Page 411 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 395

so that

.2
But, from Eq. (6.20.4) , f = 0. Therefore, the terms involving viscosity in the Navier-
n
BxjdXj
Stokes equation drop out in the case of irrotational flows so that the equations take the same
form as the Euler's equation for an inviscid fluid. Thus, if the viscous fluid has homogeneous
tne
density and if the body forces are conservative (i.e., B/ = ~"^~~)> results of the last sections
ax;
show that irrotational flows are dynamically possible also for a viscous fluid. However, in any
physical problems, there are always solid boundaries. A viscous fluid adheres to the boundary
so that both the tangential and the normal components of the fluid velocity at the boundary
should be those of the boundary. This means that both velocity components at the boundary
are to be prescribed. For example, if v = 0 is a solid boundary at rest, then on the boundary,
the tangential components, v x ~v z = 0, and the normal components v y = 0. For irrotational
flow, the conditions to be prescribed for <p on the boundary are <f> = constant aty = 0 (so that
d<p
v = v
x z ~ 0) and -^- = 0 at y - 0. But it is known (e.g., see Example 6.18.1, or from the
potential theory) that in general there does not exist solution of the Laplace equation satisfying
both the conditions <p - constant and V#> • n = -*- = 0 on the complete boundaries. There-
fore, unless the motion of solid boundaries happens to be consistent with the requirements of
irrotationality, vorticity will be generated on the boundary and diffuse into the flow field
according to vorticity equations to be derived in the next section. However, in certain
problems under suitable conditions, the vorticity generated by the solid boundaries is confined
to a thin layer of fluid in the vicinity of the boundary so that outside of the layer the flow is
irrotational if it originated from a state of irrotationality. We shall have more to say about this
in the next two sections.

Example 6.22.1

For the Couette flow between two coaxial infinitely long cylinders, how should the ratio of
the angular velocities of the two cylinders be, so that the viscous fluid will be having irrotational
flow?
Solution. From Example 19.2 of Section 6.19, the only nonzero vorticity component in the
Couette flow is

where Q, denotes the angular velocities.If fi^-Qiri = 0, the flow is irrotational. Thus,