Θεωρία διαφορικών αναπαραστάσεων στη ροή Stokes

Abstract

Particle–in–cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple but reliable platform for the analytical or semianalytical solution of heat and mass transport problems. Most of the analytical models in this realm consider either spherical or, in latter versions, non–spherical but still axisymmetric shapes. Despite of the fact that many practical applications involve particles with axial symmetry, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in three and two–dimensional spherical, spheroidal or ellipsoidal domains. Four different complete representations of the solutions for flows that follow the Stokes description are considered here. The first one, named Stokes representation, is obtained, expressing equation of motion in 2–D spherical or spheroidal coordinates, according to which the stream function is expanded in terms of separable or semiseparable eigenmodes, respectively. The other three, valid in non–axisymmetric geometries as well, are the Papkovich – Neuber, the Boussinesq – Galerkin and the Palaniappan et al. differential representations, where the velocity and total pressure fields are expressed in terms of harmonic and biharmonic eigenfunctions. These complete differential solutions hold true also for 2–D flow problems. Connection formulae are obtained for the case of axisymmetric and three–dimensional flows, which relate the harmonic and the stream potential functions. The interrelation is a consequence of the equation of the flow fields and specifies the exact relations of the connection between the corresponding constant coefficients of the potentials. The inversion of this procedure depends on the geometry and the complexity of the differential solutions. It seems that the Papkovich – Neuber differential representation offers us certain important advantages and forms a more complete way in order to solve 2–D and mostly 3–D cell models, where either stationary particles are embedded within a uniformly moving fluid (Kuwabara model) or the particles are moving with a constant uniform velocity and / or rotate with a constant angular velocity in an otherwise quiescent fluid (Happel model, self–sufficient in mechanical energy). The flexibility of the representation, inherited by its degrees of freedom, helps us to confront certain indeterminacies in complicated geometries. This is demonstrated by solving the problem of the flow in a fluid cell filling the space between the surface of the solid particle and the fictitious outer boundary with Kuwabara or Happel–type boundary conditions in several geometries. Thus, we obtain analytical expressions for the velocity, the total pressure, the angular velocity and the stress tensor fields for different particle–in–cell system models. The laborious task of reducing the results to simpler geometries is also included.