Louisiana Tech UniversityRuston, LA 71272
Forces
• Body Forces
• Pressure
• Normal Stresses
• Shear Stresses
Louisiana Tech UniversityRuston, LA 71272
What is a Fluid?
Solid: Stress is proportional to strain (like a spring).
Fluid: Stress is proportional to strain rate.
Louisiana Tech UniversityRuston, LA 71272
The Stress Tensor (Fluids)
For fluids:
11 12 13
21 22 23
31 32 33
31 1 2 1
1 2 1 3 1
32 1 2 2
1 2 2 3 2
3 31 2
1 3 2 3
1 0 0
0 1 0
0 0 1
1 1
2 2
1 12
2 2
1 1
2 2
P
uu u u u
x x x x x
uu u u u
x x x x x
u uu u
x x x x
3
3
u
x
v
y
One dimensional
Three-Dimensional, where u is velocity
Louisiana Tech UniversityRuston, LA 71272
The Stress Tensor (Solids)
For solids:
11 12 13
31 221 22 23
1 2 331 32 33
31 1 2 1
1 2 1 3 1
32 1 2 2
1 2 2 3 2
3 1
1 3
1 0 0
0 1 0
0 0 1
1 1
2 2
1 12
2 2
1
2
uu u
x x x
uu u u u
x x x x x
uu u u uG
x x x x x
u u
x x
3 32
2 3 3
1
2
u uu
x x x
E One dimensional
Three-Dimensional, where u is displacement
Louisiana Tech UniversityRuston, LA 71272
Non-Newtonian Fluid
For fluids:
11 12 13
21 22 23
31 32 33
31 1 2 1
1 2 1 3 1
32 1 2 2
1 2 2 3 2
3 31 2
1 3 2 3
1 0 0
0 1 0
0 0 1
1 1
2 2
1 12
2 2
1 1
2 2
P
uu u u u
x x x x x
uu u u u
x x x x x
u uu u
x x x x
3
3
u
x
v
y
(One dimensional)
is a function of strain rate
Louisiana Tech UniversityRuston, LA 71272
Apparent Viscosity of Blood
meff
3.5 g/(cm-s)
1 dyne/cm2
Non-Newtonian Region
Rouleau Formation
Louisiana Tech UniversityRuston, LA 71272
The Sturm Liouville Problem
Be able to reduce the Sturm Liouville problem to special cases.
bxaxxwxqdx
xdxp
dx
d
,0,,
Or equivalently:
2
2
, ,, 0,
d x d xp x p x q x w x x
dx dxa x b
Louisiana Tech UniversityRuston, LA 71272
Orthogonality
From Bessel’s equation, we have w(x) = x, and the derivative is zero at x = 0, so it follows immediately that:
nmdxxJxJx
nmdxxxxw
mn
b
a nm
for
for
0
0
1
0 00
Provided that m and n are values of for which the Bessel function is zero at x = 1.
Louisiana Tech UniversityRuston, LA 71272
Lagrangian vs. Eulerian
Consider the flow configuration below:
The velocity at the left must be smaller than the velocity in the middle.
A. What is the relationship?
B. If the flow is steady, is v(t) at any point in the flow a function of time?
0x 1x
Louisiana Tech UniversityRuston, LA 71272
Lagrangian vs. Eulerian
Conceptually, how are these two viewpoints different?
Why can’t you just use dv/dt to get acceleration in an Eulerian reference frame?
Give an example of an Eulerian measurement.
Be able to describe both viewpoints mathematically.
Louisiana Tech UniversityRuston, LA 71272
Exercise
Consider the flow configuration below:
Assume that along the red line: 0 1 0.5v x v x
0x 1x
a. What is the velocity at x = -1 and x = 0?
b. Does fluid need to accelerate as it goes to x = 0?
c. How would you calculate the acceleration at x = -0.5?
d. How would you calculate acceleration for the more general case v = f(x)?
e. Can you say that acceleration is a = dv/dt?
Louisiana Tech UniversityRuston, LA 71272
Streamlines, Streaklines, Pathlines and Flowlines
Be able to draw each of these for a given (simple) flow.
Understand why they are different when flow is unsteady.
Be able to provide an example in which they are different.
If shown a picture of lines, be able to say which type of line it is.
Be able to write down a differential equation for each type of line.
So:
Louisiana Tech UniversityRuston, LA 71272
Flow Lines
Louisiana Tech UniversityRuston, LA 71272
Flow Lines
Louisiana Tech UniversityRuston, LA 71272
Conservation Laws: Mathematically
CV CS
d ddV dA
dt dt
v n
All three conservation laws can be expressed mathematically as follows:
Increase of “entity per unit volume”
Production of the entity (e.g. mass, momentum, energy)
Flux of “entity per unit volume” out of the surface of the volume
(n is the outward normal)
is some property per unit volume. It could be density, or specific energy, or momentum per unit volume.
is some entity. It could be mass, energy or momentum.
Louisiana Tech UniversityRuston, LA 71272
Reynolds Transport Theorem: Momentum
xx xCV CS
d mv dv dV v dA
dt dt v n
If we are concerned with the entity “mass,” then the “property” is mass per unit volume, i.e. density.
Increase of momentum within the volume.
Production of momentum within the volume
Flux of momentum through the surface of the volume
Momentum can be produced by:
External Forces.
Louisiana Tech UniversityRuston, LA 71272
Mass Conservation in an Alveolus
CV CS
dm ddV dA
dt dt v n
Density remains constant, but mass increases because the control volume (the alveolus) increases in size. Thus, the limits of the integration change with time.
Term 1: There is no production of mass.
Term 2: Density is constant, but the control volume is growing in time, so this term is positive.
Term 3: Flow of air is into the alveolus at the inlet, so this term is negative and cancels Term 2.
Control Volume (CV)
Control Surface CS
Louisiana Tech UniversityRuston, LA 71272
Vector and Tensor Analysis
• In the material derivative in Gibbs notation, we introduced some new mathematical operators
• Gradient– In Cartesian, cylindrical, and spherical coordinates
respectively:
At
A
dt
Adm
v What is this operation?
1 2 31 2 3
1 2 3
1 2 3
1
1 1
sin
A A AA
z z z
A A AA
r r zA A A
Ar r r
e e e
e e e
e e e
Louisiana Tech UniversityRuston, LA 71272
Coordinate Systems
Cartesian
Cylindrical
Spherical
What is an area element in each system?
What is a volume element in each system?
Louisiana Tech UniversityRuston, LA 71272
Momentum Equation
2bp v
t
v
v v f
bpt
vv v τ f
Navier-Stokes
General Form (varying density and viscosity)
Louisiana Tech UniversityRuston, LA 71272
Momentum Equation
• Be able to translate each vector term into its components. E.g.
• Be able to state which terms are zero, given symmetry conditions. E.g. what does “no velocity in the radial direction” mean mathematically. What does “no changes in velocity with respect to the direction” mean mathematically?
yx zvv v
x y z
v
Louisiana Tech UniversityRuston, LA 71272
Continuity Equation
0t
v
0 v
Compressible
Incompressible
Louisiana Tech UniversityRuston, LA 71272
Last Word of Advice
• Always check your units.