The 3D Happel model for complete isotropic Stokes flow

Abstract

The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere-in-cell model. The mathematical treatment is based on the two-concentric-spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non-axisymmetric formulation are similar to those of the 2D sphere-in-cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich-Neuber differential representation of the solutions for Stokes flow, which is valid in non-axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self-sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided

On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation

“…Dassios et al [21] demonstrated how Papkovich-Neuber representations of the flow fields were correlated to the general solution of Stokes equations in spheroidal geometry. Furthermore, Dassios and Vafeas [22] corroborated the powerfulness and usability of the above 3D Papkovich-Neuber representation by obtaining a closed-form solution for a Happel-type 3D Stokes flow through a swarm of translating and rotating spherical particles.…”

“…, n and q = e, o with p = int, ext, specifies the system's solution, achieved up to any precise level, where the targeted degree of accuracy is attained. Thereupon, the flow fields follow from Equations ( 19)-( 21) with (22), where the relationships of Equations ( 23)-( 25) are readily incorporated.…”

Interactions between Proteins and the Membrane Surface in Multiscale Modeling of Organic Fouling

“…The computed q′′ was equal to −0.021 (C/m 2 ) in good agreement with the values reported in the literature. 39,40 In addition, q′′ and the equivalent diameter were used to calculate the protein surface potential obtaining −7.5 mV. Interestingly, the surface charges calculated by the Loẅdin method yielded a value of q′′ equal to −0.012 C/m 2 and a BSA surface potential equal to −4.9 mV 12 .…”

Connection Formulae between Ellipsoidal and Spherical Harmonics with Applications to Fluid Dynamics and Electromagnetic Scattering

“…The fields are provided in a closed form fashion as full series expansions of ellipsoidal harmonic eigenfunctions. The velocity, to the first degree, which represents the leading term of the series, is sufficient for most engineering applications and also provides us with the corresponding full 3D solution for the sphere [10] after a proper reduction. The whole analysis is based on the Lamé functions and the theory of ellipsoidal harmonics.…”