This paper demonstrates a numerical study of heat transfer in a square porous cavity filled with non-Newtonian power-law fluid. A Graphics Processing Unit (GPU) has been used to accelerate the numerical simulation, which uses the Multiple-Relaxation-Time (MRT) Lattice Boltzmann Method. A modified power-law model has been employed to characterize the flow of non-Newtonian fluids. The simulations have been conducted for the power-law index $n$ ranging from $(0.6 \leq n \leq 1.0)$, the Darcy number $Da$ ranging from $(10^{-3} \leq Da \leq 10^{-1})$ and the Rayleigh number $Ra$ ranging from $(10^3 \leq Ra \leq 10^5)$. Results show that the average Nusselt number ($\overline{Nu}$) decreases with an increase in the value of $n$ while $\overline{Nu}$ increases with an increase in the value of $Da$. Moreover, an increment in the value of $Ra$ leads to an increase in the average Nusselt number.
In this paper, global sensitivity analysis (GSA) and uncertainty quantification (UQ) have been applied to the problem of natural convection (NC) in a porous square cavity. This problem is widely used to provide physical insights into the processes of fluid flow and heat transfer in porous media. It introduces however several parameters whose values are usually uncertain. We herein explore the effect of the imperfect knowledge of the system parameters and their variability on the model quantities of interest (QoIs) characterizing the NC mechanisms. To this end, we use GSA in conjunction with th…
We model the flow of a bi-viscous non-Newtonian fluid in a porous medium by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of the being the correlation length exponent. We find that both critical exponents …
The study of flow of non-Newtonian fluids in porous media is very important and serves a wide variety of practical applications in processes such as enhanced oil recovery from underground reservoirs, filtration of polymer solutions and soil remediation through the removal of liquid pollutants. These fluids occur in diverse natural and synthetic forms and can be regarded as the rule rather than the exception. They show very complex strain and time dependent behavior and may have initial yield-stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality …
Single fluid porous medium systems are typically modeled at an averaged length scale termed the macroscale using Darcy's law. Standard approaches for modeling macroscale single fluid phase flow of non-Newtonian fluids extend Darcy's law, using an effective viscosity and assuming that the permeability is invariant. This approach results in a need to determine the effective viscosity for every fluid and flow rate. We use the thermodynamically constrained averaging theory (TCAT) to examine the formulation and closure of a macroscale model for non-Newtonian flow that is consistent with microscale …
The thesis investigates the flow of non-Newtonian fluids in porous media using pore-scale network modeling. Non-Newtonian fluids show very complex time and strain dependent behavior and may have initial yield stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. They are generally classified into three main categories: time-independent, time-dependent and viscoelastic. Two three-dimensional networks representing a sand pack and Berea sandstone were used. An iterative numerical technique is used to solve th…
The injection and in-situ generation of foam in porous media successfully control gas mobility and improve the fluids' sweep efficiency inside porous media. Mathematical models describing this problem use two phases, foamed gas and fluid, and usually have a term for foam generation and destruction. Moreover, the non-Newtonian foam behavior is frequently modeled using the Hirasaki and Lawson's formula for foamed gas viscosity. In this paper, we detail how the traveling wave analysis can be used to estimate the propagation profiles and velocity for a range of non-Newtonian foam models in porous …
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
We analyzed the effect of viscoelasticity on the dynamics of fluids in porous media by studying the flow of a Maxwell fluid in a circular tube, in which the flow is induced by a wave traveling on the tube wall. The present study investigates novelties brought about into the classic peristaltic mechanism by inclusion of non-Newtonian effects that are important, for example, for hydrocarbons. This problem has numerous applications in various branches of science, including stimulation of fluid flow in porous media under the effect of elastic waves. We have found that in the extreme non-Newtonian …
We investigate the flow of various non-Newtonian fluids through three-dimensional disordered porous media by direct numerical simulation of momentum transport and continuity equations. Remarkably, our results for power-law (PL) fluids indicate that the flow, when quantified in terms of a properly modified permeability-like index and Reynolds number, can be successfully described by a single (universal) curve over a broad range of Reynolds conditions and power-law exponents. We also study the flow behavior of Bingham fluids described in terms of the Herschel-Bulkley model. In this case, our sim…
Unlike the power-law model, the Ellis model describes the apparent viscosity of a shear-thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular, this model matches the Newtonian behaviour when the shear stresses are very small. The emergence of the Rayleigh-Bénard instability is studied when a horizontal pressure gradient, yielding a basic throughflow, is prescribed in a horizontal porous layer. The threshold conditions for the linear instability of this system are obtained both analytically and numerically. In the case of a negligible flow rate, the …
We demonstrate through numerical simulations and a mean field calculation that immiscible two-phase flow in a porous medium behaves effectively as a Bingham viscoplastic fluid. This leads to a generalized Darcy equation where the volumetric flow rate depends quadratically on an excess pressure difference in the range of flow rates where the capillary forces compete with the viscous forces. At higher rates, the flow is Newtonian.
The flow of power law fluids, which include shear thinning and shear thickening as well as Newtonian as a special case, in networks of interconnected elastic tubes is investigated using a residual based pore scale network modeling method with the employment of newly derived formulae. Two relations describing the mechanical interaction between the local pressure and local cross sectional area in distensible tubes of elastic nature are considered in the derivation of these formulae. The model can be used to describe shear dependent flows of mainly viscous nature. The behavior of the proposed mod…
Present work attempts to identify the roles of flow- and geometric-variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. Present study al…
In the above paper the author treats the boundary layer flow of an electrically conducting, non-Newtonian fluid along a vertical plate embedded in a Darcy-Brinkman-Forchheimer porous medium. The fluid is under the action of a constant transverse magnetic field and the radiation is taken into account in the energy equation. The boundary layer equations are transformed into ordinary ones and subsequently are solved numerically. However, there is a fundamental error in this paper which is presented below:
In this work, we study non-Newtonian fluid flow in heterogeneous porous media. We are interested in fluids presenting a specific change in rheology: Newtonian below a certain shear rate and power law above. Since porous media generally exhibit strong spatial heterogeneity at large geological scales, we study the interaction between such inhomogeneity and the nonlinear rheology of the fluid. The coupling between permeability heterogeneity and nonlinear rheology significantly affects the flow. We are particularly in the statistical properties of the velocity field (mean, variance, correlation, e…
The creeping flow of a generalized Newtonian fluid in a non--uniform pore throat is investigated analytically. The analytical solution determines the flow regimes and the transition point from Newtonian to the power law flow regime. As an application of the new model, a regular lattice--based pore network model is constructed to simulate the flows of shear--thinning and shear--thickening fluids through porous media. The pore throats have convergent--divergent geometries.
The flow of non-Newtonian fluids is ubiquitous in many applications in the geological and industrial context. We focus here on yield stress fluids (YSF), i.e. a material that requires minimal stress to flow. We study numerically the flow of yield stress fluids in 2D porous media on a macroscopic scale in the presence of local heterogeneities. As with the microscopic problem, heterogeneities are of crucial importance because some regions will flow more easily than others. As a result, the flow is characterized by preferential flow paths with fractal features. These fractal properties are charac…
We experimentally examine pressure-driven flows of 1%, 3%, and 5% dilute suspensions over and through a porous media model. The flow of non-colloidal, non-Brownian suspensions of rigid and spherical particles suspended in a Newtonian fluid is considered at very low Reynolds numbers. The model of porous media consists of square arrays of rods oriented across the flow in a rectangular channel. Systematic experiments using high-spatial-resolution planar particle image velocimetry (PIV) and index-matching techniques are conducted to accurately measure the velocity measurements of both very dilute …
We present an experimental and numerical study of immiscible two-phase flow in 3-dimensional (3D) porous media to find the relationship between the volumetric flow rate ($Q$) and the total pressure difference ($ΔP$) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold, making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium, introducing a threshold pressure $P_t$. In this regime, $Q$ depends quadratically on an excess press…
In this article, the extensional flow and viscosity and the converging-diverging geometry were examined as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista-Manero model, which successfully describes shearthinning, elasticity and thixotropic time-dependency, was used for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm, originally proposed by Philippe Tardy, that employs this model to simulate steadystate time-dependent flow was implemented in a non-Newtonian flow simulation code using pore-scale modeling and …
Yield-stress is a problematic and controversial non-Newtonian flow phenomenon. In this article, we investigate the flow of yield-stress substances through porous media within the framework of pore-scale network modeling. We also investigate the validity of the Minimum Threshold Path (MTP) algorithms to predict the pressure yield point of a network depicting random or regular porous media. Percolation theory as a basis for predicting the yield point of a network is briefly presented and assessed. In the course of this study, a yield-stress flow simulation model alongside several numerical algor…
Newtonian fluid invasion into a Yield-Stress Fluid-Saturated Medium
Université Paris-Saclay, CNRS, FAST（巴黎萨克雷大学，法国国家科学研究中心，FAST）；PoreLab, Department of Physics, Norwegian University of Science and Technology（孔隙实验室，挪威科技大学物理系）；Université Paris-Saclay, CNRS, LPTMS（巴黎萨克雷大学，法国国家科学研究中心，LPTMS）；PoreLab, Department of Physics, University of Oslo（孔隙实验室，奥斯陆大学物理系）
We investigate the elastoviscoplastic flow through porous media by numerical simulations. We solve the Navier-Stokes equations combined with the elastoviscoplastic model proposed by Saramito for the stress tensor evolution. In this model, the material behaves as a viscoelastic solid when unyielded, and as a viscoelastic Oldroyd-B fluid for stresses higher than the yield stress. The porous media is made of a symmetric array of cylinders, and we solve the flow in one periodic cell. We find that the solution is time-dependent even at low Reynolds numbers as we observe oscillations in time of the …
Significant improvements in the computational performance of the lattice-Boltzmann (LB) model, coded in FORTRAN90, were achieved through application of enhancement techniques. Applied techniques include optimization of array memory layouts, data structure simplification, random number generation outside the simulation thread(s), code parallelization via OpenMP, and intra- and inter-timestep task pipelining. Effectiveness of these optimization techniques was measured on three benchmark problems: (i) transient flow of multiple particles in a Newtonian fluid in a heterogeneous fractured porous do…
Solute dispersion in magnetically influenced multiphase flow through a porous tube: axial transport and microrotational effects
Cooch Behar Panchanan Barma University（古奇比哈尔潘查南·巴尔马大学）；SRM Institute of Science and Technology（SRM科学技术研究所）；University of Galway（高威大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
The Saffman-Taylor instability occurs when { a} Stokes fluid is displaced by a less viscous one { in} a Hele-Shaw cell. { This model is useful to study the secondary oil recovery from a pororus medum.} Since 1960, { polymer solutions} were used as displacing fluids; moreover, the oil in a porous reservoir can often be considered a non-Newtonian fluid. Motivated by this fact, in this paper we study the linear stability of the displacement of two Oldroyd-B fluids in a rectilinear Hele-Shaw cell, even if the direct relevance for the flow through porous media is not so evident. We get an { approxi…
Zhu & Granick [Phys. Rev. Lett. 87, 096105 (2001)] have recently experimentally established existence of a boundary slip in a Newtonian liquid. They reported typical values of the slip length of the order of few micro-meters. In this light, the effect of introduction of the boundary slip into the theory of propagation of elastic waves in a fluid-saturated porous medium formulated by Biot is investigated. The new model should allow to fit the experimental seismic data in circumstances when Biot's theory fails, as the introduction of phenomenological dependence of the slip velocity upon frequenc…
