Page 296 - Intro to Tensor Calculus

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and are given the physical interpretation of an internal force per unit volume. These internal forces arise
~
from the shearing stresses in the moving fluid. If f viscous is zero the vector equation in (2.5.28) is called
Euler’s equation.
If the viscosity coefficients are nonconstant, then the Navier-Stokes equations can be written in the
Cartesian form

∂v i ∂v i ∂ ∂v k ∂v i ∂v j
∗
%[ + v j ]=%b i + −pδ ij + λ δ ij + µ ∗ +
∂t ∂x j ∂x j ∂x k ∂x j ∂x i

∂p ∂ ∂v k ∂ ∂v i ∂v j
=%b i − + λ ∗ + j µ ∗ +
∂x i ∂x i ∂x k ∂x ∂x j ∂x i
2
which can also be written in terms of the bulk coefficient of viscosity ζ = λ + µ as
∗
∗
3

∂v i ∂v i ∂p ∂ 2 ∂v k ∂ ∂v i ∂v j
%[ + v j ]=%b i − + (ζ − µ ) + µ ∗ +
∗
∂t ∂x j ∂x i ∂x i 3 ∂x k ∂x j ∂x j ∂x i

∂p ∂ ∂v k ∂ ∂v i ∂v j 2 ∂v k
=%b i − + ζ + j µ ∗ + − δ ij
∂x i ∂x i ∂x k ∂x ∂x j ∂x i 3 ∂x k
These equations form the basics of viscous flow theory.
2
In the case of orthogonal coordinates, where g (i)(i) = h (no summation) and g ij =0 for i 6= j, general
i
expressions for the Navier-Stokes equations in terms of the physical components v(1),v(2),v(3) are:
Navier-Stokes-Duhem equations for compressible fluid in terms of physical components: (i 6= j 6= k)

h
∂v(i) v(1) ∂v(i) v(2) ∂v(i) v(3) ∂v(i)
% + + +
∂t h 1 ∂x 1 h 2 ∂x 2 h 3 ∂x 3

v(j) ∂h j ∂h i v(k) ∂h i ∂h k
− v(j) − v(i) + v(i) − v(k) =
h i h j ∂x i ∂x j h i h k ∂x k ∂x i

b(i) 1 ∂p 1 ∂ µ ∗ h j ∂ v(j) h i ∂ v(i) ∂h i
% − + λ ∇· V ~ + +
∗
h i h i ∂x i h i ∂x i h i h j h i ∂x i h j h j ∂x j h i ∂h j

µ ∗ h h i ∂ v(i) h k ∂ v(k) i ∂h i 2µ ∗ 1 ∂v(j) v(k) ∂h j v(i) ∂h j
+ + − + +
h i h k h k ∂x k h i h i ∂x i h k ∂x k h i h j h j ∂x j h j h k ∂x k h i h j ∂x i

2µ ∗ 1 ∂v(k) v(i) ∂h k v(k) ∂h k ∂h k 1 ∂ 1 ∂v(i) v(j) ∂h i v(k) ∂h i
∗
− + + + 2µ h j h k + +
h i h k h k ∂x k h i h k ∂x i h k h j ∂x i ∂x i h i h j h k ∂x i h i ∂x i h i h j ∂h j h i h k ∂x k

∂ h j ∂ v(j) h i ∂ v(i) ∂ n h i ∂ v(i) h k ∂ v(k) o
+ µ h i h k + + µ h i h j +
∗
∗
∂x j h i ∂x i h j h j ∂x j h i ∂x k h k ∂x k h i h i ∂x i h k
(2.5.31)
where ∇· ~v is found in equation (2.1.4).
In the above equation, cyclic values are assigned to i, j and k. That is, for the x 1 components assign
the values i =1,j =2,k =3; for the x 2 components assign the values i =2,j =3,k = 1; and for the x 3
components assign the values i =3,j =1,k =2.
The tables 5.2, 5.3 and 5.4 show the expanded form of the Navier-Stokes equations in Cartesian, cylin-
drical and spherical coordinates respectively.