CONTINUUM MECHANICS for ENGINEERS

This system, Eqs 7.2-1 through 7.2-7, together with the definition of the rate
of deformation tensor,
( )
1
D = v + v
2
ij i, j j, i
(7.2-8)
represents 22 equations in the 22 unknowns, σ ij, ρ, v i, D ij, u, q i, p, and θ. If
thermal effects are neglected and a purely mechanical problem is proposed,
we need only Eqs 7.2-1 through 7.2-3 as well as Eq 7.2-8 and a temperature
independent form of Eq 7.2-5, which we state as
p= p(
) ρ
(7.2-9)
This provides a system of 17 equations in the 17 unknowns, σ ij, ρ, v i, D ij, and p.
Certain of the above field equations may be combined to offer a more
compact formulation of viscous fluid problems. Thus, by substituting
Eq 7.2-3 into Eq 7.2-2 and making use of the definition Eq 7.2-8, we obtain
( ) +
˙ i i , i j, ji i, jj
* *
*
ρv = ρb − p + λ + µ v µ v
(7.2-10)
which are known as the Navier-Stokes equations for fluids. These equations,
along with Eqs 7.2-4, 7.2-5, and 7.2-6, provide a system of seven equations
for the seven unknowns, v i, ρ, p, u, and θ. Notice that even though Eq 7.2-3
is a linear constitutive equation, the Navier-Stokes equations are nonlinear
because in the Eulerian formulation
˙v
i
∂ vi
= + vv
∂ t
If Stokes condition
⎛ * 2 * ⎞
λ =− µ is assumed, Eq 7.2-10 reduces to the form
⎝ 3 ⎠
(7.2-11)
Also, if the kinetic equation of state has the form of Eq 7.2-9, the Navier-
Stokes equations along with the continuity equation form a complete set of
four equations in the four unknowns, v i and ρ.
In all of the various formulations for viscous fluid problems stated above,
the solutions must satisfy the appropriate field equations as well as boundary
and initial conditions on both traction and velocity components. The boundary
conditions at a fixed surface require not only the normal, but also the
tangential component of velocity to vanish because of the “boundary layer”
effect of viscous fluids. It should also be pointed out that the formulations
j i,j
1 *
ρv˙ i = ρ bi − p,
i + µ (vj,ji + 3vi,jj)
3