Louisiana Tech University Ruston, LA 71272 Vectors 1.What is the projection of the vector (1, 3, 2) onto the plane described by ? n v

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


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:


Forces

• Body Forces

• Pressure

• Normal Stresses

• Shear Stresses


What is a Fluid?

Solid: Stress is proportional to strain (like a spring).

Fluid: Stress is proportional to strain rate.


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


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


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


Apparent Viscosity of Blood

meff

3.5 g/(cm-s)

1 dyne/cm2

Non-Newtonian Region

Rouleau Formation


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


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.


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


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.


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?


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:


Flow Lines


Flow Lines


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.

Steven A. Jones

Sometimes we denote vectors as v with an arrow above it.


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.

Steven A. Jones

Sometimes we denote vectors as v with an arrow above it.


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


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


Coordinate Systems

Cartesian

Cylindrical

Spherical

What is an area element in each system?

What is a volume element in each system?


Momentum Equation

2bp v

t

v

v v f

bpt

vv v τ f

Navier-Stokes

General Form (varying density and viscosity)


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


Continuity Equation

0t

v

0 v

Compressible

Incompressible


Last Word of Advice

• Always check your units.

Documents

Louisiana Tech University Ruston, LA 71272 Vectors 1.What is the projection of the vector (1, 3, 2) onto the plane described by ? n v