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02 Vector Calculus and Indicial Notation

02 Vector Calculus and Indicial Notation - 2 Vector...

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2 Vector Calculus and Indicial Notation 1 To analyze fluid dynamics, we need to formally define and discuss some properties of scalar, vectors, and tensors. The goal is to gain familiarity and comfort indicial notation and vector calculus. 2.1 Definitions & Notation 2 Advanced notation can benefit and shorten representations of complex fluid dynamics. Two forms of notation are reviewed/introduced here. Symbolic: A ur (benefit of symbolic notation is that you don’t have to choose a coordinate system) Symbolic (Gibbs) notation (such as the following) is NOT a function of a coordinate system. b DV p f Dt ρ ρ τ = - + + �� r uur t Examples of symbolic: , , A τ Α ur r Indicial: i A where i is any index (limited to the Cartesian coordinate system) Indicial notation (such as the following) is limited to Cartesian coordinate systems i ji i b i j Du p f Dt x x τ ρ ρ = - + + o Free Index Any index that appears only once in an additive term “for 1,2,3 j = ” is implied for a free index free indices must balance for each additive term in an equation except for constants o Dummy Index (or Summation Index) Any index that appears twice in the same additive term 3 1 n = ” or “from 1 to 3” are implied o Order of the expression “Rank of the Tensor”= # of free indices Zero order: Scalar ( T , p , φ ) First Order: “vector” 1 Panton Chapter 3 2 Panton pg 33-41
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, , i i i j j u u x u x , where j is a dummy index and i is a free index Second Order: “Tensor” 1 2 j i ij j i u u x x ε = + where i and j are free indices Third Order ijk ε Note: Do not confuse dimensionality with rank Example: ( 29 ( 29 ( 29 , V u x y i v x j w x k = + + ur r r r is 2-D but Rank 1 , U Ci = ur r C is a constant is 0-D but still Rank 1 o Alternative forms of indicial notation Comma notation ( 29 ( 29 ( 29 , 2 2 , 2 , , : free :free, :dummy , : free : free i i j j i i jj j j ij i j i i u u V i j x u u V i j x x i j x x u u V t i t φ φ ur ur ur & Subscripted del notation (Panton) 2 2 i j i j i j j i j j i j i j i i u u x u u x x x x u u t φ φ 2.2 Vector Transformation Law 2.2.1 “Define a Vector” Consider a right-handed Cartesian coordinate system (RHCCS) where p ur has the position (a vector) coordinates 1 2 3 , , ( , , ) x x x x y z
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that can be generally expressed as for 1,2,3 i x i = If RHCCS is rotated to a new direction the coordinates of p ur will change to ' for ' 1,2,3 j x j = 2 x 2' x 3 x 3' x 1 x 1' x 11' c 2'3 c 23' c Define: ( 29 ' ' cos , ij i j c x x = ' ' ' ' but ij j i ij i j c c c c = By the vector transformation law 3 ' ' 1 for ' 1,2,3 j j n n n x c x j = = = (Helpful hints: free index is “ ' j ” in this example) Proof of the vector transformation law Note: treat 1 2 3 , , x x x as components of an arbitrary vector in the 1,2,3 directions
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2 x 2' x 3 x 1 x 1' x 11' c 2'3 c 23' c 3 x 2' x 1 x 1' x 11' c 3 x = + 2 x 2' x 1' x 23' c 3 x 2' x 3 x 1' x 2'3 c 3 x + 13' c 12' c 22' c 21' c 3'3 c 1'3 c 1 11 1 2 12 1 3 13 1 component: component: component: x c x x c x x c x 1 21 2 2 22 2 3 23 2 component: component: component: x c x x c x x c x
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