A stochastic model for thermal transport of nanofluid in porous media: Derivation and applications

Abstract In this paper, a stochastic thermal transport model is developed for nanofluid flowing through porous media. This model incorporates the influences of nanoparticle migration on convective heat transfer of the colloidal solution. We show that Levy flight movement patterns of nanoparticles result in the derived model using fractional derivative for the diffusion term. The new thermal transport model is then applied to the mixed convective problem which is solved using finite difference method. Numerical results show that the smaller values of Levy index γ lead to larger Nusselt numbers, thus the occurrence of long jumps for nanoparticles increases the heat transport of nanofluids. The effects of other involved physical parameters are also presented and discussed.

[1]  Qiang Li,et al.  Stochastic thermal transport of nanoparticle suspensions , 2006 .

[2]  O. Bénichou,et al.  Mean first-passage times of non-Markovian random walkers in confinement , 2016, Nature.

[3]  Richard L. Magin,et al.  Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation , 2016 .

[4]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[5]  H. Qi,et al.  Solutions of the space-time fractional Cattaneo diffusion equation , 2011 .

[6]  Martin Stynes,et al.  A finite difference method for a two-point boundary value problem with a Caputo fractional derivative , 2013, 1312.5189.

[7]  Walter Zimmermann,et al.  Convection in nanofluids with a particle-concentration-dependent thermal conductivity. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  I. Podlubny,et al.  Modelling heat transfer in heterogeneous media using fractional calculus , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  S L Wearne,et al.  Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  S. Phillpot,et al.  Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids) , 2002 .

[11]  Yulong Ding,et al.  Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions , 2004 .

[12]  Hong Wang,et al.  A fast finite volume method for conservative space-fractional diffusion equations in convex domains , 2016, J. Comput. Phys..

[13]  Robert A. Van Gorder,et al.  Convective heat transfer in the flow of viscous Ag–water and Cu–water nanofluids over a stretching surface , 2011 .

[14]  Clinton S. Willson,et al.  The impact of immobile zones on the transport and retention of nanoparticles in porous media , 2015 .

[15]  R. Zheng,et al.  Experimental investigation of heat conduction mechanisms in nanofluids. Clue on clustering. , 2009, Nano letters.

[16]  D Brockmann,et al.  Lévy flights in inhomogeneous media. , 2003, Physical review letters.

[17]  X. Zhu,et al.  Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains , 2016, J. Comput. Phys..

[18]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[19]  John W. Crawford,et al.  The impact of boundary on the fractional advection dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives , 2007 .

[20]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[21]  Fawang Liu,et al.  A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain , 2017 .

[22]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .

[23]  Weihua Deng,et al.  Boundary Problems for the Fractional and Tempered Fractional Operators , 2017, Multiscale Model. Simul..

[24]  George Em Karniadakis,et al.  Stochastic heat transfer enhancement in a grooved channel , 2006, Journal of Fluid Mechanics.

[25]  Compte,et al.  Stochastic foundations of fractional dynamics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  Fawang Liu,et al.  Fast Finite Difference Approximation for Identifying Parameters in a Two-dimensional Space-fractional Nonlocal Model with Variable Diffusivity Coefficients , 2016, SIAM J. Numer. Anal..

[27]  C. Yang,et al.  On the Anomalous Convective Heat Transfer Enhancement in Nanofluids: A Theoretical Answer to the Nanofluids Controversy , 2013 .

[28]  M. Shlesinger,et al.  Stochastic pathway to anomalous diffusion. , 1987, Physical review. A, General physics.

[29]  Xiaoyun Jiang,et al.  Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative , 2012 .

[30]  Ioan Pop,et al.  Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate , 2011 .

[31]  Marie-Christine Néel,et al.  Space-fractional advection-diffusion and reflective boundary condition. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Patricia E. Gharagozloo,et al.  A Benchmark Study on the Thermal Conductivity of Nanofluids , 2009 .

[33]  Richard L. Magin,et al.  Using spectral and cumulative spectral entropy to classify anomalous diffusion in Sephadex™ gels , 2017, Comput. Math. Appl..

[34]  Antonio Politi,et al.  A stochastic model of anomalous heat transport: analytical solution of the steady state , 2008, 0809.0453.

[35]  J. Buongiorno.  Convective Transport in Nanofluids , 2006 .

[36]  Igor M. Sokolov,et al.  ANOMALOUS TRANSPORT IN EXTERNAL FIELDS : CONTINUOUS TIME RANDOM WALKS AND FRACTIONAL DIFFUSION EQUATIONS EXTENDED , 1998 .

[37]  Rohit Jain,et al.  Lévy flight with absorption: A model for diffusing diffusivity with long tails. , 2017, Physical review. E.

[38]  F. Mainardi,et al.  Fractional Calculus: Quo Vadimus? (Where are we Going?) , 2015 .

[39]  Chengming Huang,et al.  An energy conservative difference scheme for the nonlinear fractional Schrödinger equations , 2015, J. Comput. Phys..

[40]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[41]  Mihaela Osaci,et al.  Study about the nanoparticle agglomeration in a magnetic nanofluid by the Langevin dynamics simulation model using an effective Verlet-type algorithm , 2017 .

[42]  Ravi Radhakrishnan,et al.  Nanoparticle stochastic motion in the inertial regime and hydrodynamic interactions close to a cylindrical wall. , 2016, Physical review fluids.

[43]  Yuriy Povstenko,et al.  Theory of thermoelasticity based on the space-time-fractional heat conduction equation , 2009 .

[44]  Y. Xuan,et al.  Investigation on Convective Heat Transfer and Flow Features of Nanofluids , 2003 .

[45]  Andrey G. Cherstvy,et al.  Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes , 2013, 1303.5533.

[46]  D. Cahill,et al.  Nanofluids for thermal transport , 2005 .

[47]  Xiaohui Lin,et al.  Nanofluids transport model based on Fokker-Planck equation and the convection heat transfer calculation , 2013 .

[48]  Yulong Ding,et al.  Effect of particle migration on heat transfer in suspensions of nanoparticles flowing through minichannels , 2005 .

[49]  Wen Chen,et al.  Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete , 2017 .

[50]  Andrew J. Majda,et al.  Stochastic models for convective momentum transport , 2008, Proceedings of the National Academy of Sciences.

[51]  Mehdi Bahiraei,et al.  Particle migration in nanofluids: A critical review , 2016 .

[52]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[53]  O. P. Singh,et al.  An analytic algorithm for the space-time fractional advection-dispersion equation , 2011, Comput. Phys. Commun..

[54]  Massimiliano Zingales.  Fractional-order theory of heat transport in rigid bodies , 2014, Commun. Nonlinear Sci. Numer. Simul..

[55]  O. K. Crosser,et al.  Thermal Conductivity of Heterogeneous Two-Component Systems , 1962 .

[56]  Melvin Lax,et al.  Stochastic Transport in a Disordered Solid. I. Theory , 1973 .

[57]  Jan S. Hesthaven,et al.  Special Issue on "Fractional PDEs: Theory, Numerics, and Applications" , 2015, J. Comput. Phys..