×

Spatially-dependent and nonlinear fluid transport: coupling framework. (English) Zbl 1254.35189

From the author’s abstract: We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas.
The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods.
In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport. At the end of the article we present some results of numerical experiments for different benchmark problems.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K15 Initial value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
76M12 Finite volume methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992; · Zbl 0171.38503
[2] Bear J., Dynamics of Fluids in Porous Media, Enviromental Science Series, American Elsevier, New York, 1972; · Zbl 1191.76001
[3] Bear J., Bachmat Y., Introduction to Modeling of Transport Phenomena in Porous Media, Theory Appl. Transp. Porous Media, 4, Kluwer, Dordrecht, 1991; · Zbl 0780.76002
[4] Davies B., Integral Transforms and their Applications, Appl. Math. Sci., 25, Springer, New York-Heidelberg, 1978; · Zbl 0381.44001
[5] Eykholt G.R., Analytical solution for networks of irreversible first-order reactions, Water Research, 1999, 33(3), 814-826 http://dx.doi.org/10.1016/S0043-1354(98)00273-5;
[6] Frolkovič P., Geiser J., Discretization methods with discrete minimum and maximum property for convection dominated transport in porous media, In: Numerical Methods and Applications, Borovets, August 20-24, 2002, Lecture Notes in Comput. Sci., 2542, Springer, Berlin, 2003, 445-453; · Zbl 1119.76356
[7] Geiser J., Discretisation Methods for Systems of Convective-Diffusive Dispersive-Reactive Equations and Applications, PhD thesis, Universität Heidelberg, 2004;
[8] Geiser J., Discretization methods with embedded analytical solutions for convection-diffusion dispersion-reaction equations and applications, J. Engrg. Math., 2007, 57(1), 79-98 http://dx.doi.org/10.1007/s10665-006-9057-y; · Zbl 1148.65067
[9] Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877-922 http://dx.doi.org/10.1002/fld.2225; · Zbl 1444.76104
[10] Geiser J., Zacher T., Time dependent fluid transport: analytical framework. preprint available at http://webdoc.sub.gwdg.de/ebook/serien/e/preprint_HUB/P-11-05.pdf;
[11] Higashi K., Pigford T.H., Analytical models for migration of radionuclides in geologic sorbing media, Journal of Nuclear Science and Technology, 1980, 17(9), 700-709 http://dx.doi.org/10.3327/jnst.17.700;
[12] Kelley C.T., Iterative Methods for Linear and Nonlinear Equations, Frontiers Appl. Math., 16, SIAM, Philadelphia, 1995 http://dx.doi.org/10.1137/1.9781611970944; · Zbl 0832.65046
[13] LeVeque R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002 http://dx.doi.org/10.1017/CBO9780511791253; · Zbl 1010.65040
[14] Van Genuchten M.T, Convective-dispersive transport of solutes involved in sequential first-order decay reactions, Computers & Geosciences, 1985, 11(2), 129-147 http://dx.doi.org/10.1016/0098-3004(85)90003-2;
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.