Handout on Dimensional Analysis 2

Fundamental Mathematics for Fluid Mechanics Herein A, B, C, and D are vectors, and f and g are scalar quantites. 1 Coordinate Systems and Transformations • Cartesian rP = xex + yey + zez = xi + yj + zk (1) (2) • Cylindrical rP = rc = θ = x = y er eθ = = = rer + θeθ + zez � x2 + y 2 y tan−1 x rc cos θ rc sin θ cos θ ex + sin θ ej − sin θ ex + cos θ ej (3) (4) (5) (6) (7) (8) (9) • Spherical = rP rs rer + θeθ + φeφ � = x2 + y 2 + z 2 (10) (11) (12) • Earth Centered Coordinates 2 Operators • Gradient: grad(f ) = �f = ∂f ∂f ∂f ex + ey + ez ∂x ∂y ∂z (Cartesian) (13) Figure 1: Coordinates and transformations between Cartesian and cylindrical (left panel) and Cartesian and spherical (right panel). 1 = = = ∂f 1 ∂f ∂f er + eθ + ez ∂r r ∂θ ∂z (Spherical) (Cylindrical) (14) (15) (Earth) (16) • Divergence: div(f ) = � · A = = = = ∂Ax ∂Az ∂Ay + + (Cartesian) ∂x ∂y ∂z 1 ∂(rAr ) 1 ∂Aθ ∂Az + + (Cylindrical) r ∂r r ∂θ ∂z (Spherical) (Earth) (17) (18) (19) (20) • Curl: curl(f ) = � × A = = � � � e x e y ez � � � � ∂ ∂ ∂ � � ∂x ∂y ∂x � � � � Ax Ay Az � � e � ez � � r eθ r � � r � ∂ ∂ ∂ � � ∂r ∂θ ∂z �� � � Ar rAθ Az � = = (Cartesian) (Cylindrical) (Spherical) (Earth) (21) (22) (23) (24) • Laplacian: �2 f = = = = 3 3.1 ∂2f ∂2f ∂2f + + (Cartesian) ∂x2 ∂y 2 ∂y 2 � � ∂f ∂2f 1 ∂ 1 ∂2f r + 2 2 + 2 (Cylindrical) ∂z r ∂r ∂r r ∂θ (Spherical) (Earth) (25) (26) (27) (28) Vector Identities Vector Products A·B = B·A A × B = −B × A = B · (C × A) = C · (A × B) � � Ax Ay Az � = �� Bx By Bz � Cx Cy Cz (29) (30) (31) (32) A · (B × C) � � � � � � A × (B × C) = B(A · C) − C(A · B) (A × B) · (C × D) = (A · C)(B · D) − (B · C)(A · D) 2 (33) (34) (35) 3.2 Differentiation with Respect to a Scalar Herein, A = A(x), B = B(x), and f = f (x). d dA dB + (A + B) = dx dx dx d dA df (f A) = f +A dx dx dx d dB dA (A · B) = A · +B· dx dx dx d dB dA (A × B) = A × + ×B dx dx dx 3.3 (37) (38) (39) Identities with the � Operator �(f g) = g�f + f �g (A · �)f = A · �f (A × �)f = A × (�f ) � � � · An �· An = n �× � (40) (41) (42) (43) n An = n � � × An (44) n (A × �) · B � · fA � × fA � · (A × B) � × (A × B) �(A · B) = = = = = = (A · �)A = � × (�f ) = � · (� × A) = � × (� × A) = 4 (36) A · (� × B) f (� · A) + A · �f f (� × A) + A × (�f ) B · (� × A) − A · (� × B) A(� · B) + (B · �)A − B(� · A) − (A · �)B (A · �)B + (B · �)A + A × (� × B) + B × (� × A) 1 �A2 − A × (� × A) 2 0 0 �(� · A) − �2 A (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) Integration Theorems • Gauss’ theorem or divergence theorem. Herein, F is a vector function, V is any volume, and A is the area that encloses the volume: � � � · F dV (55) F · en dA = A V • Stokes’ theorem. Herein, F is a vector function and A is any area with s the line which bounds it and en normal to the area: � � F · ds = (� × F) · en dA (56) A 3 5 Notes • All identities should be cross­referenced in a couple other books. • References to proofs of some or all of the identities. • Introduction, description? • What are core Fluid mechanics texts, including and excluding GFD. • Assuming mixed partials equal? • Spherical, Earth coordinates, description (notation, order). • grad() versus �. • Align text to right of equations. • Quick references. • Include long versions of curl() operator? • Script version of Laplacian operator, i.e. curl(). • Expand to include things such as Taylor series, Euler’s formula, hyperbolic function defini­ tions • Finish spherical and earth centered coordinates. Look up common notation. • Define volume integrals in different coordinate systems? References [1] Granger, Robert A. “Fluid Mechanics.” Dover Publications, Inc. 1995. 4