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62 2. Conservation Equations

and
(i) « ^fe)"
M <1 1 (2A27b)
l l •« \ ^ n »,2
it can be expressed in the linearized form (see Problem 2.5)

<l-*Og g-0 (,4, )
8
+

2.4.2 Stokes Flow

A second simplification of the Navier-Stokes equations arises when the inertia
terms are small enough to neglect. This situation arises when the Reynolds
number R^ is much less than unity, because the velocity u e is very small, or
the scale of the flow L is very small, or the fluid is very viscous. In this case,
the resulting equations are known as the Stokes equations, written in tensor
notation as
V V = ^ (2.4.29)

These linear equations provide a good approximation to flows with Reynolds
numbers less than unity such as in some lubrication problems; for simple bound-
aries they can be solved analytically, as discussed by Schlichting [9].

2.4.3 Boundary Layers

A third simplification of the Navier-Stokes equations occurs when (6/L <C 1
strictly, dS/dx <C 1, d6/dz < 1 ) ; this includes both laminar and turbulent flows
at high Reynolds numbers. In this case, the continuity equation remains the
same as Eq. (2.2.1), but the ^/-compcnent of the momentum equation is elim-
inated and the pressure p is assumed only to be a function of x and z. For
three-dimensional incompressible laminar and turbulent flow, the momentum
and energy equations can be written as
2
Du dp d u d ,—7—7, „ ,^ 4 _ x
e = +tt iur/)+efx (2A30)
Di -£ w-%
2
Dw dp d w d —7-r N r , n . 0 ^ x
/
2
DT . ,d T „ d
QC V— = q w + k - ^ - QC p-T'v' (2.4.32)
However, unlike the thin-layer Navier-Stokes equations given by Eqs. (2.4.4)
to (2.4.6) and the Stokes equations, Eq. (2.4.29), the pressure is not computed as
part of the solution but is specified in the solution procedure. As will be shown
in Section 2.6, while the Navier-Stokes equations for incompressible flows are