CEE 262A H YDRODYNAMICS

1. CEE 262A HYDRODYNAMICS Lecture 2 Mathematical preliminaries and tensor analysis

2. Unit vectors Position vector Where are components of Right-handed, Cartesian coordinate system

3. Vector: Kundu- “…Any quantity whose components change like the components of a position vector under the rotation of the coord. system.” Scalar: Any quantity that does NOT change with rotation or translation of the coord. system e.g. density (r) or temperature (T)

4. e.g. Rows Columns Tensor: Assigns a vector to each direction in space ( 2nd order) • Isotropic – Components are unchanged by a rotation of frame of reference (i.e. independent of direction - e.g. “Kronecker Delta dij”) • Symmetric : Aij = Aji (in general Aij = ATji) • Anti-symmetric: Aij = -Aji • Useful result: Aij = 1/2 (Aij+Aij)+1/2 (Aji-Aji) • = 1/2 (Aij+Aji)+1/2 (Aij-Aji) • = Symmetric+ Anti-symmetric

5. * Result is a scalar quantity When a summation occurs over a repeated index contraction Einstein summation convention A) If an index occurs twice in a term a summation over the repeated index is implied e.g. B) Higher-order tensors can be formed by multiplying lower order tensors: a) If Ui and Vj are 1st-order tensors then their product Ui Vj = Wij is a 2nd-order tensor. Also known as vector outer product ( ). b) If Aij and Bkl are 2nd-order tensors then their product Aij Bkl = Pijkl is a 4th-order tensor.

6. C) Lower-order tensors are obtained from contractions (a) Contraction of two 2nd-order tensors (b) Tensor multiplied by a vector (c) Double-contraction of two 2nd-order tensors

7. D) Kronecker delta =1 if i=j =0 otherwise Isotropic tensor of 2nd order Expand: If *

8. E) Alternating tensor (c) Epsilon - Delta Relation e e = d d - d d ijk klm il jm im jl = 1 ijk in cyclic order e.g. 123,231,312 = -1 ijk in anticyclic order e.g. 321,132,213 = 0 if any two indices are equal FLOI = first(il)last (jm) - outer(im)inner(jl)

9. Vector . Vector = Scalar Basic vector operations A) Dot Product (“Inner” Product) * “Magnitude of one vector times component of other in direction of first vector” implies

10. e.g. C) Cross product “…is the vector, , whose magnitude is , and whose direction is perpendicular to the plane formed by and such that form a right-handed system” W ～ B) Outer product ( )

11. Cross-product rules (a) (b) (c) Now since If:

12. “…( ) is perpendicular to lines and gives magnitude and direction of maximum spatial rate of change of ” The “Del” ( ) operator A) Gradient – “Grad”: increase tensor order Vector If we apply  to a vector, we produce a second-order tensor

13. [ Scalar ] e.g. [ Vector ] B) Divergence – “div”: Reduce tensor order Our application will be to the divergence of a flux of various quantities.

14. C) Curl e.g. i=1: If j=1 or k=1

15. If is a scalar Now But Important div/grad/curl identities (a) (b) (c) -1 +1

16. (d) “Curl of a vector is non divergent” (see above) But: e e = d d - d d ijk klm il jm im jl (e)

17. 1 if 1 if 1 if 1 if We will make good use of this result!

18. (outward unit normal vector to surface element) (infinitesimal surface area) (infinitesimal volume) ò ò “ … Relates a vo lume integral to a surface integral ” V A Integral theorems A) Gauss’ Theorem

19. If is a scalar, vector, or any order tensor Specifically, if is a vector or “Divergence Theorem”: Integral over volume of divergence of flux = integral over surface of the flux itself

20. Examples… (a) (b) (c) or Divergence of flux within volume = Net flux across

21. A (open surface) C (bounding curve) B) Stokes' Theorem