Solution of Energy Transport Equation with Variable Material Properties by BEM

Solution of Energy Transport Equation with Variable Material Properties by BEM

Ravnik, J. Skerget, L. Tibaut, J. Yeigh, B.W.

University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, SI-2000 Maribor, Slovenia

University of Washington, 18115 Campus Way NE Bothell WA 98011-8246, USA

| |
| | Citation



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. 


boundary element method, energy transport equation, nanofluids, variable material properties


[1] Skerget, L., Zagar, I. & Alujevic, A., Three-dimensional steady-state diffusion-convec-tion. In Boundary Elements IX, 3, Springer: Berlin, 1987.

[2] Qiu, Z.H., Wrobel, L. & Power, H., Numerical solution of convection-diffusion prob-lems at high Peclet number using boundary elements. International Journal for Numer-ical Methods in Engineering, 41, pp. 899–914, 1998.<899::AID-NME314>3.0.CO;2-T

[3] DeSilva, S.J., Chan, C.L., Chandra, A. & Lim, J., Boundary element method analysis for the transient conduction convection in 2-D with spatially variable convective veloc-ity. Applied Mathematical Modelling, 22(12), pp. 81–112, 1998.

[4] Rap, A., Elliott, L., Ingham, D.B., Lesnic, D. & Wen, X., DRBEM for Cauchy convec-tion-diffusion problems with variable coefficients. Engineering Analysis with Boundary Elements, 28(11), pp. 1321–1333, 2004.

[5] Ravnik, J. & Skerget, L., A gradient free integral equation for diffusion - convection equation with variable coefficient and velocity. Engineering Analysis with Boundary Elements, 37, pp. 683–690, 2013.

[6] Ravnik, J. & Skerget, L., Integral equation formulation of an unsteady diffusionconvec-tion equation with variable coefficient and velocity. Computers and Mathematics with Applications, 66(12), pp. 2477–2488, 2014.

[7] Epstein, P., Zur theorie des radiometers. Zeitschrift fur Physik, 54, pp. 537–563, 1929.

[8] McNab, G.S. & Meisen, A., Thermophoresis in liquids. Journal of Colloid and Inter-face Science, 44(2), pp. 339–346, 1973.

[9] Michaelides, E.E., Brownian movement and thermophoresis of nanoparticles in liquids. International Journal of Heat and Mass Transfer, 81, pp. 179–187, 2015.

[10] Ravnik, J. & Skerget, L., A numerical study of nanofluid natural convection in a cubic enclosure with a circular and an ellipsoidal cylinder. International Journal of Heat and Mass Transfer, 89, pp. 596–605, 2015.

[11] Ravnik, J., Skerget, J. & Hribersek, M., Analysis of three-dimensional natural convec-tion of nanofluids by BEM. Engineering Analysis with Boundary Elements, 34, pp. 1018–1030, 2010.