A Complex Variable Circle Theorem for Plane Stokes Flows

Two dimensional steady Stokes flow around a circular cylinder is examined in the light of complex variable theory and a circle theorem for the flow, are established. The theorem gives a complex variable expression of the velocity for a Stokes flow external to a circular cylinder, in terms of the same variable expression of the velocity for a slow and steady irrotational flow in unbounded incompressible viscous fluid, and also gives a formula for the steam function for the flow. A few illustrative solutions of Stokes flow around a circular cylinder are presented.


Introduction
It is well known (1) that for a two -dimensional uniform streaming motion past a circular cylinder there exists no solution of the governing Stokes equations. It is in great contrast to the corresponding three-dimensional problem of a uniform stream disturbed by a sphere. This situation for the motion past a circular cylinder is known as the Stokes paradox. (2) Recently, Avudainayagam, Jotiram and Ramakrishna (3) have established a necessary condition (called the consistency condition) for the existence of plane Stokes flow past a circular cylinder and the authors have also given, for the first time, an explanation of the Stokes paradox with the aid of the same condition. Sen (4) has given a circle theorem for the stream function for two-dimensional steady Stokes flow past a rigid circular cylinder in terms of the stream function for a slow flow in an unbounded incompressible viscous fluid. Usha and Hemalatha (5) have also established a circle theorem for the stream function for plane Stokes flow past a shear free impermeable circular cylinder.
In the present paper, we have found it much convenient to a study on Stokes flow past a circular cylinder in the light of complex variable theory is considerably convenient here for the formulae for the same flow are mathematically concise, and have "initial conditions" which are very simple. In Section 3,we have given the first theorem for the complex velocity and the stream function for plane Stokes flow external to the circular cylinder, when the primary flow in an unbounded incompressible viscous fluid is irrotational everywhere, and this theorem corresponds to Milne-Thomson's circle theorem for potential flow (6) By making use of Taylor's series which is well known in complex variable theory, the second theorem which is similar to the first one stands for the flow internal to a circular cylinder. We also easily show that the Stokes flow problem only for a two dimensional source or sink, etc. outside a circular cylinder or a two dimensional uniform streaming function past the same cylinder does not exist; but we have found that their combination taken two or more at a time, by choosing their "strengths" and "positions" suitably, gives Stokes flow around a circular cylinder. This phenomenon is illustrated by the exact solutions of a number of the Stokes flow past a circular cylinder with the aid of the "Cauchy Integral formula for multiply-connected region", well known in complex variable theory ; and the same theorem gives the formulae for the velocity and the stream function for the flow. In Section 2 we begin with a discussion of the fundamental singular solutions of the Stokes equations with external force and give the formulation of the velocity and pressure of the two dimensional fundamental singularities, such as Stokeslet, Stoke doublet, rotlet, potential doublet, etc in order to use the theorem in the application of theorem.

Fundamental Solutions of the Two Dimensional Stokes Equations
The complex variable formed by the two dimensional Stokes equations for a steady motion in an incompressible viscous fluid can be obtained in a quite easy manner and these are And.
Where = √−1 , u and v are the Cartesian velocity components, p the pressure, the constant viscosity coefficient and ϑ is kinematic viscosity where 1 2 1 2 , F F iF F and F = + being the Cartesian components of external force per unit volume. In this note for two dimensional flow in a viscous fluid we denote the combination u-iv by the symbol υ so that.
Here we shall call the complex velocity after Milne-Thomson (6) who first called , having expression (3) without, ̅ , the complex velocity in connection with potential flow.
We then define the complex conjugate of the complex velocity as 2i z We then give below the complex variable forms of the fundamental solutions to the two dimensional Stokes Equations (6) and (7), which corresponds to the vector forms of the fundamental solutions to the three dimensional Stokes equations in Chwang and Wu (9). The primary fundamental solution of Equations (6) and (7), is concerned with a singular point force located, say at the origin, α being a constant complex quantity, and ( ) ( ) x and y ξ ξ one-dimensional Dirrac delta functions. α characterizes its strength (in magnitude α and direction arg α ).
In fact, expression (9) is the rwo-dimensional Stokeslet in Clearly, a derivative of any order of s s and P υ is also a solution of equation (2.7), the corresponding.
F being the derivative of the same order of the conjugate of the Stokeslet. We may now introduce two-dimensional potential doublet, potential quadrapole, rotlet, stresslet etc. as follows. A two-dimesnional potential doublet corresponding to its three dimensional as analogue in (9) has the simple complex velocity representation.
Where α (a constant complex quantity) is the doublet strength. It is of interest to note that a potential doublet is related to the Stokeslet by.
The corresponding pressure is given by.
The potential quadrapole and potential octupole may be introduced respectively as the complex velocities. And.
Again the symmetric component (with respect to an interchange of the complex quantities ) and α β of the Stokes doublet is itself a physical quantity called a stresslet after Batchelor (11). Its complex velocity and pressure are respectively.

The Circle Theorems
In the case f conservative forces the Stokes equation (6) reduces to the form.
( ) Where Ω is the potential function due to the forces. The complex conjugate form of this equation is.
Wherein we note p p and = Ω = Ω , since p and Ω are real (scalar) functions. Now a differential equation satisfied by ( ) p + Ω without the complex velocity υ can be obtained from equations (27) and (28) with the aid of the mass conservation equation (7) as.

( )
And therefore by using this result in (27) we have the following equation for the complex velocity. ( ) This equation for Stokes flow is in agreement with that in Milne-Thomson (6, p.683), and the solution of it for 2iΨ , due to him is stated here in a slight different form for future reference as follows.
Where w (z) and w (z) are arbitrary complex functions. By applying formula (5) to this expression, we then obtain the general complex velocity for two-dimensional steady Stokes flow as follows.
Where the prime in second term of the R.H.S of (33) denotes differentiation with respect to z. The expression for ( ) p + Ω corresponding to the complex velocity (33) is given by.
Where 0 p is an arbitrary real constant. First we present relatively simple expressions for he complex velocity and the stream function for a twodimensional steady Stokes flow external (or internal) to a circular cylinder in terms of the complex velocity for a slow irrotational flow in an incompressible viscous fluid with no rigid boundaries.
Circle Theorem: Let there be steady, slow and twodimensional irrotational flow in incompressible viscous fluid with no rigid boundaries, in the z-plane. Let the flow be characterized by the complex velocity 0 0 ( ) z υ υ = , whose singularities are all at a distance greater than a from the origin, and let 0 ( ) k z υ ο ≈ for 1 k ≥ , at the origin. If a circular cylinder of radius a (whose intersection with the z-plane is the circle : z a Γ = ), be introduced into the flow, the complex velocity and the stream function for the Stokes flow past the circular cylinder become respectively.
( ) is the perturbation complex velocity and where the prime denotes differentiation with respect to z. Proof The proof essentially consists in showing that the complex velocity (35) must satisfy the following four conditions, and in deriving the stream function (36) out of the complex velocity.