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The purpose of this paper is to consider double‐diffusive convection in a heated porous medium saturated with a fluid. Of particular interest is the case where the fluid has a stabilizing concentration gradient and small diffusivity.

A fully‐coupled stabilized finite element scheme and adaptive mesh refinement (AMR) methodology are introduced to solve the resulting coupled multiphysics application and resolve fine scale solution features. The code is written on top of the open source finite element library LibMesh, and is suitable for parallel, high‐performance simulations of large‐scale problems.

The stabilized adaptive finite element scheme is used to compute steady and unsteady onset of convection in a generalized Horton‐Rogers‐Lapwood problem in both two and three‐dimensional domains. A detailed study confirming the applicability of AMR in obtaining the predicted dependence of solutal Nusselt number on Lewis number is given. A semi‐permeable barrier version of the generalized HRL problem is also studied and is believed to present an interesting benchmark for AMR codes owing to the different boundary and internal layers present in the problem. Finally, some representative adaptive results in a complex 3D heated‐pipe geometry are presented.

This work demonstrates the feasibility of stabilized, adaptive finite element schemes for computing simple double‐diffusive flow models, and it represents an easily‐generalizable starting point for more complex calculations since it is based on a highly‐general finite element library. The complementary nature of h‐adaptivity and stabilized finite element techniques for this class of problem is demonstrated using particularly simple error indicators and stabilization parameters. Finally, an interesting double‐diffusive convection benchmark problem having a semi‐permeable barrier is suggested.

A mathematical model to predict the concentration polarisation in nanofiltration/reverse osmosis is described. It incorporates physical modelling for mass transfer, laminar hydrodynamics and the membrane rejection coefficient. The SIMPLE algorithm solves the discretised equations derived from the governing differential equations. The convection and diffusive terms of those equations are discretised by the upwind, the hybrid and the exponential schemes for comparison purposes. The hybrid scheme appears as the most suitable one for the type of flows studied herein. The model is first applied to predict the concentration polarisation in a slit, for which mathematical solutions for velocities and concentrations exist. Different grids are used within the hybrid scheme to evaluate the model sensitivity to the grid refinement. The 55×25 grid results agree excellently for engineering purposes with the known solutions. The model, incorporating a variation law for the membrane intrinsic rejection coefficient, was also applied to the predictions of a laboratory slit where experiments are performed and reported, yielding excellent results when compared with the experiments.

When a multi component alloy solidifies the redistribution of solute components leads to the formation of macrosegregation patterns. Blending ideas from a number of recent publications the purpose of this paper is to provide a “best practice” on how grid convergence of a given macrosegregation simulation can be measured and determined.

The best practice is arrived at by considering a benchmark problem consisting of a 2D-casting simulation of an idealized Al-4.5%Cu alloy in a side cooled square (76×76 mm) cavity. The model for this simulation is based on a mixture treatment of the relevant heat and mass transfer equations. Simulations are made using three increasingly refined grid sizes.

The best practice to determine grid resolution involves two steps: first, a visual evaluation of predicted segregation images leading to the evaluation of solute profiles along selected transects; and second, the construction of a cumulative distribution function (CDF) of the predicted segregation field. On application to the benchmark problem, it is concluded that current computer resources are insufficient to grid resolve macrosegregation patterns but that the CDF provides a useful signal of the nature of macrosegregation in a given system.

The benchmark is chosen to be representative. Exact convergence behavior, however, may depend on the system chosen.

In addition to establishing a best practice for measuring grid resolution of macrosegregation simulations the work also highlights, even in the absence of complete grid convergence, how the recently proposed CDF treatment can inform solidification modeling and process understanding.

The purpose of this paper is used to study the combined effect of solute gradient and magnetic field on dusty couple-stress fluid in the presence of rotation through a porous medium.

The perturbation technique (experimental method) is applied in this study.

For the case of stationary convection, solute gradient and rotation have stabilizing effect, whereas destabilizing effect is found in dust particles in the system. Couple stress and medium permeability both have dual character to its stabilizing effect in the absence of magnetic field and rotation. Magnetic field succeeded in establishing a stabilizing effect in the absence of rotation.

The results are discussed by allowing one variable to vary and keeping other variables constant, as well as by drawing graphs.

This paper aims to investigate the onset of convection, heat and mass transports in a sparse porous layer saturated with chemically reactive binary fluid mixture heated and salted from below under the influence of Soret and Dufour effects.

The Brinkman model is used for momentum equation. Linear stability analysis based on normal mode technique is used to evaluate the onset threshold for stationary and oscillatory convection. In weak-nonlinear theory, the truncated Fourier series method is used. The resulting system of differential equations is solved numerically by using the Runge–Kutta fourth-order method.

Because of the competition between the processes of thermal, solute diffusions, chemical reaction and cross-diffusions, the onset of instability is via oscillatory mode instead of stationary. The effect of dissolution/precipitation of reactive component and the cross-diffusions on the stability, heat and mass transports is investigated.

By the proper adjustment of underlying parameters, the onset of convection can either be advanced or delayed as per the requirement. Therefore, the present investigation forms a useful tool for regulating the onset of convection.

The purpose of this paper is to numerically study the dissolution of solid particles using the smoothed particle hydrodynamics (SPH) method.

To implement dissolution, an advection–diffusion mass transport equation is solved over computational particles. Subsequently, these particles disintegrate from the solute when their concentration falls below a certain threshold.

It is shown that the implementation of dissolution is in good agreement with available data in the literature. The dissolution of solid particles is studied for a wide range of Reynolds and Schmidt numbers. Two-dimensional (2D) results are compared with three-dimensional (3D) cases to identify where 2D results are accurate for modelling 3D dissolution phenomena.

The present numerical model is capable of addressing related problems in pharmaceutical, biochemical, food processing and detergent industries.

The purpose of this study is to apply Buongiorno’s two phase model to analyse double diffusion natural convection in a square enclosure filled with nanofluids.

A computational code based on the SIMPLE algorithm and finite volume method is used to solve the non-dimensional governing equations.

The nanoparticle plays a crucial role when thermal and solutal buoyancy forces are equal and opposing.

This is the first paper to apply Buongiorno’s two phase model for double diffusion natural convection in enclosures filled with nanofluids.

The problem aims to find the effects of coupled cross-diffusion in micropolar fluid oversaturated porous medium, subjected to Double-Diffusive Chandrasekhar convection.

Normal mode and perturbation technique have been employed to determine the critical Rayleigh number. Non-linear analysis is carried out by deriving the Lorenz equations using truncated Fourier series representation. Heat and Mass transport are quantified by Nusselt and Sherwood numbers, respectively.

Analysis related to the effects of various parameters is plotted, and the results for the same are interpreted. It is observed from the results that the Dufour parameter and Soret parameter have an opposite influence on the system of cross-diffusion.

The effect of the magnetic field on the onset of double-diffusive convection in a porous medium coupled with cross-diffusion in a micropolar fluid is studied for the first time.

The purpose of this paper is to numerically study the unsteady double-diffusive mixed convective stagnation-point flow of a water-based nanofluid accompanied with one salt past a vertical flat plate. The effects of Brownian motion and thermophoresis parameters are also introduced through Buongiorno’s two-component nonhomogeneous equilibrium model in the governing equations.

In the present explanation of double-diffusive mixed convective model, there are four boundary layers entitled: velocity, thermal, solutal concentration and nanoparticle concentration. The resulting basic equations are solved numerically via an efficient Runge–Kutta fourth-order method with shooting technique after the governing nonlinear partial differential equations are converted into a system of nonlinear ordinary differential equations by the use of similarity transformations.

To avail the physical insight of problem, the effects of the mixed convection parameter, unsteadiness parameter and salt/nanoparticle parameters on the boundary layers behavior are investigated. Moreover, four possible types of diffusion problems entitled: double-diffusive nanofluid (DDNF), double-diffusive regular fluid (DDRF), mono-diffusive nanofluid (MDNF) and mono-diffusive regular fluid (MDRF) are considered to analyze and compare them in concepts of heat and mass transfer.

The results demonstrate that, for a regular fluid, without nanoparticle and salt (MDRF), the dimensionless heat transfer rate is smaller than other diffusion cases. As we include nanoparticle and salt (DDNF), the rate of heat transfer increases due to an increase in thermal conductivity and rate of diffusion of salt. Moreover, it is observed that the highest heat transfer rate is obtained for the situation that the thermophoretic effect of nanoparticles is negligible. Besides, the heat transfer rate enhances with the increase in the regular double-diffusive buoyancy parameter of salt.

The objective of this paper is to establish a solution strategy for obtaining dual solutions, namely trivial (conventional) and nontrivial (unconventional) solutions, of coupled pore-fluid flow and chemical dissolution problems in heterogeneous porous media.

Through applying a perturbation to the pore-fluid velocity, original governing partial differential equations of a coupled pore-fluid flow and chemical dissolution problem in heterogeneous porous media are transformed into perturbed ones, which are then solved by using the semi-analytical finite element method. Through switching off and on the applied perturbation terms in the resulting perturbed governing partial differential equations, both the trivial and nontrivial solutions can be obtained for the original governing partial differential equations of the coupled pore-fluid flow and chemical dissolution problem in fluid-saturated heterogeneous porous media.

When a coupled pore-fluid flow and chemical dissolution system is in a stable state, the trivial and nontrivial solutions of the system are identical. However, if a coupled pore-fluid flow and chemical dissolution system is in an unstable state, then the trivial and nontrivial solutions of the system are totally different. This recognition can be equally used to judge whether a coupled pore-fluid flow and chemical dissolution system involving heterogeneous porous media is in a stable state or in an unstable state. The proposed solution strategy can produce dual solutions for simulating coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media.

A solution strategy is proposed to obtain the nontrivial solution, which is often overlooked in the computational simulation of coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media. The proposed solution strategy provides a useful way for understanding the underlying dynamic mechanisms of the chemical damage effect associated with the stability of structures that are built on soil foundations.