Modeling of 2D density-dependent flow and transport in porous media using finite volume method

A two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular mesh is developed for modeling density-dependent flow and transport through saturated-unsaturated porous media. The combined flow and transport model can handle a wide range of real-word problems, including the simulation of flow alone, contaminant transport alone, and combined flow and transport. Saltwater intrusion problems and instability caused by denser water on the top were investigated in this paper. Because the fundamental mechanism causing saltwater intrusion most likely is caused by density-induced convection and dispersion, the developed model is used to assess the interplay between density-driven flow and subsurface media through which the saltwater intrusion occurs.The mathematical formulation of the model is comprised of fluid flow and solute transport equations, coupled by fluid density. In the specific case of saltwater intrusion and unstable brine transport problems, this set of governing equation is non-linear and requires iterative methods to solve them simultaneously. Three case studies, which include a wide range of physical conditions, are used for verification of presented model and comparison with previously published solutions from other researchers presents encouraging agreements.     

Homogenous–heterogeneous reactions in MHD flow of Powell–Eyring fluid over a stretching sheet with Newtonian heating

This article addresses the effects of homogenous–heterogeneous reactions on electrically conducting boundary layer fluid flow and heat transfer characteristics over a stretching sheet with Newtonian heating are examined. Using similarity transformations, the governing equations are transformed into nonlinear ordinary differential equations. The constricted ordinary differential equations are solved computationally by shooting technique. The impact of pertinent physical parameters on the velocity, concentration and temperature profiles is discussed and explored via figures and tables. It is clear from figures that the velocity profile reduces for large values of fluid parameter B and Hartmann number H. Skin friction coefficient decreases for large values of Hartmann number H and fluid parameter B. Also, heat transfer rate monotonically enhances with conjugate parameter of Newtonian heating γ and Prandtl number Pr.

     

Numerical solution of shear-thinning and shear-thickening boundary-layer flow for Carreau fluid over a moving wedge

This paper investigates the linear stability of the flow in the two-dimensional boundary-layer flow of the Carreau fluid over a wedge. The corresponding rheology is analysed using the non-Newtonian Carreau fluid. Both mainstream and wedge velocities are approximated in terms of the power of distance from the leading edge of the boundary layer. These forms exhibit a class of similarity flows for the Carreau fluid. The governing equations are derived from the theory of a non-Newtonian fluid which are converted into an ordinary differential equation. We use the Chebyshev collocation and shooting techniques for the solution of governing equations. Numerical results show that the viscosity modification due to Carreau fluid makes the boundary layer thickness thinner. Numerical results predict an additional solution for the same set of parameters. Thus, a further aim was to assess the stability of dual solutions as to which of the solutions can be realized. This leads to an eigenvalue problem in which the positive eigenvalues are important and intriguing. The results from eigenvalues form tongue-like structures which are rather new. The presence of the tongue means that flow becomes unstable beyond the critical value when the velocity ratio is increased from the first solution.

     

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

     

Coupled model of surface water flow, sediment transport and morphological evolution

This paper presents a mathematical model coupling water flow and sediment transport dynamics that enables calculating the changing surface morphology through time and space. The model is based on the shallow water equations for flow, conservation of sediment concentration, and empirical functions for bed friction, substrate erosion and deposition. The sediment transport model is a non-capacity formulation whereby erosion and deposition are treated independently and influence the sediment flux by exchanging mass across the bottom boundary of the flow. The resulting hyperbolic system is solved using a finite volume, Godunov-type method with a first-order approximate Riemann solver. The model can be applied both to short time scales, where the flow, sediment transport and morphological evolution are strongly coupled and the rate of bed evolution is comparable to the rate of flow evolution, or to relatively long time scales, where the time scale of bed evolution associated with erosion and/or deposition is slow relative to the response of the flow to the changing surface and, therefore, the classical quasi-steady approximation can be invoked. The model is verified by comparing computed results with documented solutions. The developed model can be used to investigate a variety of problems involving coupled flow and sediment transport including channel initiation and drainage basin evolution associated with overland flow and morphological changes induced by extreme events such as tsunami.     

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

A conservative TVD scheme is adopted to solve the equations governing the three-dimensional flow of a nonideal compressible conducting fluid in a magnetic field.The eight-wave equations for magnetohydrodynamics(MHD) are proved to be a non-strict hyperbolic system,therefore it is difficult to develop its eigenstructure.Powell developed a new set of equations which cannot be numerically simulated by conservative TVD scheme directly due to its non-conservative form.A conservative TVD scheme augmented with a ne...     

Deposition of suspensions in the entrance of a parallel-plate channel with gravity effect

The deposition of aerosols near the inlet region of a horizontal parallel channel (for a distance of 7 channel widths) was investigated by solving numerically the governing equations for both the fluid and particulate phases with boundary layer assumptions. Surface adhesion between the channel walls and the aerosol particles along with the gravitational influence was varied.This analysis took into consideration the simultaneous development of both the fluid and particle phases. The deposition for the present analysis for high surface adhesion was found to be greater for $\text{X ≥ 2}$ than the deposition obtained by Ingham for Plug flow who used only the diffusion equation with zero particulate density at the wall.     

A model for the metal deposition process

In IC industry, the manufacturing technology trends toward more precision and more miniaturization, such as from 28 to 14 nm process. This nano-manufacturing, metal deposition process must be studied to improve higher quality and more reliability production rate of produce. Hence, a model for the metal deposition process is necessary. This model of a vacuum metal film deposition system for substrate application with normal metal material is considered. In this model, the finite difference method is employed to discretize the solution domain into meshes and nodes. The numerical solution of electric potential or electric intensity of the metal film sputtering system can be obtained by MATLAB numerical simulation software, and to solve these equations of the discrete nodes. Numerical analysis results indicate that this simulation model can be employed to study the coating quality of the metal deposition process system.

     

Dusty flow with different water based nanoparticles along a paraboloid revolution: thermal analysis

The thermal analysis on hydromagnetic two-dimensional flow of dusty nano fluid along an upper horizontal surface of paraboloid revolution have been scrutinized. The governing flow are derived under the assumptions of Boussinesq’s boundary layer approximation theory. The effects of Cattaneo-Christov heat flux, variable thermal conductivity, joule heating and viscous dissipation are incorporated in the energy equation. The governing PDE’s for the flow and energy transfer for both the phases are transformed into ODE’S by employing the suitable similarity transformations. The final dimensionless governing coupled ordinary differential equations are resolved with the aid of bvp5c procedure in computational Matlab software. The effects of dimensionless governing controlled flow parameters on velocity, micropolar velocity, and temperature profiles for both the phases are reported and discussed elaborately through plots and tables. The emerging three nanoparticles namely gold, silver and platinum (\(Au,Ag\) and \(Pt\)) are considered throughout graphical analysis along with \(H_{2} O\) is used as a base liquid. It is revealed that the flow velocity declined for strengthen of the applied magneticfield. It is worthy note that the larger values of thermal relaxation parameter \(\gamma\) declines the fluid temperature for both phases. Also, the rate of heat transfer is an increasing function to the escalating values of variable thermal conductivity \(\varepsilon\), while it is reverse trend for the thermal relaxation parameter \(\gamma\). The observations exhibit the prominent features in the field of an advanced bio-medical and thermal engineering.

     

Diffusion of a chemically reactive species of a power-law fluid past a stretching surface

A numerical solution for the steady magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a continuously moving surface with species concentration and chemical reaction has been obtained. The viscous flow is driven solely by the linearly stretching sheet, and the reactive species emitted from this sheet undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. Using a similarity transformation, the governing non-linear partial differential equations are transformed into coupled nonlinear ordinary differential equations. The governing equations of the mathematical model show that the flow and mass transfer characteristics depend on six parameters, namely, the power-law index, the magnetic parameter, the local Grashof number with respect to species diffusion, the modified Schmidt number, the reaction rate parameter, and the wall concentration parameter. Numerical solutions for these coupled equations are obtained by the Keller-Box method, and the solutions obtained are presented through graphs and tables. The numerical results obtained reveal that the magnetic field significantly increases the magnitude of the skin friction, but slightly reduces the mass transfer rate. However, the surface mass transfer strongly depends on the modified Schmidt number and the reaction rate parameter; it increases with increasing values of these parameters. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

Thermophoresis and MHD mixed convection three-dimensional flow of viscoelastic fluid with Soret and Dufour effects

Heat and mass transfer effects in three-dimensional mixed convection flow of viscoelastic fluid over a stretching surface with convective boundary conditions are investigated. The fluid is electrically conducting in the presence of constant applied magnetic field. Conservation laws of energy and concentration are based upon the Soret and Dufour effects. First order chemical reaction effects are also taken into account. By using the similarity transformations, the governing boundary layer equations are reduced into the ordinary differential equations. The transformed boundary layer equations are computed for the series solutions. Dimensionless velocity, temperature, and concentration distributions are shown graphically for different values of involved parameters. Numerical values of local Nusselt and Sherwood numbers are computed and analyzed. It is found that the behaviors of viscoelastic, mixed convection, and concentration buoyancy parameters on the Nusselt and Sherwood numbers are similar. However, the Nusselt and Sherwood numbers have qualitative opposite effects for Biot number, thermophoretic parameter, and Soret-Dufour parameters.

     

Analysis of magnetohydrodynamic flow and heat transfer of Cu–water nanofluid between parallel plates for different shapes of nanoparticles

The present study analyzes the heat transfer in the flow of copper–water nanofluids between parallel plates. For effective thermal conductivity of nanofluids, Hamilton and Crosser's model has been utilized to examine the flow by considering different shape factors. By employing the suitable similarity transformations, the equations governing the flow are transformed into a set of nonlinear ordinary differential equations. The resulting set of equations is solved numerically with the help of Runge–Kutta–Fehlberg numerical scheme. The graphical simulation presents the analysis of variations, in velocity and temperature profiles, for emerging parameters. A comprehensive discussion also accompanies the graphical results. Moreover, the effects of relevant parameters, on skin friction coefficient and Nusselt number, are highlighted graphically. It is noticed that the velocity field is an increasing function of all the parameters involved. Furthermore, the temperature of the fluid is maximum for the platelet-shaped particles followed by the cylinder- and brick-shaped particles.

     

Field equations for incompressible non-viscous fluids using artificial intelligence

In this article, we introduce new field equations for incompressible non-viscous fluids, which can be treated similarly to Maxwell’s electromagnetic equations based on artificial intelligence algorithms. Lagrangian and Hamiltonian formulations are used to arrive at field equations that are solved using convolutional neural networks. Four linear differential equations, which describe the two fields, namely, the dynamic pressure and the vortex fields, are derived, and these can be used in place of Euler’s equation. The only assumption while deriving this equation is that the dynamic pressure and vortex fields obey the superposition principle. The important finding to be noted is that Euler’s fluid equations can be converted into field equations analogous to Maxwell’s electromagnetic equations. We solve the flow problem for laminar flow past a cylinder, sphere, and cone in two dimensions similar to the conduction in a uniform electric field and arrive at closed-form expressions. These closed-form expressions, which are obtained for the potentials of fluid flow, are similar to the streamline potential functions in the case of fluid dynamics.

     

Real-time simulation of physically based on-surface flow

Although many papers have been published in the field of fluid simulation, little attention has been paid to on-surface flow involving wetting and stain transportation as well as erosion and deposition phenomena. In this paper, we introduce nonzero divergence in the mass equation of Navier–Stokes equations to simulate water penetration from on-surface flow into substrate material. Also, the volume of fluid method is adopted to track the free surface. With a computation of the actual amount of absorbed water we render the wetting effects with fully dry and fully wet texture images simultaneously. Using our model, the on-surface flow that accompanies water absorption can be simulated realistically in real time with OpenGL preview rendering. Experimental results illustrate that our model can be widely applied to solve various problems related to on-surface flow.     

A comprehensive theoretical model of capillary transport in rectangular microchannels

A detailed theoretical model of capillary transport in rectangular microchannels is proposed. Two important aspects of capillary transport are revisited, which are considered with simplified assumption in the literature. The capillary flow is assumed as a low Reynolds number flow and hence creeping flow assumptions are considered for majority of analyses. The velocity profile used with this assumption results into a steady state fully developed velocity profile. The capillary flow is inherently a transient process. In this study, the capillary flow analysis is performed with transient velocity profile. The pressure field expression at the entrance of the microchannel is another aspect which is not often accurately represented in the literature. The approximated pressure field expression at the entrance of the rectangular microchannel is widely used in the literature. An appropriate entrance pressure field expression for a rectangular microchannel is proposed. For both analyses, the governing equation of the capillary transport in rectangular microchannel is derived by applying the momentum equation to the fluid control volume along the microchannel. The non-dimensional governing equations are obtained, each for a transient velocity profile and a newly proposed pressure field, for analyzing the importance of such velocity profile and pressure field expression.     

Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity

The effect of temperature gradient orientation on the fluid flow and heat transfer in a lid-driven differentially heated square cavity is investigated numerically. The transport equations are solved using the high-order compact scheme. Four cases are considered depending on the direction of temperature gradient imposed. The differentially heated top and bottom walls result in gravitationally stable and unstable temperature gradients. While the differentially heated left and right side walls lead to assisting and opposing buoyancy effects. The governing parameters are Pr = 0.7 and Ri = 0.1, 1, and 10. It is found that both Richardson number and direction of temperature gradient affect the flow patterns, heat transport processes, and heat transfer rates in the cavity. Computed average Nusselt number indicates that the heat transfer rate increases with decreasing Ri regardless the orientation of temperature gradient imposed. And the assisting buoyancy flows have best performance on heat transport over the other three cases.     

Magnetohydrodynamic three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions

The present research focuses on three-dimensional nonlinear convective flow of viscoelastic nanofluid. Here, the flow is generated due to stretching of a impermeable surface. The phenomenon of heat transport is analyzed by considering thermal radiation and prescribed heat flux condition. Nanofluid model comprises of Brownian motion and thermophoresis. An electrically conducting fluid is accounted due to consideration of an applied magnetic field. The dimensionless variables are introduced for the conversion of partial differential equations into sets of ordinary differential systems. The transformed expressions are explored through homotopic algorithm. Behavior of different dimensionless parameters on the non-dimensional velocities, temperature and concentration are scrutinized graphically. The values of skin friction coefficients, Nusselt and Sherwood numbers are also calculated and elaborated. It is visualized that the heat transfer rate increases with Prandtl number and radiation parameter is higher.

     

Heat transfer effects on carbon nanotubes suspended nanofluid flow in a channel with non-parallel walls under the effect of velocity slip boundary condition: a numerical study

The present article is dedicated to analyze the flow and heat transfer of carbon nanotube (CNT)-based nanofluids under the effects of velocity slip in a channel with non-parallel walls. Water is taken as a base fluid, and two forms of CNTs are used to perform the analysis, namely the single- and multi-walled carbon nanotubes (SWCNTs and MWCNTs, respectively). Both the cases of narrowing and widening channel are discussed. The equations governing the flow are obtained by using an appropriate similarity transform. Numerical solution is obtained by using a well-known algorithm called Runge–Kutta–Fehlberg method. The influence of involved parameters on dimensionless velocity and temperature profiles is displayed graphically coupled with comprehensive discussions. Also, to verify the numerical results, a comparative analysis is carried out that ensures the authenticity of the results. Variation of skin friction coefficient and the rate of heat transfer at the walls are also performed. Some already existing solutions of the particular cases of the same problem are also verified as the special cases of the solutions obtained here.