Introduction to Tensor Index Notation in Fluid Mechanics, Lecture notes of Fluid Mechanics

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Introduction to Tensor Index Notation

AAE 511 Introduction to Fluid Mechanics

Jonathan Poggie Fall 2020

School of Aeronautics and Astronautics Purdue University

Copyright

This presentation contains material copyrighted by the instructor and by other authors. These materials are intended only for personal educational use in the context of this class.

Gradient, Divergence, and Curl

See references for more information

For reference see Appendix A of Currie (2013), Chapter 3 of Panton (1984), or Aris (1962)

The divergence of a vector field is a scalar field

∇·v = ∂v^1 ∂x 1

• ∂v^2 ∂x 2

• ∂v^3 ∂x 3

Griffiths (1999), copyrighted

The curl of a vector field is a vector field

• c 1 = ∂v c = ∇×v
  • ∂x
    • − ∂v
      • ∂x
• c 2 = ∂v
  • ∂x
    • − ∂v
      • ∂x
• c 3 = ∂v
  • ∂x
    • − ∂v
      • ∂x

Curl requires the permutation tensor

The curl is:

ω = ∇×v

ωi = ijk^ ∂vk ∂xj Think of the curl as an operator:

ωi =

[

ijk^ ∂ ∂xj

]

vk

Identities with the gradient become easy in index notation

These are easy to prove using index notation, symmetry, and the properties of the permutation tensor:

∇(fg ) = f ∇g + g ∇f (1) ∇(A·B) = A×(∇×B) + B ×(∇×A) + (A·∇)B + (B ·∇)A (2)

Treat ijk and δij as constants in taking derivatives.

Identities with curl become easy in index notation

These are easy to prove using index notation, symmetry, and the properties of the permutation tensor:

∇×(f A) = f ∇×A − A×∇f (5) ∇×(A×B) = (B ·∇)A − (A·∇)B + A ∇·B − B ∇·A (6)

Second derivative identities become easy in index notation

These are easy to prove using index notation, symmetry, and the properties of the permutation tensor:

∇·(∇×v ) = 0 (7) ∇×∇φ = 0 (8) ∇×(∇×v ) = ∇(∇·v ) − ∇^2 v (9)

Example: simplify the curl of the curl

Using index notation, show that:

∇×(∇×v ) = ∇(∇·v ) − ∇^2 v

This is very useful in vorticity derivations.

Coordinate Transformations

Rotation is the only transformation in Cartesian coordinates

The matrix of direction cosines is Cij = cos θij , where θij is the angle between the x i′ -axis and the xj -axis. Some examples of tensor transformations are:

p′^ = p v (^) i′ = Cij vj τ (^) ij′ = Cik Cjl τkl T (^) ijk′ ...m = Cis Cjt Cku · · · Cmv Tstu...v

Some quantities transform slightly differently

A pseudotensor (or relative tensor) transforms according to:

T (^) ijk′...m = |C |Cis Cjt Cku · · · Cmv Tstu...v

where |C | = det C is the determinant. The vorticity (curl of velocity) is a pseudotensor.