Singularity Induced Interior Stokes Flows

Three complex variable circle theorems for studying the two-dimensional Stokes flows interior to a circular cylinder are presented. These theorems are formulated in terms of the complex velocities of the fundamental singularities in an unbounded incompressible viscous fluid. Illustrative examples are given to demonstrate their usefulness.


Introduction
The solutions of fluid mechanical problems involving fundamental singularities in the presence of rigid boundaries are of considerable interest in practice. Except in a few simple cases the solution of such a problem involving an arbitrary boundary is still, in general exceedingly difficult. In the case of two-dimensional slow flow theory there is a complex variable circle theorem [1] for the solutions of Stokes flows due to singularities outside a circular cylinder which corresponds to Milne-Thomson's circle theorem [2,3] for potential flow outside the same cylinder, in the inviscid flow theory. Again it is notable that the same complex variable circle theorem can solve, in particular, some particular Stokes flow problems which cannot be done by the application of the real variable circle theorem [4,5]. Moreover, the "condition for zero perturbation velocity" referred to in the former theorem may suggest, in many cases, relatively easily the strengths or positions or both, of the singularities of the basic flow so that the viscous flows outside the circular boundary exist. Again the literature for Stokes flows in the region interior to a circular cylinder is not as wide as that for the Stokes flows outside the same cylinder; and its mathematical treatment is mostly known in terms of the real variables, nearly polar co-ordinates ( ) r, θ .
Notably, Ranger [6] and Meleshko and Arof [7] have independently solved the problem for a single rotlet within a circular cylinder. Sen [5] has pointed out a method to solve some problems of slow viscous fluid flow within a circular cylinder with the aid of his circle theorems for the flows outside a circular boundary. Chowdhury and Sen [8,9] have solved the problem of viscous fluid motion due to a Stokeslet within a circular cylinderical container. Daripa and Palaniappan [10] have extensively studied the singularity (rotlet or Stokeslet) driven Stokes flows interior and exterior to a circular cylinder. Here our object is to study Stokes flow interior to a circular cylinder in the light of the complex variable theory and to establish a number of complex variable circle theorems for slow viscous fluid motion within a circular cylinder in terms of the complex velocities of the fundamental singularities found in Chowdhury and Sen [11]; these theorems also correspond to the complex variable circle theorems for potential flow [12,13] in an inviscid fluid within the same cylinder.

The Interior Circle Theorems
In this section we attempt to study in some detail the twodimensional slow viscous flows interior to a circular cylinder with the help of the complex analysis and we establish three complex variable circle theorems for the determination of such flows. This is done by making use of the fundamental solutions of the Stokes equations presented somewhat differently in Chowdhury and Sen [11], Milne-Thomson [3] and Langlois [14] and these solutions governing generally the two-dimensional Stokes flows are the complex velocity and the stream function which are respectively where W(z) and ω (z) are arbitrary functions of z. Theorem 1. Let there be an irrotational two-dimensional flow in a viscous fluid with no rigid boundaries. Let the flow be represented by the complex velocity whose singularities are all in the region z a ≤ and let number. If a circular cylinder z a = is now inserted into flow field, the complex velocity and the stream function for the flow inside the cylinder becomes respectively and ( ) Proof. The proof consists in satisfying the following conditions.
(2) The same function satisfies the no-slip conditions on z a. = It is to be shown that 0 (z, z) * υ introduces no singularities in the region z a, where the terms, excluding the first one, constitute the image system exterior to the cylinder z a, = which thus consists of (1) a rotlet of strength, -ik, (2) a stresslet of strength  6) and (11) in the formula (4); but it is convenient to use the complex velocity (14) in the same formula (see the appendix) in order to visualize ψ as the algebraic form of the stream functions of the individual singularities referred in (14). Thus expressing the stream function ψ in terms of the polar co-ordinates ( ) r, θ one gets from which Ranger's [6] stream function for the flow pattern due to a rotlet within circular cylinder is at once recovered by choosing = ω − ω − ω be the complex velocity for a motion in an unbounded viscous fluid, the singularities of ( ) 0 z, z υ being all at a distance less than a from the origin. Let constant, α its complex conjugate and iλ a pure complex constant. If a circular cylinder z a = is now introduced into the flow field, the complex velocity ( ) z, z υ and the stream function ψ for the flow inside the cylinder become and ( ) Proof. The proof of the theorem is complete if the following four conditions are satisfied.
(2) The same expression satisfies the no-slip conditions on the boundary z a. =  (20) and (21) in the general complex velocity (1), we obtain the following particular complex velocity.
( ) Thus the condition (1) which implies that 0 (z, z) * υ has no singularities within the space z a ≤ and, in particular, gives finite velocity at the origin; this satisfies the above two conditions (3) and (4). Thus the proof of the theorem is complete. Example. A Stokeslet inside a circular cylinder Let the primary flow field in a viscous fluid be due to a Stokeslet of strength β at the point 0 z , where 0 z a; < and the corresponding complex velocity [11] is given by Ln z z z z 2 which may be expressed in the form and ( ) We then easily calculate that the complex functions (29) and (30) are respectively The expression (31), after an appropriate simplification, takes the standard form where the terms, excepting 0 (z, z), υ constitute the image system in the region outside the boundary z a, = which thus consists of (1) a Stokeslet of strength, −β , (2) a stresslet of strength where the last two constant terms are added to make ψ vanish on r = a ,and where And thus one may be interested in verifying that the stream function (2.34) satisfies no slip conditions, 0 r ∂ ψ ψ = = ∂ on the boundary r = a . Theorem 3. Let there be a two-dimensional flow in an unbounded incompressible viscous fluid in the z -plane. Let the flow be characterized by the complex velocity − ω whose singularities are all at a distance less than a from the origin; and let and where Proof: "Account into (1) the value of the complex functions W(z) and (z) ω by the expressions (38) and (39) respectively, one gets the function (35) which is clearly a particular complex velocity." Next, making use of the transformation 2 a z z = in the complex velocity (35) yields ( ) z, z 0, υ = on z a; = and thus the no-slip condition is satisfied.
Since, by hypothesis, which implies that 0 (z, z) * υ has no singularities inside the region z a, ≤ and has in particular, finite velocity at the origin; and all conditions being satisfied the theorem is therefore proved.
Example. A stresslet interior to a circular cylinder Let the primary flow be due to a stresslet of strength α at the point 0 z , with 0 z a, < in an incompressible viscous fluid; the complex velocity [11] generated by this singularity in the fluid is given by ( ) which may be written in the form where 0 0

Conclusion
Except in a few simple cases the solution of fluid mechanical problems involving an arbitrary boundary is still, in general exceedingly difficult. In the case of twodimensional slow flow theory there is a complex variable circle theorem [1] for the solutions of Stokes flows due to singularities outside a circular cylinder which corresponds to Milne-Thomson's circle theorem [2,3] for potential flow outside the same cylinder, in the inviscid flow theory. Again it is notable that the same complex variable circle theorem can solve, in particular, some particular Stokes flow problems which cannot be done by the application of the real variable circle theorem [4,5]. Moreover, the "condition for zero perturbation velocity" referred to in the former theorem may suggest, in many cases, relatively easily the strengths or positions or both, of the singularities of the basic flow so that the viscous flows outside the circular boundary exist. Besides these type of solutions, many relevant solutions regarding this topic are also provided in this paper. Here our object is to study Stokes flow interior to a circular cylinder in the light of the complex variable theory and to establish a number of complex variable circle theorems for slow viscous fluid motion within a circular cylinder in terms of the complex velocities of the fundamental singularities found in Chowdhury and Sen [11]; these theorems also correspond to the complex variable circle theorems for potential flow [12,13] in an inviscid fluid within the same cylinder.