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

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 170
Author(s):  
Panayiotis Vafeas ◽  
Eleftherios Protopapas ◽  
Maria Hadjinicolaou

Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions.

2004 ◽  
Vol 2004 (4) ◽  
pp. 347-360 ◽  
Author(s):  
George Dassios ◽  
Panayiotis Vafeas

Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and Usha (PNAU) proposed two different representations of the velocity and the pressure fields in Stokes flow, in terms of harmonic and biharmonic functions, which form a practical tool for many important physical applications. One is the particle-in-cell model for Stokes flow through a swarm of particles. Most of the analytical models in this realm consider spherical particles since for many interior and exterior flow problems involving small particles, spherical geometry provides a very good approximation. In the interest of producing ready-to-use basic functions for Stokes flow, we calculate the PNAU and the PN eigensolutions generated by the appropriate eigenfunctions, and the full series expansion is provided. We obtain connection formulae by which we can transform any solution of the Stokes system from the PN to the PNAU eigenform. This procedure shows that any PNAU eigenform corresponds to a combination of PN eigenfunctions, a fact that reflects the flexibility of the second representation. Hence, the advantage of the PN representation as it compares to the PNAU solution is obvious. An application is included, which solves the problem of the flow in a fluid cell filling the space between two concentric spherical surfaces with Kuwabara-type boundary conditions.


2005 ◽  
Vol 105 ◽  
pp. 271-276 ◽  
Author(s):  
Wei Tong

A new mathematical formulation is presented for describing the three-dimensional anisotropic plastic flow behavior of a heterogeneous polycrystalline solid. By using three principal stresses, three loading orientation angles, and a generally non-quadratic, real-valued stress exponent, a mathematical theory of anisotropic plasticity is formulated as two coupled orthogonal series expansions in both the 3D principal stress space (the p–plane) and the 3D loading orientation space. A geometrical interpretation of the new mathematical representation of anisotropic plasticity is offered. Specific examples are given to illustrate the application of the proposed theory for modeling the plastic anisotropy of orthotropic polycrystalline sheets under uniaxial and biaxial tension.


1990 ◽  
Vol 112 (2) ◽  
pp. 199-204 ◽  
Author(s):  
T. C. Vu ◽  
W. Shyy

Three-dimensional turbulent viscous flow analyses for hydraulic turbine elbow draft tubes are performed by solving Reynolds averaged Navier-Stokes equations closed with a two-equation turbulence model. The predicted pressure recovery factor and flow behavior in the draft tube with a wide range of swirling flows at the inlet agree well with experimental data. During the validation of the Navier-Stokes flow analysis, particular attention was paid to the effect of grid size on the accuracy of the numerical result and the importance of accurately specifying the inlet flow condition.


2018 ◽  
Vol 25 (6) ◽  
pp. 061208 ◽  
Author(s):  
Francesco Taccogna ◽  
Pierpaolo Minelli

2009 ◽  
Vol 631 ◽  
pp. 363-373 ◽  
Author(s):  
P. N. SHANKAR

A general method is suggested for deriving exact solutions to the Stokes equations in spherical geometries. The method is applied to derive exact solutions for a class of flows in and around a sphere or between concentric spheres, which are generated by meridional driving on the spherical boundaries. The resulting flow fields consist of toroidal eddies or pairs of counter-rotating toroidal eddies. For the concentric sphere case the exact solution when the inner sphere is in instantaneous translation is also derived. Although these solutions are axisymmetric, they can be combined with swirl about a different axis to generate fully three-dimensional fields described exactly by simple formulae. Examples of such complex fields are given. The solutions given here should be useful for, among other things, studying the mixing properties of three-dimensional flows.


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