Page 288 - Intro to Tensor Calculus

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§2.5 CONTINUUM MECHANICS (FLUIDS)

Let us consider a fluid medium and use Cartesian tensors to derive the mathematical equations that
describe how a fluid behaves. A fluid continuum, like a solid continuum, is characterized by equations
describing:
1. Conservation of linear momentum
σ ij,j + %b i = %˙v i (2.5.1)

2. Conservation of angular momentum σ ij = σ ji .
3. Conservation of mass (continuity equation)
∂% ∂% ∂v i D%
~
+ v i + % =0 or + %∇· V =0. (2.5.2)
∂t ∂x i ∂x i Dt
In the above equations v i ,i =1, 2, 3 is a velocity field, % is the density of the fluid, σ ij is the stress tensor
and b j is an external force per unit mass. In the cgs system of units of measurement, the above quantities
have dimensions
2 2 3
[˙v j ]= cm/sec , [b j ]= dynes/g, [σ ij ]= dyne/cm , [%]= g/cm . (2.5.3)

The displacement field u i ,i =1, 2, 3 can be represented in terms of the velocity field v i ,i =1, 2, 3, by
the relation
Z t
u i = v i dt. (2.5.4)
0
The strain tensor components of the medium can then be represented in terms of the velocity field as

Z t Z t
1 1
e ij = (u i,j + u j,i )= (v i,j + v j,i ) dt = D ij dt, (2.5.5)
2 0 2 0
where
1
D ij = (v i,j + v j,i ) (2.5.6)
2
is called the rate of deformation tensor , velocity strain tensor,or rate of strain tensor.
Note the difference in the equations describing a solid continuum compared with those for a fluid
continuum. In describing a solid continuum we were primarily interested in calculating the displacement
field u i ,i =1, 2, 3 when the continuum was subjected to external forces. In describing a fluid medium, we
calculate the velocity field v i ,i =1, 2, 3 when the continuum is subjected to external forces. We therefore
replace the strain tensor relations by the velocity strain tensor relations in all future considerations concerning
the study of fluid motion.

Constitutive Equations for Fluids

In addition to the above basic equations, we will need a set of constitutive equations which describe the
material properties of the fluid. Toward this purpose consider an arbitrary point within the fluid medium
and pass an imaginary plane through the point. The orientation of the plane is determined by a unit normal
(n)
n i , i =1, 2, 3 to the planar surface. For a fluid at rest we wish to determine the stress vector t acting
i
(n)
on the plane element passing through the selected point P. We desire to express t in terms of the stress
i
tensor σ ij . The superscript (n) on the stress vector is to remind you that the stress acting on the planar
element depends upon the orientation of the plane through the point.