Page 295 - Intro to Tensor Calculus

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In the cgs system of units the above quantities have the following units of measurements in Cartesian
coordinates
v i is the velocity field ,i =1, 2, 3, [v i ]= cm/sec
is the stress tensor,i, j =1, 2, 3, [σ ij ] = dyne/cm 2
σ ij
% is the fluid density [%]= gm/cm 3
i
b i is the external body forces per unit mass [b ] = dyne/gm
is the rate of deformation tensor [D ij ]= sec −1
D ij
p is the pressure [p] = dyne/cm 2

λ ,µ ∗ are coefficients of viscosity [λ ]= [µ ]= Poise
∗
∗
∗
where 1 Poise = 1gm/cm sec
If we assume the external body forces per unit mass are known, then the equations (2.5.24), (2.5.25),
(2.5.26), and (2.5.27) represent 16 equations in the 16 unknowns

%, v 1 ,v 2 ,v 3 ,σ 11 ,σ 12 ,σ 13 ,σ 22 ,σ 23 ,σ 33 ,D 11 ,D 12 ,D 13 ,D 22 ,D 23,D 33 .

Navier-Stokes-Duhem Equations of Fluid Motion

Substituting the stress tensor from equation (2.5.27) into the linear momentum equation (2.5.25), and
assuming that the viscosity coefficients are constants, we obtain the Navier-Stokes-Duhem equations for fluid
motion. In Cartesian coordinates these equations can be represented in any of the equivalent forms

∗ ∗ ∗
%˙v i = %b i − p ,j δ ij +(λ + µ )v k,ki + µ v i,jj
∂v i
% + %v j v i,j = %b i +(−pδ ij + τ ij ) ,j
∂t
(2.5.28)
∂%v i
+(%v i v j + pδ ij − τ ij ) ,j = %b i
∂t
D~v 2
~
∗
∗
∗
% = %b −∇ p +(λ + µ )∇ (∇· ~v)+ µ ∇ ~v
Dt
D~v ∂~v
where = +(~v ·∇) ~v is the material derivative, substantial derivative or convective derivative. This
Dt ∂t
derivative is represented as
∂v i ∂v i dx j ∂v i ∂v i j ∂v i j
˙ v i = + = + v = + v i,j v . (2.5.29)
∂t ∂x dt ∂t ∂x j ∂t
j
In the vector form of equations (2.5.28), the terms on the right-hand side of the equation represent force
~
terms. The term %b represents external body forces per unit volume. If these forces are derivable from a
potential function φ, then the external forces are conservative and can be represented in the form −%∇ φ.
The term −∇ p is the gradient of the pressure and represents a force per unit volume due to hydrostatic
pressure. The above statement is verified in the exercises that follow this section. The remaining terms can
be written
~
2
∗
f viscous =(λ + µ )∇ (∇· ~v)+ µ ∇ ~v (2.5.30)
∗
∗