Hybrid solutions obtained via integral transforms for magnetohydrodynamic flow with heat transfer in parallel-plate channels

Abstract

Purpose The purpose of this study is to show the procedure, application and main features of the hybrid numerical-analytical approach known as generalized integral transform technique by using it to study magnetohydrodynamic flow of electrically conductive Newtonian fluids inside flat parallel-plate channels subjected to a uniform and constant external magnetic field. Design/methodology/approach The mathematical formulation of the analyzed problem is given in terms of a streamfunction, obtained from the Navier–Stokes and energy equations, by considering steady state laminar and incompressible flow and constant physical properties. Findings Convergence analyses are performed and presented to illustrate the consistency of the integral transformation technique. The results for the velocity and temperature fields are generated and compared with those in the literature as a function of the main governing parameters. Originality/value A detailed analysis of the parametric sensibility of the main dimensionless parameters, such as the Reynolds number, Hartmann number, Eckert number, Prandtl number and electrical parameter, for some typical situations is performed.

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

“…The integral transform analysis of fluid flow problems governed by the Navier-Stokes equations has required the proposition of new eigenfunction expansions, other than those normally employed in diffusion or convection-diffusion problems, directly derived from the general Sturm-Liouville eigenvalue problem. Along the years, in the present methodological context, the Navier-Stokes equations have been mostly dealt with in the streamfunction-only formulation [25,[35][36][37][38][39][40][41][42][43][44][45][46], and less frequently in the primitive variables formulation [47,48]. In two-dimensional problems, the streamfunction formulation offers the advantages of automatically satisfying the continuity equation and eliminating the pressure field.…”

“…However, the extension of this concept to three-dimensional flows, leading to vector and scalar potentials, has been shown to be less advantageous when dealt with by the same hybrid approach [49]. Nevertheless, the integral transform method under the two-dimensional streamfunction formulation has been applied to various classes of problems, including cavity and channel flows, rectangular and cylindrical geometries, regular and irregular domains, laminar and turbulent flows, steady and transient states, natural and forced convection, as well as on magnetohydrodynamics [25,[35][36][37][38][39][40][41][42][43][44][45][46]. The integral transformation of the streamfunction formulation is the first one to be here reviewed, for both steady and transient state situations, in light of its popularity among the contributions that employed this hybrid approach so far.…”

Vector eigenfunction expansion in the integral transform solution of transient natural convection

“…The GITT has been applied to the solution of heat and fluid flow problems under the Navier–Stokes formulation in both primitive variables (Lima et al , 2007; Curi et al , 2014; Souza et al , 2016) and streamfunction-only (Perez-Guerrero and Cotta, 1992; Perez-Guerrero et al , 2000; Silva et al , 2010; An et al , 2013; Matt et al , 2017; Pontes et al , 2018; Fu et al , 2018) formulations. The latter approach has been preferred due to the enhanced convergence, the elimination of the pressure field, and automatic satisfaction of the continuity equation granted by the definition of the streamfunction, though restricted to two-dimensional fluid flow problems.…”