Non-local methods in transport phenomena




Yousefzadeh Tabasi, Rambod

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Peridynamics is a well-established nonlocal method for modeling the deformation of solid bodies. The concepts introduced by the peridynamic (PD) theory have demonstrated special utility for problems in solid mechanics which include the evolution of spatial discontinuities, i.e. cracks. While the PD theory has been extensively studied on problems of solid mechanics, its capabilities as a multi-scale modeling theory also make it a candidate for modeling problems of heat and mass transport in fluids. Fluid mechanics has so far been an under-explored area of peridynamic research. The aim of this dissertation is to lay the foundation for the use of peridynamics in fluid mechanics by presenting viable techniques for modeling heat and mass transport problems using peridynamic-based nonlocal models. This investigation starts with a nonlocal advection-diffusion model for immiscible two-phase flow in porous media. The proposed nonlocal formulation is shown to be capable of naturally handling the sharp and irregular changes in the concentration at the interphase of the fluids. As a result, the proposed model can capture the formation and evolution of instabilities at the fluid interface. An important feature of peridynamics models is an influence function which governs the strength of nonlocality and can be used as a tool for multiscale modeling. The local limit of a peridynamics model is independent of the choice of the peridynamic influence function; however, the correct nonlocal mechanics cannot be modeled unless a physically meaningful influence function is used. This dissertation presents a systematic approach for the calculation of a nonlocal kernel that has been homogenized from molecular dynamics (MD) calculations of heat transfer in nanofluids. The MD calculations fully resolve the individual constitutes of the nanofluid. The peridynamic continuum model is then shown to be accurate in demonstrating the enhanced heat transfer properties of nanofluids on domains larger than what can be practically solved using MD. Finally, a nonlocal extension of the Navier-Stokes equations is developed along with a penalty formulation for their numerical solution. Mathematical convergence is shown to recover the classical Navier-Stokes partial differential equations as limiting case. Computational simulation results are then presented for several test cases demonstrating that the formulation is stable and can recover important features of the classical theory for several test cases.


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