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CHEE 6333 Handout 1
Transport Processes (University of Houston)
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Handout 1: Introduction and
mathematics review
CHEE 6333
Relevant reading: BSL Appendix A and/or Deen Appendix
conventions for parentheses, indexing follow BSL
Learning objective: review vector and tensor manipulations and calculus
needed for transport processes.
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Physical quantities as mathematical objects
• Scalar:
- Zeroth order:
- Examples:
• Vector:
- First order:
- Examples:
• Tensor:
- Second order:
- Examples:
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Vectors: definition
Geometrically: magnitude and direction
Mathematically: sum of projections onto an orthonormal basis
Unit vector:
Example: Cartesian coordinates:
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Vectors: properties
Vectors satisfy:
Commutative addition
Associative addition
Commutative scalar multiplication:
Associative scalar multiplication:
Distributive scalar multiplication:
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Dot product: definitions
Geometric meaning:
In terms of components:
Kronecker delta:
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Dot product: properties
The dot product is:
Commutative
Distributive
Note: the dot productive is NOT ASSOCIATIVE
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Cross product: definitions
Geometric meaning:
In terms of components:
Permutation symbol:
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Cross product: properties
The cross product is:
Distributive
Note: the cross product is NOT COMMUTATIVE
Note: the cross productive is NOT ASSOCIATIVE
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Break 1: exercises
v = (3, 2, 4)
w = ( 1, 0, 2)
Calculate:
(a) v · w
(b) |v|
(c) [v ⇥ w]
(d) [ x ⇥ w]
(note: there is a channel on Teams for discussing lectures/questions!)
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Tensor: definition
Mathematically: sum of dyadic products
Geometrically:
General comments:
(a) Dyadic products are not commutative
(b) Physical example:
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Tensors: components
Unit tensor:
Tensor in terms of dyadic products:
Tensor as a matrix:
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Tensor: operations (1)
The transpose of a tensor is:
The dyadic product of two vectors (not commutative) is:
Operating on dyadic products: take the dot or cross product of the
nearest unit vector on each side
Example of a dot product:
Example of a cross product:
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Tensors: operations (2)
The double dot product of two tensors is:
The tensor product (or single dot product) of two tensors is:
The vector product (dot product) of a tensor and a vector is:
The tensor product (cross product) of a tensor and a vector is:
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Break 2: exercises
0
3 2
⌧ =@ 2 2
1 1
1
1
1A
4
v = (4, 3, 2)
Calculate (using the index notation previously given):
(a) [⌧ · v]
(b) [v · ⌧ ]
(c) (⌧ : ⌧ )
(d) vv
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Vector calculus: differential operators (1)
Vector differential operator:
Gradient of a scalar (distributive only):
Divergence of a vector field (distributive only):
Curl of a vector field (distributive only):
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Vector calculus: differential operators (2)
Gradient of a vector field:
Divergence of a tensor field:
Laplacian operator (distributive only):
Laplacian of a scalar field:
Laplacian of a vector field:
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Three useful theorems
Divergence theorem (Gauss):
Curl theorem (Stokes):
Leibniz formula:
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Break 3: exercises
For vector fields
v1 = (by, bx, 0) and v2 = ( by, bx, 0)
Calculate (using the index notation previously given):
(a) (r · v)
(b) rv
(c) [r · vv]
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Cylindrical coordinates: definitions
Coordinates: (r, θ, z)
Useful for: pipe flow problems
Cartesian to cylindrical and cylindrical to Cartesian:
Unit vectors:
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Cylindrical coordinates: operations
Dot and cross products of vectors and tensors:
Differential operator:
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Spherical coordinates: definition
Coordinates: (r, θ, φ) (note differences from cylindrical)
Useful for: flows around spheres
Cartesian to spherical and spherical to Cartesian:
Unit vectors:
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Spherical coordinates: operations
Dot and cross products of vectors and tensors:
Differential operator:
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