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§ 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
σij,j + %bi = %v˙i ()
2. Conservation of angular momentum σij = σji.
3. Conservation of mass (continuity equation)
∂% ∂% ∂vi D%
+ vi + % =0 or + %∇·V~ =0. ()
∂t ∂xi ∂xi Dt
In the above equations vi,i=1, 2, 3 is a velocity field, % is the density of the fluid, σij is the stress tensor
and bj is an external force per unit mass. In the cgs system of units of measurement, the above quantities
have dimensions
2 2 3
[˙vj]=cm/sec , [bj]=dynes/g, [σij ]=dyne/cm , [%]=g/cm . ()
The displacement field ui,i =1, 2, 3 can be represented in terms of the velocity field vi,i =1, 2, 3, by
the relation Z
t
ui = vi dt. ()
0
The strain ponents of the medium can then be represented in terms of the velocity field as
Z Z
1 t 1 t
eij = (ui,j + uj,i)= (vi,j + vj,i) dt = Dij dt, ()
2 0 2 0
where
1
D = (v + v )()
ij 2 i,j j,i
is called the rate of deformation tensor , velocity strain tensor,orrate of strain tensor.
Note the difference in the equations describing a solid pared with those for a fluid
continuum. In describing a solid continuum we were primarily interested in calculating the displacement
field ui,i =1, 2, 3 when the continuum was subjected to external forces. In describing a fluid medium, we
calculate the velocity field vi,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 flu