A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers. (English) Zbl 1471.74014

Summary: Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scale” variant of the same problem involving only ten layers.


74E30 Composite and mixture properties
74H05 Explicit solutions of dynamical problems in solid mechanics
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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[1] Hickson, R. I., Critical Times of Heat and Mass Transport Through Mulitple Layers, (2010), The University of New South Wales, Ph.D. thesis
[2] Hickson, R. I.; Barry, S. I.; Mercer, G. N., Critical times in multilayer diffusion. part 1: exact solutions, Int. J. Heat Mass Transf., 52, 5776-5783, (2009) · Zbl 1177.80022
[3] E.J. Carr, P. Perré, I.W. Turner, The extended distributed microstructure model for gradient-driven transport: A two-scale model for bypassing effective parameters, (2016) in review. · Zbl 1373.76333
[4] Carr, E. J.; Turner, I. W., Two-scale computational modelling of water flow in unsaturated soils containing irregular-shaped inclusions, Int. J. Numer. Methods Eng., 98, 157-173, (2014) · Zbl 1352.76110
[5] Carslaw, H. S.; Jaegar, J. C., Conduction of Heat in Solids, (1959), Oxford University Press New York
[6] Mulholland, G. P.; Cobble, M. H., Diffusion through composite media, Int. J. Heat Mass Transf., 15, 147-160, (1972)
[7] Johnston, P. R., Diffusion in composite media: solution with simple eigenvalues and eigenfunctions, Math. Comput. Model., 15, 10, 115-123, (1991) · Zbl 0754.35049
[8] M.R. Rodrigo, A.L. Worthy, Solution of multilayer diffusion problems via the Laplace transform, (2016) in review. · Zbl 1358.35052
[9] Fokas, A. S., A unified transform method for solving linear and certain nonlinear pdes, Proc. R. Soc. Lond. A, 453, 1411-1443, (1997) · Zbl 0876.35102
[10] Deconinck, B.; Pelloni, B.; Sheils, N., Non-steady state heat conduction in composite walls, Proc. R. Soc. A Math. Phys., 470, 2165, 20130605, (2014)
[11] Mantzavinos, D.; Papadomanolaki, M. G.; Saridakis, Y. G.; Sifalakis, A. G., Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in 1+1 dimensions, Appl. Numer. Math., (2014) · Zbl 1338.92049
[12] Asvestas, M.; Sifalakis, A. G.; Papadopoulou, E. P.; Saridakis, Y. G., Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity, J. Phys. Conf. Ser., 490, 012143, (2014)
[13] Tittle, C. W., Boundary value problems in composite media: quasi-orthogonal functions, J. Appl. Phys., 36, 4, 1486-1488, (1965) · Zbl 0137.43502
[14] Bulavin, P. E.; Kashcheev, V. M., Solution of nonhomogeneous heat conduction equation for mulilayered bodies, Int. Chem. Eng., 5, 1, 112-115, (1965)
[15] de Monte, F., Transient heat conduction in one-dimensional composite slab. A ‘natural’ analytic approach, Int. J. Heat Mass Transf., 43, 3607-3619, (2000) · Zbl 0964.80003
[16] de Monte, F., An analytic approach to the unsteady heat conduction process in one-dimensional composite media, Int. J. Heat Mass Transf., 45, 1333-1343, (2002) · Zbl 0992.80006
[17] Sun, Y.; Wichman, I. S., On transient heat conduction in a one-dimensional composite slab, Int. J. Heat Mass Transf., 47, 1555-1559, (2004)
[18] Ilic, M.; Turner, I. W.; Liu, F.; Anh, V., Analytical and numerical solutions of a one-dimensional fractional-in-space diffusion equation in a composite medium, Appl. Math. Comput., 216, 8, 2248-2262, (2010) · Zbl 1193.65168
[19] Trefethen, L. N.; Weideman, J. A.C.; Schmelzer, T., Talbot quadratures and rational approximations, BIT Numer. Math., 46, 653-670, (2006) · Zbl 1103.65030
[20] Ozisik, M. N., Boundary Value Problems of Heat Conduction, (1968), International Textbook Company Scranton, Pennsylvania
[21] (Driscoll, T. A.; Hale, N.; Trefethen, L. N., (2014), Pafnuty Publications Oxford)
[22] Hickson, R. I.; Barry, S. I.; Mercer, G. N.; Sidhu, H. S., Finite difference schemes for multilayer diffusion, Math. Comput. Model., 54, 210-220, (2011) · Zbl 1225.76215
[23] Hickson, R. I.; Barry, S. I.; Mercer, G. N., Critical times in multilayer diffusion. part 2: approximate solutions, Int. J. Heat Mass Transf., 52, 5784-5791, (2009) · Zbl 1177.80023
[24] de Monte, F.; Beck, J. V.; Amos, D. E., Solving two-dimensional Cartesian unsteady heat conduction problems for small values of the time, Int. J. Therm. Sci., 60, 106-113, (2012)
[25] Schimmel Jr., W. P.; Beck, J. V.; Donaldson, A. B., Effective thermal diffusivity for a multimaterial composite laminate, J. Heat. Trans T ASME, 99, 466-470, (1977)
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