Basics of Fluid Mechanics, 2014a

8.7. EXAMPLES FOR DIFFERENTIAL EQUATION (NAVIER-STOKES) 259
If the fluid was solid material, pulling the side
will pull all the material. In fluid (mostly liquid)
shear stress pulling side (surface) will have limited
effect and yet sometime is significant and more
rarely dominate. Consider, for example, the case
shown in Figure 8.14. The shear stress carry the
material as if part of the material was a solid material.
For example, in the kerosene lamp the burning
occurs at the surface of the lamp top and the liquid
is at the bottom. The liquid does not move up due
the gravity (actually it is against the gravity) but
because the surface tension.
The physical conditions in Figure 8.14 are
used to idealize the flow around an inner rode to
understand how to apply the surface tension to the
boundary conditions. The fluid surrounds the rode
and flows upwards. In that case, the velocity at the
surface of the inner rode is zero. The velocity at
the outer surface is unknown. The boundary condition
at outer surface given by a jump of the shear
stress. The outer diameter is depends on the surface
tension (the larger surface tension the smaller
the liquid diameter). The surface tension is a function
of the temperature therefore the gradient in
surface tension is result of temperature gradient.
Fig. -8.14. Kerosene lamp.
U(r i )=0
μ ∂U
∂r = ∂σ
∂h
} temperature
gradent
mix zone
}
constant
} T
Fig. -8.15. Flow in a kendle with a
surfece tension gradient.
In this book, this effect is not discussed. However, somewhere downstream the temperature
gradient is insignificant. Even in that case, the surface tension gradient remains.
It can be noticed that, under the assumption presented here, there are two principal
radii of the flow. One radius toward the center of the rode while the other radius is
infinite (approximatly). In that case, the contribution due to the curvature is zero in
the direction of the flow (see Figure 8.15). The only (almost) propelling source of the
flow is the surface gradient ( ∂σ
∂n ).
8.7 Examples for Differential Equation (Navier-Stokes)
Examples of an one-dimensional flow driven by the shear stress and pressure are presented.
For further enhance the understanding some of the derivations are repeated.
First, example dealing with one phase are present. Later, examples with two phase are
presented.
Example 8.6:
Incompressible liquid flows between two infinite plates from the left to the right (as
shown in Figure 8.16). The distance between the plates is l. The static pressure per