A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

Abstract

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form r 2 v, where v is a harmonic function. Previously Johnson (J Fluid Mech 110:217-238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum. (2000). 76D07. Mathematics Subject Classification

Steady symmetric low-Reynolds-number flow past a film-coated cylinder

“…Considering the contact angle to be O (1), he presented theoretical and experimental results in which the film either forms a single lobe covering the front or rear portion of the sphere or forms a belt around the sphere. Johnson's [14] work has been recently extended by Choudhuri and Padmavati [5] who considered an arbitrary Stokes flow and assumed the surface tension to be large (so that the film interface remains spherical). Johnson's [14] work has been recently extended by Choudhuri and Padmavati [5] who considered an arbitrary Stokes flow and assumed the surface tension to be large (so that the film interface remains spherical).…”

“…While previous studies of flow past a film-coated sphere have applications in industrial processes involving microcapsules [5], many microcapsules bear little resemblance to simple spheres, therefore considering flow past a film-coated cylinder provides insight into the effect of microcapsule shape. Using an asymptotic approach, we determine the dynamics of the film and how the film's presence perturbs the bulk flow.…”

A Novel Application of the Classical Banach Fixed Point Theorem

“…But T is linear and no matter what it always fixes the '0' vector thereby allowing us to conclude that the only solution to the problem is the trivial solution-similar to 'Hopf's principle. We will now give few examples of partial differential equations of the type given in (6) to guarantee the existence of a solution using the condition given in (12).…”

“…It is also today one of the richly enhanced field of research in Mathematics. The applications of elliptic partial differential equations are almost unrestricted spreading across fields like Fluid Mechanics, Electro-magnetics, Biological systems and finance ( [3][4][5][6][7][8] and the references therein). Though many a times, one would attempt analytical solutions, the real life problems may strictly demand numerical treatment.…”

Axisymmetric Creeping Flow of a Micropolar Fluid over a Sphere Coated with a Thin Fluid Film

“…Such flows for Newtonian fluid have been modeled earlier by Gupalo andRyazantsev (1974), O'Neill andRanger (1981), Johnson (1981), Kawano and Hashimoto (1997). Recently, a method was developed to solve the problem of an arbitrary Stokes flow of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity by Choudhuri and Padmavati (2010). The purpose of this study is to extend the work of Johnson (1981) by taking micropolar fluid in place of Newtonian fluid for outer region.…”