Lecture-2b - APPH 4200 Physics of Fluids Cartesian Tensors...

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Unformatted text preview: APPH 4200 Physics of Fluids Cartesian Tensors (Ch. 2) September 8, 2011 1. Geometric Identities 2. Vector Calculus 1 Scalars, Vectors, &Tensors • Scalars: mass density (ρ), temperature (T), concentration (S), charge density (ρq) • Vectors: flow (U), force (F), magnetic field (B), current density (J), vorticity (!) • Tensors: stress (τ), strain rate (ε), rotation (R), identity (I) How to work and operate with tensors… 2 Vector Identities VECTOR IDENTITIES4 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) (f g ) = (7) · (f A) = f ·A+A· (8) × (f A) = f ×A+ (9) · (A × B) = B · ×A−A· (10) × (A × B) = A( · B) − B( (11) A × ( (gf ) = f g+g f f f ×A ×B · A) + (B · × B) = ( B) · A − (A · (12) (A · B) = A × ( (13) 2 f= · (14) A= ( · A) − )B )B f 2 )A − (A · × B) + B × ( × × A) + (A · )B + (B · )A ×A Let r = ix + jy + kz be the radius vector of magnitude r, from the origin to (15) the point x,× z . Then y, f = 0 (16) · r ·= 3 A = 0 × (21) If e (22) 1 , e2 ,r e3 0 orthonormal unit vectors, a second-order tensor T can be × = are written in the dyadic form (23) rT =r/r = (17) i,j 3 Tij ei ej (24) (1/r) coordinates the divergence of a tensor is a vector with components = −r/r 3 In cartesian Green’s Theorem Variants 3 (25) = ( ( Tji (18) · (r/r)i)= 4πδ∂r) /∂ xj ) ( ·T j (26) definition is required for consistency with Eq. (29)]. In general [This r = I If V is a volume ) = ( · Aby + (A · )B and dS = ndS , where n is the unit (19) · (AB enclosed )B a surface S normal outward from V, (20) · (f T ) = f ·T +f ·T dV (27) f= V dV (28) ·A= V dV (29) V (30) ·T = dV V V (32) V 4 S S ×A= dV (f (31) dSf S dV (A · 2 g−g × dS · A d S ·T S dS × A 2 f) = S dS · (f ×B−B· = S dS · (B × × g − g f) × A) ×A−A× × B) If S is an open surface bounded by the contour C , of which the line element is dl, 4 2 dV (f (31) V (32) dV (A · g−g × 2 f) = S dS · (f ×B−B· g − g f) × × A) Stokes’ Theorem Variants V dS · (B × = S ×A−A× × B) If S is an open surface bounded by the contour C , of which the line element is dl, (33) S (34) S (35) S (36) S dS × dS · f= dlf C ×A= (dS × C )×A= dS · ( f × dl · A 5 dl × A C g) = C f dg = − gdf C DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 5 Cylindrical Coordinates Divergence What is a Tensor? ·A= Gradient ( f )r = 1∂ ∂ Az 1 ∂ Aφ (rAr ) + + r ∂r r ∂φ ∂z ∂f ; ∂r ( f )φ = 1 ∂f ; r ∂φ ( f )z = ∂f ∂z Curl ( × A)r = ∂ Aφ 1 ∂ Az − r ∂φ ∂z ( × A)φ = ∂ Az ∂ Ar − ∂z ∂r ( × A)z = 1∂ 1 ∂ Ar (rAφ ) − r ∂r r ∂φ In other words: a vector has a direction and magnitude … Laplacian 2 f= 1∂ r ∂r r ∂f ∂r + 1 ∂2f ∂2f + r2 ∂φ2 ∂ z2 6 6 Rotation about the z-Axis 7 Rotation Matrix is Orthogonal 8 Vector Examples 9 Scalar Product 10 Vector Product 11 Permutation Tensor 12 CoPlanar and Not 13 Triple Scalar Product 14 εijk Identity 15 Triple Vector Product 16 Rigid Rotation 17 Tensors 18 Symmetric and Antisymmetric 19 Antisymmetric Tensor 20 Time Derivative 21 Vector Fields and Trajectory Lines 22 Gradient Operator (Scalar) 23 Divergence of a Vector Field 24 Laplacian 25 Green’s (or Gauss’) Theorem 26 Curl of a Vector Field 27 Stokes’ Theorem 28 Classification of Vector Fields 29 Irrotational Field 30 Solenoidal Field 31 Summary • Vectors & tensors transform under coordinate rotation like position vector • Vector operators: scalar product, vector product, triple scalar product, triple vector product • Tensors: isotropic, symmetric, antisymmetric, orthogonal • Calculus of vectors: derivative, gradient, divergence, curl • Gauss’ & Stokes’ Theorems • Classification of Vector Fields • Next Lecture: Kinematics of fluids 32 ...
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