Lecture-3 - APPH 4200 Physics of Fluids Cartesian...

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APPH 4200 Physics of Fluids Cartesian Tensors (Ch. 2) September 13, 2011 1. Geometric Identities 2. Vector Calculus 1
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Scalars, Vectors, &Tensors Scalars: mass density ( ρ ), temperature (T), concentration (S), charge density ( ρ q ) Vectors: flow (U), force (F), magnetic field (B), current density (J), vorticity ( Ω ) Tensors: stress ( τ ), strain rate ( ε ), rotation (R), identity (I) How to work and operate with tensors… 2
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Vector Examples 3
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Scalar Product 4
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Vector Product 5
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Gradient Operator (Scalar) 6
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Vector Identities Notation: f, g, are scalars; A , B , etc., are vectors; T is a tensor; I is the unit dyad. (1) A · B × C = A × B · C = B · C × A = B × C · A = C · A × B = C × A · B (2) A × ( B × C ) = ( C × B ) × A = ( A · C ) B - ( A · B ) C (3) A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 (4) ( A × B ) · ( C × D ) = ( A · C )( B · D ) - ( A · D )( B · C ) (5) ( A × B ) × ( C × D ) = ( A × B · D ) C - ( A × B · C ) D (6) ( fg ) = ( gf ) = f g + g f (7) · ( f A ) = f · A + A · f (8) ∇ × ( f A ) = f ∇ × A + f × A (9) · ( A × B ) = B · ∇ × A - A · ∇ × B (10) ∇ × ( A × B ) = A ( · B ) - B ( · A ) + ( B · ) A - ( A · ) B (11) A × ( ∇ × B ) = ( B ) · A - ( A · ) B (12) ( A · B ) = A × ( ∇ × B ) + B × ( ∇ × A ) + ( A · ) B + ( B · ) A (13) 2 f = · f (14) 2 A = ( · A ) - ∇ × ∇ × A (15) ∇ × ∇ f = 0 (16) · ∇ × A = 0 7
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What is a Tensor? In other words: a vector has a direction and magnitude … 8
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Rotation about the z-Axis 9
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Rotation Matrix is Orthogonal 10
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Ch 2: Problem 4 4. Show that c . CT = CT . C = 8, ,.; where C is the direction cosine matrx and 8 is the matrx of the Kronecker delta.
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