MSE327

Transport Phenomena

9:05-9:55 MWF

Olin Hall, Room 200

Math overview

Approximations with Taylor series

0

f(x)

x

f(x)=f(0)

f(x)=f(0)+f(0)x

(1)

(2)

(3) f(x)=f(0)+f(0)x+f(0)x2/2!

f(x)=f(0)+f(0)x+f(0)x2/2!

,

Example

X=0.1

Regular Expansions

Matrix algebra

Addition

Scalar multiplication

Transpose

Matrix determinant

minor cofactor

n x m matrix has n columns and m rows

Cofactor aij is positive if the sum i+j is even and negative otherwise

Matrix algebra

Inverse matrix

Matrix multiplication

ONLY SQUARE MATRIX HAS INVERSE

i=3, k=4, j=1,..4

http://mathworld.wolfram.com/, wikipedia.org

When square matrix has inverse?

Space

Coordinates of the point: x1, x2, x3

x1

x2

x3

x=(x1,x2,x3)Position:

http://english.sxu.edu/sites/wordpress/libraryblog/?attachment_id=749

O

Geographical coordinate systemCartesian coordinate system

xLatitude(Equator)

Longitude(Greenwich)

Height(Sea level)

x=(x1,x2,x3)

x1=(x1,x2,x3) ,

Coordinate transformations:

(a1,a2,a3)

(b1,b2,b3)

Norm:

Vector AlgebraLinear operations:

a

b

a sa

a+bscaling addition

O

a

b

a-b

subtractions

O

sa=(sa1, sa2, sa3) a+b=(a1+b1, a2+b2, a3+b3) a b=(a1 b1, a2 b2, a3 b3)

Bilinear products:

dot product (scalar) cross product (vector) tensor product

a b=a1b1+ a2b2+ a3b3a b=|a||b| cos

a

b

a b, a b=0

a x b=(a2b3a3b2, a3b1a1b3, a1b2a2b1,)

a b

a x b |a x b|=|a|x|b|sin

a || b, a x b = 0

parallelogram

Basis vectors

Set of normalized and mutually orthogonal vectors that represents each vector in a given space:

1

2

3

x1

x2

x3

O

x

e1

e2

e3

|e1|= |e2|= |e3|=1

e1e2= e2e3= e3e1=0

(normalized)

(mutually orthogonal)

x=x1e1+ x2e2+ x3e3

Any vector is a combination of the basis vectors :

Vector coordinates:

x1=e1x , x2=e2x , x3=e3x

Completeness of basis: x=e1(e1x )+ e2(e2x )+ e2(e2x )

I=e1e1+ e2e2+ e3e3=

UNIT MATRIX

Simple coordinate transformationsSimple translation:

Simple rotation:

Simple reflection (in -yz plane):

x=x cy=yz=z

x=x cos+y siny= x sin+y cosz=z

x= xy=yz=z

translation

rotation

reflectionx2x1

General Coordinate transformations

1O e1

e2

e3

a1

a2a3

c

x

x

x=x-c=x1a1+ x2a2+ x3a3

x1= a1 (xc)

x= x1e1+ x2e2+ x3e3c= c1e1+ c2e2+ c3e3

a1= a11e1+ a12e2+ a13e3

aij are the coordinates of new basis vector

x1= a1 (xc)=a11 (x1c1)+ a12 (x2c2)+ a13 (x3c3) x2= a2 (xc) = a21 (x1c1)+ a22 (x2c2)+ a23 (x3c3) x3= a3 (xc) = a31 (x1c1)+ a32 (x2c2)+ a33 (x3c3)

Index form: Matrix form:

transformation matrix translation vector

old system

new system

Index notationsx=(x1, x2, x3)=xi, where i=1, 2 , 3

Algebraic operations: sa=(sa1, sa2, sa3)= sai

a+b=(a1+b1, a2+b2, a3+b3)= ai+bi a b=(a1 b1, a2 b2, a3 b3)= ai bi

Scalar product: ab=a1b1+a2b2+a3b3=

The Kronecker delta:

Function of two variables i and j:

Exp:

The Levi - Civita symbol:

,

Exp:

,

Even: permutation: 1Odd : -1

a x b=(a2b3a3b2, a3b1a1b3, a1b2a2b1)

Scalars, Vectors, TensorsGeometric quantities may be classified according to their behavior under pure rotations.Scalar quantities: A single quantity S is called scalar if it is an invariant under rotation

S=S

Exp: Distance, norm, vectors dot product, mass, charge, density etc.

Vector quantities: Any triplet of quantities which transforms under rotation according to:

Tensor quantities: transforms under rotation according to

Tensor product

Continuum Physics

Exp: Radius R of Hydrogen molecule ~10 10 mAverage distance L between gas molecules ~10 7m

In CP, the fields are the functions of space and time. For example, density, temperature, concentration, pressure, etc.Thefield evolution is described by a PDE.

Continuous media is opposite to discrete media like atomic lattice. In continuous media the mass is spread over the volume. Approximation! The phenomenon can be described using a continuous media approximation if its length scale is larger then the size of the particles, R, constituting the media and distances between them, L.

0.01 mm

Air cube with the side size 0.01 mm contains ~ 27 billions molecules!

Scalar fields. Gradient.Mass, Charge, material constants: u = f(x,y,z)

Gradient of the scalar field at particular point M, is a vector defined as:

Level surface is a surface where the function f(x,y,z) has a c constant value: f (x,y,z) = const

Gradient points in the direction of increase of the scalar field u and magnitude of the gradient vector is equal to this rate of increase.

Presence of the gradients causes flows. Flux rate @ which a quantity is transferred through unit area within unit time. Heat flux, the flux of the molecules, flux of charges.

Fouriers law (Heat flow)

Ohms law (Electric flow)

First Ficks law(Mass flow)

Darcys law (Liquid flow through porous media)

K thermal conductivity, - electrical conductivity, D diffusion coeff., k permeability, liquid viscosity

T1

T2

sink

source

Nabla operatorTriplet of spatial derivatives:

Action on scalar field S(x):(gradient of the scalar field)

Dot product with vector field V(x):(Divergence of the vector field)

Cross- product with vector field V(x):(curl or rot of the vector field)

Laplace operator:

Useful relations

Scalar field, S Vector field, V

grad div rot

grad grad div V

div div grad S =S div rot V = 0

rot rot grad u=0 rot rot V = grad div V V

Curvilinear coordinate systems

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

Coordinate lines are curved.

HW: What is grad, div, rot, , Area, Volume and length in curvilinear coordinates?

Euler and Lagrange coordinatesEuler framework: t, x1, x2, x3 (Control volume is fixed in space)

Lagrange framework: t, x01, x02, x03

x=(x1, x2, x3)X=(x01, x02, x03)

Material derivative:

LE

Accumulation.

The amount of species i, that accumulates in a volume V in a Cartesian system during the time interval t can be found as:

J(x)

dx

Change in the density of species i due to flux is:

Production rate: generation rate + vanishing rate

In 3-D:

Divergence of the vector field is a measure of the magnitude of a vector field's source or sink. This is the rate at which the flux causes the density of the quantity comprisingthe flux to decrease

Conservedquantities:Energy, particles

Non-conserved quantities:Entropy