CREEPING FLOW
(7.3- 16) The boundary conditions for are
3
at at (7.3-17) (7.3-27)
(7.3-16) and applying these boundary conditions gives
radial velocity depends on r through (which remains to be determined) The applied force can be related now to the velocity and the half-spacing
on t through both and Ignoring any gravitational or buoyancy contributions, the force applied to either plate is given by
The pressure gradient and are evaluated by using the continuity equation,
1 (7.3-29)
-
(7.3-19)leads to an expression for
z
I
a
(7.3-20)
0
know how and therefore that the boundary conditions
at (7.3-21)
Assuming that t) we assume that the
a 9
(7.3-22)
(7.3-30) Using (7.3-18) and
For a constant force the maximum value of is obtained at = so that may be interpreted 1 z 3
(7.3-23) as the initial disk velocity.
f by evaluating
gives
3
7.4 STREAM FUNCTION SOLUTIONS
--
-
2
tude. One of these analyses leads to Stokes' law for the drag on a sphere. In such problems the stream function (Section 5.9) provides an effective approach for obtaining a solution. In addition to the examples, general solutions of the stream-function form of Stokes' equation are discussed briefly.
To determine the stream function we must solve with these boundary conditions. Once is known, the velocity components are calculated from (7.4-1).
Examining the boundary conditions in and (7.4-7), and in particular the no-slip condition at the moving surface, it appears that the solution may be of the form
(7.4-8) Example 7.4-1 Flow Near a Corner Consider the situation depicted in Fig. 7-3. in which a
fluid is bounded by two planar, solid surfaces which meet at an angle One surface slides past the other at a velocity Both surfaces are assumed to extend indefinitely. A flow like this might occur, for example, near the opening of a die in a coating or extrusion process. There is no fixed length scale in this problem, which suggests that the distance from the comer (r) be used in estimating the relative importance of and viscous effects. In other words, the pertinent Reynolds number is and Stokes' equation will be valid close to the comer. It is desired to characterize the Row in this region. This problem is one of involving sharp corners which are discussed in Moffat (1964).
The stream function, for a planar flow in cylindrical coordinates defined as (Table 5-11)
Substituting (7.4-8) in it is found that all terms containing r factor out, leaving
Expressed in terms of the conditions become
This ability to obtain a problem for which is consistent with Stokes' equation and all bound-
:
conditions confirms that the assumed form of the solution correct.Equation (7.4-9) is solved by rewriting it as two coupled, second-order differential equa- tions involving and another function such that
From Eq. (7.2-6), the form of Stokes' equation is simply
, After solving (7.4-12) for g, (7.4-1 1) becomes The no-slip and no-penetration conditions at the solid surfaces, and a, are ex-
pressed as
a cos
+
sinwhere a and b are constants. The general solution for this nonhomogeneous equation is
cos 14)
Using Eq. these boundary conditions are written in terms of as
where and A and B are additional constants. Using (7.4-10) to evaluate the four constants in the stream function is found to be
= sin - (sinZ - (a - sin cos sin 15) The velocity components for any angle a can be obtained now, if desired, by using
in (7.4-1).
For perpendicular surfaces (a = 15) simplifies to Figure Flow near a comer due to a moving
surface.
The velocity components for this special case are
Surprisingly, neither nor depends on this is true for any angle a. Another distinctive aspect of this problem is that and both vary as and therefore are singular at Although the stresses singular at r =0, the force on an imaginary cylindrical surface of radius r remains finite in the limit That is, the surface area varies as and the stresses as so that their product is finite. This illustrates the fact that an infinite stress at a point is not necessarily
tic, although an infinite force is not allowed.
Example 7.4-2 Flow Past a Solid Sphere The slow motion of a solid sphere relative to a stagnant, viscous fluid, first analyzed by Stokes is the most widely known problem in low Reynolds number hydrodynamics.' As shown in Fig. it is convenient to choose a reference frame in which the sphere (of radius R) is stationary and the far from the sphere is moving at a uniform velocity, = U = It is desired to determine the velocity, the pressure, and the drag on the sphere, assuming that Re = 1.
The stream function, for an flow in spherical coordinates is defined as (Table 5-11)
Yo=--
-sin
.
and the corresponding stream-function of Stokes' equation is
Figure around a stationary, solid sphere, with uniform velocity far from the sphere.
his career at Cambridge University, George Stokes made major and to many areas of science, including fluid mechanics, solid optics, use of (Parkinson. in Gillispie, 1976, pp. 74-79). He is known for his work Stokes credit (with Navier) for deriving the conservation equation for and was largely responsible for the concept of the no-slip boundary at a solid The paper in which he reported his of the drag on a sphere (Stokes is one of the in the history of fluid mechanics, in that it contains the first solutions to viscous flow problems.
by some 10 years first published of equation (Chapter 6).
sin 1
r2
ae
Equation (7.4-20) is similar to the biharmonic equation. but the operator defined in (7.4- 21) is not quite the same as the (V2) involving and 0.
In spherical 8) coordinates, the boundary conditions of no slip and no penetration at the sphere surface, and uniform flow far from the sphere, are expressed as
0) cos 0, 0) - sin (7.4-23)
The boundary conditions for the stream function at the sphere surface, corresponding to are
The stream-function conditions for the uniform flow far from the sphere, corresponding to Eq.
are
sin cos 0, (7.4-25)
Integrating (7.4-25) and (7.4-26) and comparing the results, we conclude that
sin2 (7.4-27)
, The absolute value of the stream function is arbitrary, so that in (7.4-27) the integra- tion constant was set equal to It will be seen this choice gives as the streamline corresponding to the sphere surface.
The required behavior of far from the sphere, as given by suggests that we seek a solution to (7.4-20) of the
= f(r)sin2 (7.4-28)
Substitution of (7.4-28) into (7.4-20) results in
From (7.4-24) the boundary conditions for f are
Much like the examples, the fact that all terms involving could be factored out to obtain and that all of the boundary conditions are satisfied using Eq.
confirms that (7.4-28) is
Equation (7.4-29) is another example of an equation, with solutions of the Substitution shows that can assume the values 4, 2, and