Chapter 2: Vectors and Tensors 2/5/2018

Home , Einstein notation

Review:

We: Now… • Defined rheology • Contrasted with Newtonian and non‐Newtonian behavior • Saw demonstrations (film)

Key to deformation and flow is the momentum balance: ⋅ ⋅

• Linear Newtonian fluids: • Instantaneous • Non-Newtonian fluids:

• Non-linear • Non-instantaneous • ? (missing piece)

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Key to deformation and flow is the momentum balance: ⋅ ⋅

• Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids.

• Non-linear • Non-instantaneous • ? (missing piece)

3 © Faith A. Morrison, Michigan Tech U.

Key to deformation and flow is the momentum balance: ⋅ ⋅

• Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids.

• Non-linear We’re going to need to • Non-instantaneous calculate how different • ? (missing piece) guesses affect the predicted behavior.

4 © Faith A. Morrison, Michigan Tech U.

2 Chapter 2: Vectors and Tensors 2/5/2018

Key to deformation and flow is the momentum balance: ⋅ ⋅

• Linear Newtonian fluids: • Instantaneous We’re going to be trying• to identify the constitutive Non-Newtonian fluids: equation for non- Newtonian fluids.

• Non-linear We’re going to need to • Non-instantaneous calculate how different • ? (missing piece) guesses affect the We need to understand predicted behavior. and be able to manipulate this 5 mathematical© Faith A. Morrison, notation. Michigan Tech U.

Chapter 2: Mathematics Review

CM4650 Polymer Rheology 1. Vector review Michigan Tech 2. Einstein notation 3. Tensors

Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University

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Chapter 2: Mathematics Review

1. Scalar – a mathematical entity that has magnitude only

e.g.: temperature T speed v time t density r

– scalars may be constant or may be variable

Laws of Algebra for yes commutative ab = ba Scalars: yes associative a(bc) = (ab)c yes distributive a(b+c) = ab+ac

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2. Vector – a mathematical entity that has magnitude and direction e.g.: force on a surface f velocity v

– vectors may be constant or may be variable

Definitions

magnitude of a vector – a scalar associated with a vector

v  v f  f This notation unit vector – a vector of unit length ,, is called v Gibbs notation.  vˆ v a unit vector in the direction of v

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Laws of Algebra for Vectors:

1. Addition a+b a

b 2. Subtraction

a+(-b) a

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Laws of Algebra for Vectors (continued):

3. Multiplication by scalar v yes commutative v  v yes associative   v   v   v yes distributive  v  w  v  w

4. Multiplication of vector by vector 4a. scalar (dot) (inner) product

v w  vwcos v 

Note: we can find w magnitude with dot

product 2 vv  vv cos0  v

v  v  vv 10 © Faith A. Morrison, Michigan Tech U.

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Laws of Algebra for Vectors (continued):

4a. scalar (dot) (inner) product (con’t) yes commutative v w  wv

NO associative v w z no such operation yes distributive z v  w  z v  z  w

4b. vector (cross) (outer) product

v  w  vwsin eˆ v 

eˆ is a unit vector w perpendicular to both v and w following the right-hand rule 11 © Faith A. Morrison, Michigan Tech U.

Laws of Algebra for Vectors (continued):

4b. vector (cross) (outer) product (con’t) NO commutative v  w  w  v NO associative vwz vw   z v wz  yes distributive z  v  w  z  v  z  w

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Coordinate Systems

•Allow us to make actual calculations with vectors

Rule: any three vectors that are non-zero and linearly independent (non-coplanar) may form a coordinate basis

Three vectors are linearly dependent if a, b, and g can be found such that:  a  b   c  0 for , ,  0

If , , and are found to be zero, the vectors are linearly independent.

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How can we do actual calculations with vectors?

Rule: any vector may be expressed as the linear combination of three, non-zero, non-coplanar basis vectors

coefficient of a in the eˆy direction

 a  any vector  x  a  axeˆx  ayeˆy  azeˆz  ay     a  This notation z xyz is called matrix  a1eˆ1  a2eˆ2  a3eˆ3 notation. 3 ˆ   a je j j1 14 © Faith A. Morrison, Michigan Tech U.

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Trial calculation: dot product of two vectors

ab  a1eˆ1  a2eˆ2  a3eˆ3  b1eˆ1  b2eˆ2  b3eˆ3

 a1eˆ1 b1eˆ1  b2eˆ2  b3eˆ3 

a2eˆ2 b1eˆ1  b2eˆ2  b3eˆ3 

a3eˆ3 b1eˆ1  b2eˆ2  b3eˆ3

 a1eˆ1 b1eˆ1  a1eˆ1 b2eˆ2  a1eˆ1 b3eˆ3 

a2eˆ2 b1eˆ1  a2eˆ2 b2eˆ2  a2eˆ2 b3eˆ3 

a3eˆ3 b1eˆ1  a3eˆ3 b2eˆ2  a3eˆ3 b3eˆ3

If we choose the basis to be orthonormal - mutually perpendicular and of unit length - then we can simplify.

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If we choose the basis to be orthonormal - mutually perpendicular and of unit length, then we can simplify.

eˆ1 eˆ1 1 eˆ1 eˆ2  0 eˆ1 eˆ3  0 

a b  a1eˆ1 b1eˆ1  a1eˆ1 b2eˆ2  a1eˆ1 b3eˆ3  a2eˆ2 b1eˆ1  a2eˆ2 b2eˆ2  a2eˆ2 b3eˆ3  a3eˆ3 b1eˆ1  a3eˆ3 b2eˆ2  a3eˆ3 b3eˆ3  a1b1  a2b2  a3b3

We can generalize this operation with a technique called Einstein notation.

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Einstein Notation

a system of notation for vectors and tensors that allows for the calculation of results in Cartesian coordinate systems.

a  a1eˆ1  a2eˆ2  a3eˆ3 3   ajeˆj This notation j1 called Einstein  ajeˆj  ameˆm notation.

•the initial choice of subscript letter is arbitrary •the presence of a pair of like subscripts implies a missing summation sign

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Einstein Notation (con’t)

The result of the dot products of basis vectors can be summarized by the Kronecker delta function

eˆ eˆ 1 1 1 1 i  p eˆ eˆ  0 eˆ eˆ     1 2 i p ip 0 i  p eˆ1 eˆ3  0  Kronecker delta

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Einstein Notation (con’t)

To carry out a dot product of two arbitrary vectors . . .

Detailed Notation Einstein Notation

a b  a1eˆ1  a2eˆ2  a3eˆ3  b1eˆ1  b2eˆ2  b3eˆ3 a b  ajeˆj bmeˆm  a1eˆ1 b1eˆ1  a1eˆ1 b2eˆ2  a1eˆ1 b3eˆ3  a e be  a e b e  a e b e  2ˆ2 1ˆ1 2ˆ2 2ˆ2 2ˆ2 3ˆ3  aj jmbm a3eˆ3 b1eˆ1  a3eˆ3 b2eˆ2  a3eˆ3 b3eˆ3  ab  a b  a b 1 1 2 2 3 3  ajbj

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3. Tensor – the indeterminate vector product of two (or more) vectors e.g.: stress  velocity gradient  – tensors may be constant or may be variable

Definitions

dyad or dyadic product – a tensor written explicitly as the indeterminate vector product of two vectors

a d dyad A general representation This notation of a tensor , is also part of Gibbs notation. 20 © Faith A. Morrison, Michigan Tech U.

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Laws of Algebra for Indeterminate Product of Vectors:

NO commutative a v  v a yes associative b a v  b a v  b a v yes distributive a v  w  a v  a w

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How can we represent tensors with respect to a chosen coordinate system? Just follow the rules of tensor algebra

a m  a1eˆ1  a2eˆ2  a3eˆ3 m1eˆ1  m2eˆ2  m3eˆ3 

 a1eˆ1m1eˆ1  a1eˆ1m2eˆ2  a1eˆ1m3eˆ3 

a2eˆ2m1eˆ1  a2eˆ2m2eˆ2  a2eˆ2m3eˆ3 

a3eˆ3m1eˆ1  a3eˆ3m2eˆ2  a3eˆ3m3eˆ3 3 3 ˆ ˆ  ak ek mwew kw1 1 3 3 Any tensor may be written as the ˆ ˆ sum of 9 dyadic products of basis  ak mw ek ew kw1 1 vectors

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What about ? Same.

3 3 A  Aij eˆieˆj ij1 1

Einstein notation for tensors: drop the summation sign; every double index implies a summation sign has been dropped.

A  Aij eˆieˆj  Apk eˆpeˆk

Reminder: the initial choice of subscript letters is arbitrary

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How can we use Einstein Notation to calculate dot products between vectors and tensors?

It’s the same as between vectors.

a b  a u v  b A 

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Summary of Einstein Notation

1. Express vectors, tensors, (later, vector operators) in a Cartesian coordinate system as the sums of coefficients multiplying basis vectors - each separate summation has a different index 2. Drop the summation signs 3. Dot products between basis vectors result in the Kronecker delta function because the Cartesian system is orthonormal.

Note: •In Einstein notation, the presence of repeated indices implies a missing summation sign •The choice of initial index (, , , etc.) is arbitrary - it merely indicates which indices change together

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3. Tensor – (continued) Definitions Scalar product of two tensors

A : M  Aipeˆieˆp : M kmeˆk eˆm

carry out the dot  Aip M kmeˆieˆp : eˆk eˆm products indicated

 Aip M km eˆp eˆk eˆi eˆm 

 Aip M km  pkim “p” becomes “k” “i” becomes “m”  Amk M km

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But, what is a tensor really?

A tensor is a handy representation of a Linear Vector Function

scalar function: y  f (x)  x2  2x  3

a mapping of values of x onto values of y

vector function: w  f (v)

a mapping of vectors of v into vectors w

How do we express a vector function?

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What is a linear function?

Linear, in this usage, has a precise, mathematical definition.

Linear functions (scalar and vector) have the following two properties:

f (x)  f (x) f (x  w)  f (x)  f (w)

It turns out . . . Multiplying vectors and tensors is a convenient way of representing the actions of a linear vector function (as we will now show).

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Tensors are Linear Vector Functions

Let f(a) = b be a linear vector function.

We can write a in Cartesian coordinates.

a  a1eˆ1  a2eˆ2  a3eˆ3

f (a)  f (a1eˆ1  a2eˆ2  a3eˆ3 )  b

Using the linear properties of f, we can distribute the function action:

f (a)  a1 f (eˆ1)  a2 f (eˆ2 )  a3 f (eˆ3 )  b

These results are just vectors, we will name them v, w, and m. 29 © Faith A. Morrison, Michigan Tech U.

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Tensors are Linear Vector Functions (continued)

f (a)  a1 f (eˆ1)  a2 f (eˆ2 )  a3 f (eˆ3 )  b

v w m

f (a)  a1v  a2 w  a3 m  b

Now we note that the coefficients ai may be written as,

a1  aeˆ1 a2  aeˆ2 a3  aeˆ3 The Substituting, indeterminate vector product f (a)  aeˆ1 v  aeˆ2 w  aeˆ3 m  b has appeared!

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Using the distributive law, we can factor out the dot product with a:

f (a)  a eˆ1 v  eˆ2 w  eˆ3 m  b

This is just a tensor (the sum of dyadic eˆ1 v  eˆ2 w  eˆ3 m  M products of vectors)

f (a)  a  M  b

CONCLUSION: Tensor operations are convenient to use to express linear vector functions.

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3. Tensor – (continued) More Definitions Identity Tensor

I  eˆieˆi  eˆ1eˆ1  eˆ2eˆ2  eˆ3eˆ3 1 0 0  0 1 0   0 0 1  123

A I  Aipeîeˆp eˆkeˆk  Aipeî pkeˆk  Aikeîeˆk  A

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3. Tensor – (continued) More Definitions

Zero Tensor 0 0 0 0  0 0 0   0 0 0  123 Magnitude of a Tensor Note that the book has a A: A typo on this equation: the A   2 “2” is under the square root. A: A  Aipeˆieˆp : Akmeˆkeˆm

 AipAkm eˆp eˆk eˆi eˆm products  Amk Akm across the diagonal

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3. Tensor – (continued) More Definitions Tensor Transpose Exchange the T T M  Mikeˆieˆk  Mikeˆkeˆi coefficients across the diagonal

CAUTION:

T T T AC  Aikeîeˆk Cpjeˆpeˆj  AikCpj eîeˆjkp T  AipCpj eîeˆj  AipCpj eˆjeî I recommend you always It is not equal to: T T interchange the indices AC  AipCpj eîeˆj on the basis vectors  A C eêˆ rather than on the pi jp i j coefficients.

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3. Tensor – (continued) More Definitions

Symmetric Tensor e.g. M  M T 1 2 3 2 4 5 M  M   ik ki 3 5 6  123

Antisymmetric Tensor e.g. M  M T 0  2  3 2 0  5 M  M   ik ki 3 5 0  123

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3. Tensor – (continued) More Definitions

Tensor order

Scalars, vectors, and tensors may all be considered to be tensors (entities that exist independent of coordinate system). They are tensors of different orders, however. order = degree of complexity

scalars 0th -order tensors 30 Number of vectors 1st -order tensors 31 coefficients tensors 2nd -order tensors 32 needed to express the higher- 3rd -order tensors 33 tensor in 3D order space tensors

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3. Tensor – (continued) More Definitions

Tensor Invariants

Scalars that are associated with tensors; these are numbers that are independent of coordinate system.

vectors: v  v The magnitude of a vector is a scalar associated with the vector It is independent of coordinate system, i.e. it is an invariant.

tensors:A There are three invariants associated with a second-order tensor.

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Tensor Invariants

IA  traceA  trA

For the tensor written in Cartesian coordinates:

traceA  App  A11  A22  A33

IIA  traceA A  A: A  Apk Akp

IIIA  traceA A A  Apj AjhAhp

Note: the definitions of invariants written in terms of coefficients are only valid when the tensor is written in Cartesian coordinates.

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4. Differential Operations with Vectors, Tensors

Scalars, vectors, and tensors are differentiated to determine rates of change (with respect to time, position)

•To carryout the differentiation with respect to a single variable, differentiate each coefficient individually. •There is no change in order (vectors remain vectors, scalars remain scalars, etc.

 w   B B B   1   11 21 31   t   t t t  w w B B B B    2    21 22 23  t t  t  t  t t t   w3   B31 B32 B33      t t t t  123  123

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4. Differential Operations with Vectors, Tensors (continued)

•To carryout the differentiation with respect to 3D spatial variation, use the del (nabla) operator. Del Operator •This is a vector operator •Del may be applied in three different ways •Del may operate on scalars, vectors, or tensors       x1        This is written in   eˆ  eˆ  eˆ  1 x 2 x 3 x x  Cartesian 1 2 3  2  coordinates      x3  123 3    eˆp  eˆp p1 xp xp

Einstein notation for del 40 © Faith A. Morrison, Michigan Tech U.

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4. Differential Operations with Vectors, Tensors (continued)

A. Scalars - gradient This is written in      Cartesian coordinates Gibbs  x1  notation         eˆ   eˆ   eˆ   1 x 2 x 3 x x  1 2 3  2       x3  123  Gradient of a  e ˆp The gradient of scalar field xp a scalar field is a vector The gradient operation captures the total spatial variation of a scalar, vector, •gradient operation increases the order of the or tensor field. entity operated upon

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4. Differential Operations with Vectors, Tensors (continued) B. Vectors - gradient    w  eˆ1 w  eˆ2 w  eˆ3 w This is all written in x1 x2 x3 Cartesian  coordinates (basis The basis vectors  eˆ w eˆ  w eˆ  w eˆ vectors are can move out of 1 1 1 2 2 3 3 x1 constant) the derivatives because they are   eˆ w eˆ  w eˆ  w eˆ constant (do not 2 x 1 1 2 2 3 3 change with 2 position)   eˆ3 w1eˆ1  w2eˆ2  w3eˆ3 x3

w1 w2 w3 w1  eˆ1eˆ1  eˆ1eˆ2  eˆ1eˆ3  eˆ2eˆ1  x1 x1 x1 x2

w2 w3 w1 w2 w3 eˆ2eˆ2  eˆ2eˆ3  eˆ3eˆ1  eˆ3eˆ2  eˆ3eˆ3 x2 x2 x3 x3 x3 42 © Faith A. Morrison, Michigan Tech U.

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4. Differential Operations with Vectors, Tensors (continued)

B. Vectors - gradient (continued) constants may appear on either side of the Gradient of a differential operator vector field 3 3 wk wk wk w  eˆjeˆk  eˆjeˆk  eˆjeˆk Gibbs x x x notation jk1 1 j j j

Einstein notation The gradient of for gradient of a a vector field is a vector tensor

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4. Differential Operations with Vectors, Tensors (continued)

C. Vectors - divergence

Divergence of a      vector field   w  eˆ1  eˆ2  eˆ3  w1eˆ1  w2eˆ2  w3eˆ3  x1 x2 x3 Gibbs w w w notation  1  2  3 x1 x2 x3 The Divergence 3 w w of a vector field   i  i is a scalar i1 xi xi

Einstein notation for divergence of a vector

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4. Differential Operations with Vectors, Tensors (continued)

C. Vectors - divergence (continued) This is all written in Cartesian constants may appear coordinates (basis on either side of the vectors are differential operator constant) Using Einstein notation

 wj wj   w  eˆm  wjeˆj  eˆm eˆj  mj xm xm xm w  j xj

•divergence operation decreases the order of the entity operated upon

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4. Differential Operations with Vectors, Tensors (continued) D. Vectors - Laplacian

Using      w  eˆ eˆ w eˆ  w eˆ eˆ eˆ Einstein m x p x j j x x j m p j notation: m p m p   Gibbs  wj mp eˆj notation xm xp   The Laplacian of a vector field is a Einstein  wj eˆj xp xp vector notation  2w 2w 2w   1  1  1   x1 x2 x3  2w 2w 2w  •Laplacian operation does   2  2  2  not change the order of the x x x Column entity operated upon  1 2 3  vector  2w 2w 2w  3  3  3 notation   46  x1 x2 x3  123 © Faith A. Morrison, Michigan Tech U.

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4. Differential Operations with Vectors, Tensors (continued)

(impossible; cannot E. Scalar - divergence   decrease order of a scalar) F. Scalar - Laplacian   G. Tensor - gradient A H. Tensor - divergence  A I. Tensor - Laplacian  A

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5. Curvilinear Coordinates Note: my spherical Cylindrical Spherical comes from the -axis.

r,, z eˆr ,eˆ ,eˆz r,, eˆr ,eˆ ,eˆ

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Note: my spherical comes from the -axis.

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5. Curvilinear Coordinates

Cylindrical See text r,, z eˆr ,eˆ ,eˆz figures 2.11 and Spherical r,, eˆr ,eˆ ,eˆ 2.12

These coordinate systems are ortho-normal, but they are not constant (they vary with position). This causes some non-intuitive effects when derivatives are taken.

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5. Curvilinear Coordinates (continued)

v  vreˆr  v eˆ  vzeˆz

v  vr eˆr  v eˆ  vzeˆz        eˆ  eˆ  eˆ  v eˆ  v eˆ  v eˆ  x x y y z z  r r   z z

First, we need to write this in cylindrical coordinates.

solve for eˆr  cos eˆx  sin eˆy x  r cos Cartesian basis eˆ  sin eˆ  cos eˆ  x y y  r sin substitute above vectors and using chain rule substitute eˆz  eˆz z  z (see next slide for above details) 51 © Faith A. Morrison, Michigan Tech U.

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x  r cos r  x2  y 2 eˆx  cos eˆr  sin eˆ 1 y  ˆ ˆ ˆ y  r sin   tan   ey  sin er  cos e  x  z  z

     ˆ ˆ ˆ    ex  ey  ez   x y z 

  r    z     sin      cos    x r x  x z x r   r    r    z    cos      sin    y r y  y z y r   r 

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5. Curvilinear Coordinates (continued)

     Result:    eˆ  eˆ  eˆ   x x y y z z   1    eˆ  eˆ  eˆ r r  r  z z Now, proceed:   1    v  eˆ  eˆ  eˆ  v eˆ  v eˆ  v eˆ  r r  r  z z r r   z z (We cannot use Einstein notation  because these are  eˆr vreˆr  v eˆ  vzeˆz  not Cartesian r coordinates) 1  eˆ v eˆ  v eˆ  v eˆ  Curvilinear  r  r r   z z coordinate notation  eˆ v eˆ  v eˆ  v eˆ z z r r   z z 53 © Faith A. Morrison, Michigan Tech U.

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5. Curvilinear Coordinates (continued)  v  eˆ v eˆ  v eˆ  v eˆ  r r r r   z z 1  eˆ v eˆ  v eˆ  v eˆ   r  r r   z z  eˆz vreˆr  v eˆ  vzeˆz 1  1 v eˆ z eˆ v eˆ  eˆ  r r  r  r r  r  1  eˆ v   eˆ  v r  eˆ r   r  r  r   eˆ  r  cos eˆ  sin eˆ   x y

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5. Curvilinear Coordinates (continued)

1  1 v eˆ eˆ v eˆ  eˆ  r r  r  r r  r  1  eˆ v   eˆ  v r  eˆ r   r  r  r   1  v   eˆ  v eˆ  eˆ r   r  r  r   1  vr This term is not intuitive, r and appears because the basis vectors in the curvilinear coordinate Curvilinear systems vary with position. coordinate notation 55 © Faith A. Morrison, Michigan Tech U.

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5. Curvilinear Coordinates (continued)

Final result for divergence of a vector in cylindrical coordinates:

 v  eˆ v eˆ  v eˆ  v eˆ  r r r r   z z 1  eˆ v eˆ  v eˆ  v eˆ   r  r r   z z  eˆ v eˆ  v eˆ  v eˆ z z r r   z z v 1 v v v v  r    r  r r r  r z Curvilinear coordinate notation 56 © Faith A. Morrison, Michigan Tech U.

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5. Curvilinear Coordinates (continued)

Curvilinear Coordinates (summary)

•The basis vectors are ortho-normal •The basis vectors are non-constant (vary with position) •These systems are convenient when the flow system mimics the coordinate surfaces in curvilinear coordinate systems. •We cannot use Einstein notation – must use Tables in Appendix C2 (pp464-468). Curvilinear coordinate notation

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6. Vector and Tensor Theorems and In Chapter 3 we review Newtonian fluid definitions mechanics using the vector/tensor vocabulary we have learned thus far. We just need a few more theorems to prepare us for those studies. These are presented without proof.

Gauss Divergence Theorem outwardly directed unit b dV  nˆ b dS normal Gibbs   notation V S

This theorem establishes the utility of the divergence operation. The integral of the divergence of a vector field over a volume is equal to the net outward flow of that property through the bounding surface.

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6. Vector and Tensor Theorems (continued)

Leibnitz Rule for differentiating integrals

 constant limits I   f (x,t) dx  one  dimension, dI d constant   f (x,t) dx limits dt dt   f (x,t)   dx  t

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6. Vector and Tensor Theorems (continued)

Leibnitz Rule for differentiating integrals

 (t) J   f (x,t) dx variable limits  (t) one  (t) dJ d dimension,  f (x,t) dx dt dt  variable  (t) limits  (t) f (x,t) d d   dx  f (,t)  f (,t)  (t) t dt dt

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6. Vector and Tensor Theorems (continued)

Leibnitz Rule for differentiating integrals

J   f (x, y, z,t) dV V (t) three dJ d  f (x, y, z,t) dV dimensions,  variable limits dt dt V (t) f (x, y, z,t)  dV  f v nˆ dS   surface V (t) t S (t)

velocity of the surface element dS

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6. Vector and Tensor Theorems (continued) Consider a function Substantial Derivative f (x, y, z,t)

true for any  f   f   f   f  path: df    dx    dy    dz    dt       x yzt  y xzt z xyt t xyz choose special path: df  f  dx  f  dy  f  dz  f                    dt x yzt dt  y xzt dt z xyt dt t xyz time rate of change of f x-component When the chosen path is along a chosen of velocity the path of a fluid particle, path along that path then these are the components of the particle velocities.

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6. Vector and Tensor Theorems (continued) Substantial Derivative When the chosen path is the path of a fluid particle, then the space df  f  dx  f  dy  f  dz  f  derivatives are the             components of the       dt x yzt dt  y xzt dt z xyt dt t xyz particle velocities.  df   f   f   f   f       v    v    v      along   x y   z   dt a particle x yzt  y xzt z xyt t xyz path

vf

Substantial Derivative  df  Df f    vf   along  dt  a particle Dt t path

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Notation Summary:

Gibbs—no reference to coordinate system ,,, ⋅ Einstein—references to Cartesian coordinate system (ortho-normal, constant) ̂,̂̂) Matrix—uses column or row vectors for vectors and 33

matrix of coefficients for tensors , Curvilinear coordinate—references to curvilinear coordinate system (ortho-normal, vary with

Done with Math background.

Let’s use it with Newtonian fluids

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Chapter 3: Newtonian Fluids

CM4650 Polymer Rheology Michigan Tech

Navier-Stokes Equation

 v    vv   p  2 v   g  t 