Generalized Circle and Sphere Theorems for Inviscid and Viscous Flows

Abstract

The circle and sphere theorems in classical hydrodynamics are generalized to a composite double body. The double body is composed of two overlapping circles/spheres of arbitrary radii intersecting at a vertex angle π/n, n an integer. The Kelvin's transformation is used successively to obtain closed form expressions for several flow problems. The problems considered here include two-dimensional and axisymmetric three-dimensional inviscid and slow viscous flows. The general results are presented as theorems followed by simple proofs. The two-dimensional results are obtained using complex function theory while the three-dimensional formulas are obtained using Stokes stream function.The solutions for several flows in the presence of the composite geometry are derived by the use of these theorems. These solutions are in singularity forms and the image singularities are interpreted in each case. In the case of three-dimensional axisymmetric viscous flows, a Faxen relation for the force acting on the composite bubble is derived.

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

“…30 using multivalued singularities in a Riemann space. General theorems for two circles or spheres intersecting at an angle $$\frac{\pi }{n}$$, where n is an integer, have been proved in the context of nonviscous and viscous hydrodynamics 31 . The fused spheres problem in viscous flow theory has been treated in Refs.…”

Stokes Flow between Two Cylinders

“…In this context, a theorem for shear-free sphere in Stokes flow due to Palaniappan et al (1990) is also noteworthy. The authors noted that the surface of a clean bubble is a shear-free boundary for the flow outside it in an uncontaminated viscous fluid (Daripa and Palaniappan 2001, FigueroaEspinoza et al 2008, Harper 1983, Legendre et al 2003, Palaniappan 1996, Usha and Hemalatha 1993. Daripa and Palaniappan (2001) examined the two-dimensional Stokes flow past a shear-free double circle (a two-dimensional stationary composite bubble) and presented a theorem to calculate to disturbed flow when a shear-free double circle is inserted into a given unbounded slow viscous fluid motion.…”

“…The authors noted that the surface of a clean bubble is a shear-free boundary for the flow outside it in an uncontaminated viscous fluid (Daripa and Palaniappan 2001, FigueroaEspinoza et al 2008, Harper 1983, Legendre et al 2003, Palaniappan 1996, Usha and Hemalatha 1993. Daripa and Palaniappan (2001) examined the two-dimensional Stokes flow past a shear-free double circle (a two-dimensional stationary composite bubble) and presented a theorem to calculate to disturbed flow when a shear-free double circle is inserted into a given unbounded slow viscous fluid motion. In the same paper (Daripa and Palaniappan 2001), for three-dimensional situation, they formulated a theorem to evaluate the perturbed flow when a shear-free double sphere (a composite bubble) is introduced into an arbitrary axisymmetric unbounded Stokes flow.…”

“…Daripa and Palaniappan (2001) examined the two-dimensional Stokes flow past a shear-free double circle (a two-dimensional stationary composite bubble) and presented a theorem to calculate to disturbed flow when a shear-free double circle is inserted into a given unbounded slow viscous fluid motion. In the same paper (Daripa and Palaniappan 2001), for three-dimensional situation, they formulated a theorem to evaluate the perturbed flow when a shear-free double sphere (a composite bubble) is introduced into an arbitrary axisymmetric unbounded Stokes flow. Here, it is important to note that the circle theorem or the sphere theorems are respectively for the flow fields due to the fundamental singularities outside (or, inside) a shear-free circular boundary or spherical boundary.…”

A generalized Milne-Thomson theorem

“…The solution of the corresponding boundary-value problem gives the well-known Milne-Thomson circle theorem [5, p. 159]. It is worth mentioning the paper [6], where a generalization of the circle theorem to the case of two overlapping circles is given and which contains references to closely related problems.…”