Vectors n v 1. What is the projection of the vector (1, 3, 2) onto the plane described by 4 x 2 y.
Download ReportTranscript Vectors n v 1. What is the projection of the vector (1, 3, 2) onto the plane described by 4 x 2 y.
Vectors
n v
1. What is the projection of the vector (1, 3, 2) onto the plane described by ?
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Cross Product
In Cartesian Coordiantes:
i
a
1
b
1
j
a
2
b
2
k
a
3
b
3 Used for: Moments Vorticity
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• Body Forces • Pressure • Normal Stresses • Shear Stresses
Forces
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What is a Fluid?
Solid:
Stress is proportional to strain (like a spring).
Fluid:
Stress is proportional to strain rate.
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The Stress Tensor (Fluids)
For fluids:
v
y
One dimensional 11 21 31 where
u
12 22 32 13 23 33
P
1 0 0 Three-Dimensional, is velocity 0 1 0 0 0 1 2 1 2 1 2
u
2
x
1
u x
u
1
x
1 3 1
u
1
x
2
u
1
x
3 1 2
u
1
x
2
u
2
x
1
u
2
x
2 1 2
u
3
x
2
u
2
x
3 1 2
u
1
x
3 1 2
u
2
x
3
u
3
x
3
u x
u
3
x
2 3 1
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The Stress Tensor (Solids)
For solids:
E
One dimensional 11 21 31 where
u
12 22 32 is displacement 13 23 33
u
1
x
1 Three-Dimensional,
u
x
2 2
u
3
x
3 1 0 0 2
G
1 2 1 2
u
2
x
1
u
u
1
x
1
x
1 3
u
1
x
2
u x
1 3 0 1 0 0 0 1 1 2
u
1
x
2
u
2
x
1
u
2
x
2 1 2
u
3
x
2
u
2
x
3 1 2
u
1
x
3 1 2
u
2
x
3
u
3
x
3
u x
1
u
3
x
2 3
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Non-Newtonian Fluid
For fluids:
v
y
(One dimensional) 11 21 31 12 22 32 is a function of strain rate 13 23 33
P
1 0 0 0 1 0 0 0 1 2 1 2 1 2
u
2
x
1
u x
u
1
x
1 3 1
u
1
x
2
u
1
x
3 1 2
u
1
x
2
u
2
x
1
u
2
x
2 1 2
u
3
x
2
u
2
x
3 1 2
u
1
x
3 1 2
u
2
x
3
u
3
x
3
u x
u
3
x
2 3 1
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Apparent Viscosity of Blood
Non-Newtonian Region Rouleau Formation m eff 3.5 g/(cm-s) 1 dyne/cm 2
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The Sturm Liouville Problem
Be able to reduce the Sturm Liouville problem to special cases.
d dx
p d
dx
,
x
q
w
, 0 ,
a
x
b
Or equivalently:
d
2
dx
2 ,
x
d
dx
,
x
a
b
,
x
0,
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Orthogonality
From Bessel’s equation, we have
w
(
x
) =
x
, and the derivative is zero at
x
= 0 , so it follows immediately that:
a b w
m
n dx
0 0 1
x J
0
n
0
m dx
for 0
m
n
for
m
n
Provided that
m
and
n
are values of the Bessel function is zero at
x
= 1.
for which
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Lagrangian vs. Eulerian
Consider the flow configuration below:
x
1
x
0 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?
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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.
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Exercise
Consider the flow configuration below:
x
1
x
0 Assume that along the red line:
v
0
x
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
= d
v
/d
t
?
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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:
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Flow Lines
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Flow Lines
Conservation Laws: Mathematically
All three conservation laws can be expressed mathematically as follows:
d
dt
d dt
CV
dV
CS
dA
Production of the entity (e.g. mass, momentum, energy) Increase of “entity per unit volume” Flux of “entity per unit volume” out of the surface of the volume (
n
is the outward normal) is some entity. It could be mass, energy or momentum.
is some property per unit volume. It could be density, or specific energy, or momentum per unit volume.
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Reynolds Transport Theorem: Momentum
If we are concerned with the entity “mass,” then the “property” is mass per unit volume, i.e. density.
x
dt
d dt
CV v dV x
Production of momentum within the volume Increase of momentum within the volume.
Momentum can be produced by:
External Forces.
CS v x
dA
Flux of momentum through the surface of the volume
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Mass Conservation in an Alveolus
dm dt
d dt
CV
dV
CS
dA
Control Volume ( CV ) Control Surface
CS
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.
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Vector and Tensor Analysis
• In the material derivative in Gibbs notation, we introduced some new mathematical operators
d m dt A
A
t
v
A
What is this operation?
• Gradient – In Cartesian, cylindrical, and spherical coordinates respectively:
e
1
A
z
1
e
2
e
1
e
1
A
r
A
r
e
2
e
2 1
r
1
z A
2
A
A
e
3
e
3
A
z
3
A
z
e
3 1
r
r
sin
A
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Coordinate Systems
Cartesian Cylindrical Spherical What is an area element in each system?
What is a volume element in each system?
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Momentum Equation
Navier-Stokes
v
t
v v
p
v
f
b
General Form (varying density and viscosity)
v
t
τ f
b
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Momentum Equation
• Be able to translate each vector term into its components. E.g.
v x x
v y
y
v z z
• 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?
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Continuity Equation
Compressible
t
0
Incompressible
0
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Last Word of Advice
• Always check your units.
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