Solution of Energy Transport Equation with Variable Material Properties by BEM

Abstract

In this paper, we derive a boundary-domain integral formulation for the energy transport equation under the assumption that the fluid properties, through which the energy is transported by diffusion and convection, are spatially and temporally changing. The energy transport equation is a second-order partial differential equation of a diffusion-convection type, with the fluid temperature as the independent variable. The presented formulation does not require a calculation of the temperature gradient, thus it is, for a known fluid velocity field, linear.The final boundary-domain integral equation is discretized using a domain decomposition approach, where the equation is solved on each sub-domain, while subdomains are joined by compatibility conditions. The validity of the method is checked using several analytical examples. Convergence properties are studied yielding that the proposed discretization technique is second-order accurate.The developed method is used to simulate flow and heat transfer of nanofluids, which exhibit properties that depend on the solid particle concentration. A Lagrange-Euler approach is used.

An accelerated Boundary‐Domain Integral Method for three‐dimensional fluid flow analysis

“…A description of the model, that we use to simulate the nanofluid flow, was presented by Ravnik et. al in [4]. Equations (3) and (4) are solved with the subdomain Boundary-Domain Integral Method, that was presented in [3].…”

“…where P r is the Prandtl number, Ra is the Rayleigh number, ϑ nf (ψ) and β nf (ψ) are the kinematic viscosity and thermal expansion of the nanofluid (ψ is the volume fraction of nanoparticles) and, ϑ 0 and β 0 are the kinematic viscosity and thermal expansion of the base fluid. The energy equation is of this form [4]:…”

A Review of Modelling Approaches for Flow and Heat Transfer in Nanofluids

“…Since the particles are very small compared to characteristic flow field length scales, we consider using pointwise approximation and one-way coupling between the continuous fluid phase and particles. Ravnik et al [56] have used Euler-Lagrange model. The concentration of nanoparticles is written as c( r) = N ( r)/V ( r).…”

Boundary-domain integral method for vorticity transport equation with variable viscosity

“…Variable diffusivity was considered by Grzhibovskis et al [8], Chkadua et al [5], Al Jawary [1][2][3] and Ang et al [4]. Ravnik and Skerget [12] proposed a boundary-domain integral formulation for diffusion-convection equations with variable coefficient and velocity and considered the energy equation with variable material properties, [13]. Boundary-domain integral method for compressible fluid flow was proposed by Škerget and co-workers [15,16].…”