Partial Differential Conservation Equations Research Articles

NUMERICAL STUDY OF SLIP EFFECTS ON UNSTEADY ASYMMETRIC BIOCONVECTIVE NANOFLUID FLOW IN A POROUS MICROCHANNEL WITH AN EXPANDING/CONTRACTING UPPER WALL USING BUONGIORNO’S MODEL

In this paper, the unsteady fully developed forced convective flow of viscous incompressible biofluid that contains both nanoparticles and gyrotactic microorganisms in a horizontal micro-channel is studied. Buongiorno’s model is employed. The upper channel wall is either expanding or contracting and permeable and the lower wall is static and impermeable. The plate separation is therefore a function of time. Velocity, temperature, nanoparticle species (mass) and motile microorganism slip effects are taken into account at the upper wall. By using the appropriate similarity transformation for the velocity, temperature, nanoparticle volume fraction and motile microorganism density, the governing partial differential conservation equations are reduced to a set of similarity ordinary differential equations. These equations under prescribed boundary conditions are solved numerically using the Runge–Kutta–Fehlberg fourth-fifth-order numerical quadrature in the MAPLE symbolic software. Excellent agreement between the present computations and solutions available in the literature (for special cases) is achieved. The key thermofluid parameters emerging are identified as Reynolds number, wall expansion ratio, Prandtl number, Brownian motion parameter, thermophoresis parameter, Lewis number, bioconvection Lewis number and bioconvection Péclet number. The influence of all these parameters on flow velocity, temperature, nanoparticle volume fraction (concentration) and motile microorganism density function is elaborated. Furthermore, graphical solutions are included for skin friction, wall heat transfer rate, nanoparticle mass transfer rate and microorganism transfer rate. Increasing expansion ratio is observed to enhance temperatures and motile microorganism density. Both nanoparticle volume fraction and microorganism increases with an increase in momentum slip. The dimensionless temperature and microorganism increases as wall expansion increases. Applications of the study arise in advanced nanomechanical bioconvection energy conversion devices, bio-nano-coolant deployment systems, etc.

Modeling hyperelasticity in non-equilibrium multiphase flows

The aim of this article is the construction of a multiphase hyperelastic model. The Eulerian formulation of the hyperelasticity represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. This model is constructed with an elegant approach where the specific energy is given in separable form. The system admits 14 eigenvalues with 7 characteristic eigenfields. The associated Riemann problem is not easy to solve because of the presence of 7 waves. The shear waves are very diffusive when dealing with the full system. In this paper, we use a splitting approach to solve the whole system using 3 sub-systems. This method reduces the diffusion of the shear waves while allowing to use a classical approximate Riemann solver. The multiphase model is obtained by adapting the discrete equations method. This approach involves an additional equation governing the evolution of a phase function relative to the presence of a phase in a cell. The system is integrated over a multiphase volume control. Finally, each phase admits its own equations system composed of three sub-systems. One and three dimensional test cases are presented.

Influence of Hall current and microrotation on the boundary layer flow of an electrically conducting fluid: Application to Hemodynamics

In this study, a problem of micropolar fluid flow is analyzed taking into account the effect of Hall current. An external magnetic field is applied transverse to that of fluid flow. The partial differential equations of conservation of mass, linear momentum, angular momentum, energy and concentration are considered along with suitable boundary conditions in formulating the model for the physical problem and analyzing it theoretically. The problem is motivated towards exploring some novel information in the dynamics of blood flow, when the influence of Hall current and rotation of the microparticles of blood, for e.g. the erythrocytes and thrombocytes co-exist at the same platform. The governing equations are reduced to a system of non-linear ordinary differential equations by making use of similarity transformations. The resulting system of coupled non-linear ordinary differential equations is then solved by using a suitable numerical technique that involves the use of finite differences and Newton's linearization method. A parametric study illustrating the influences of the magnetic field, Hall current and the micropolarity of suspended microparticles has been carried out to investigate the variations in velocity, temperature and concentration profiles. The problem has an important bearing on some bio-engineering problems, where the biological conduits, cells and membranes are typically surrounded by fluids, which are electrically conducting (as in the case of blood) and the conduits, cells and membranes that are stretched constantly. The computational results reveal that the microrotation of erythrocytes in blood is enhanced due to the effect of Hall current, when blood flows in the arterial system.

Lie group analysis for bioconvection MHD slip flow and heat transfer of nanofluid over an inclined sheet: Multiple solutions

This paper considers numerical analysis of bioconvection boundary layer flow and heat transfer of electrically conducting nanofluid containing nanoparticles and gyrotactic microorganism over an inclined permeable sheet. Lie symmetry group transformations are used to convert the governing non-linear partial differential equations for continuity, momentum, energy, nanoparticles and microorganisms conservation into non-linear ordinary differential equations. The influences of important physical parameters such as mass transfer parameter s, Richardson number Ri, Buoyancy ratio Nr, bioconvection Rayleigh number Rb, velocity and thermal slip parameters, Brownian motion parameter Nb, thermophoresis parameter Nt, the bioconvection Schmidt number Scb and the bioconvection Peclet number Pe, on the skin friction, the rate of heat transfer and microorganisms flux are discussed numerically in this study. The dual solutions are obtained for some critical range of mass transfer parameter s and stretching/shrinking parameter χ.

Numerical investigation of radiative optically-dense transient magnetized reactive transport phenomena with cross diffusion, dissipation and wall mass flux effects

High temperature electromagnetic materials fabrication systems in chemical engineering require ever more sophisticated theoretical and computational models for describing multiple, simultaneous thermophysical effects. Motivated by this application, the present paper addresses transient magnetohydrodynamic heat and mass transfer in chemically-reacting fluid flow from an impulsively-started vertical perforated sheet. Thermal radiation flux, internal heat generation (heat source), Joule magnetic heating (Ohmic dissipation), thermo-diffusive and diffuso-thermal (i.e. cross-diffusion) effects and also viscous dissipation are incorporated in the mathematical model. To facilitate numerical solutions of the coupled, nonlinear boundary value problem, non-similar transformations are employed and the partial differential conservation equations are normalized into a dimensionless system of momentum, energy and concentration equations with associated boundary thermal conditions. An implicit finite difference method (FDM) is utilized to solve the unsteady equations. Verification of the FDM solutions for dimensionless velocity, temperature and concentration functions is achieved with a variational finite element method code (MAGNETO-FEM) and also a network simulation method code (MAG-PSPICE). The influence of the emerging thermo-physical parameters on transient velocity, temperature, concentration, wall shear stress, Nusselt number and Sherwood number is elaborated. The flow is accelerated with increasing thermal radiative flux, Eckert number, heat generation and Soret number whereas the flow is decelerated with greater wall suction, heat absorption, magnetic field and Prandtl number. Temperatures are also observed to be elevated with magnetic parameter, radiation heat transfer, Dufour number, heat generation (source) and Eckert number with the contrary effects computed for increasing suction parameter or Prandtl number. The species concentration is enhanced with Soret number and generative chemical reaction whereas it is depressed with greater wall suction, Schimidt number and destructive chemical reaction parameter

Numerical solution of thermo-solutal mixed convective slip flow from a radiative plate with convective boundary condition

A mathematical model for mixed convective slip flow with heat and mass transfer in the presence of thermal radiation is presented. A convective boundary condition is included and slip is simulated via the hydrodynamic slip parameter. Heat generation and absorption effects are also incorporated. The Rosseland diffusion flux model is employed. The governing partial differential conservation equations are reduced to a system of coupled, ordinary differential equations via Lie group theory method. The resulting coupled equations are solved using shooting method. The influences of the emerging parameters on dimensionless velocity, temperature and concentration distributions are investigated. Increasing radiative-conductive parameter accelerates the boundary layer flow and increases temperature whereas it depresses concentration. An elevation in convection-conduction parameter also accelerates the flow and temperatures whereas it reduces concentrations. Velocity near the wall is considerably boosted with increasing momentum slip parameter although both temperature and concentration boundary layer thicknesses are decreased. The presence of a heat source is found to increase momentum and thermal boundary layer thicknesses but reduces concentration boundary layer thickness. Excellent correlation of the numerical solutions with previous non-slip studies is demonstrated. The current study has applications in bioreactor diffusion flows and high-temperature chemical materials processing systems.

MHD free convection flow of Eyring–Powell fluid from vertical surface in porous media with Hall/ionslip currents and ohmic dissipation

A mathematical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flow, heat and mass transfer of non-Newtonian Eyring–Powell fluid from a vertical surface in a non-Darcy, isotropic, homogenous porous medium, in the presence of Hall currents and ionslip currents. The governing nonlinear coupled partial differential equations for momentum conservation in the x, and z directions, heat and mass conservation, in the flow regime are transformed from an (x,y,z) coordinate system to a (ξ,η) coordinate system in terms of dimensionless x-direction velocity (f′) and z-direction velocity (G), dimensionless temperature and concentration functions (θ and ϕ) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also heat and mass transfer rates. It is observed that with increasing ɛ, primary velocity, secondary velocity, heat and mass transfer rates are decreased whereas, the temperature, concentration and skin friction are increased. An increasing δ is found to increase primary and secondary velocities, skin friction, heat and mass transfer rates. But the temperature and concentration decrease. Increasing βe and βi are seen to increase primary velocity, skin friction, heat and mass transfer rates whereas secondary velocity, temperature and concentration are decreased. Excellent correlation is achieved with a Nakamura tridiagonal finite difference scheme (NTM). The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Computational Study of MHD Free Convection Flow of Non-Newtonian Tangent Hyperbolic Fluid from a Vertical Surface in Porous Media with Hall/Ionslip Currents and Ohmic Dissipation

A numerical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flows of non-Newtonian tangent hyperbolic fluid past a vertical surface in a non-Darcy, isotropic, homogenous porous medium in the presence of Hall currents and Ionslip currents. The governing nonlinear coupled partial differential equations for momentum conservation in x and z directions, heat and mass conservation in the flow regime are transformed from an (x, y, z) coordinate system to (\(\upxi ,\upeta \)) coordinate system in terms of dimensionless x-direction velocity (\(f^{\prime }\)) and z-direction velocity (G), dimensionless temperature and concentration functions (\(\uptheta \) and \(\upphi \)) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also heat and mass transfer rates. Increasing Weissenberg number (We) is observed to decrease primary and secondary velocity and concentration but increase temperature. It is found that the primary and secondary velocity is increased with increasing power law index (n) whereas the temperature and concentration are decreased. Increasing hall and Ionslip current (\(\beta _{e}\) and \(\beta _{i}\)) is observed to increase primary velocity but decreases secondary velocity, temperature and concentration. Increasing magnetic parameter (\(N_{m}\)) is seen to decrease primary velocity but decreases secondary velocity, temperature and concentration. Increasing Darcy number (Da) is found to increase primary and secondary velocity whereas decreases temperature and concentration. Increasing Forchheimer parameter (Fs) is seen to decrease primary and secondary velocity but increases temperature and concentration. The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Forced convection heat and mass transfer of MHD nanofluid flow inside a porous microchannel with chemical reaction on the walls

Purpose – The purpose of this paper is to focus on convective heat and mass transfer characteristics of Cu-water nanofluid inside a porous microchannel in the presence of a uniform magnetic field. The walls of the microchannel are subjected to constant asymmetric heat fluxes and also the first order catalytic reaction. To represent the non-equilibrium region near the surfaces, the Navier’s slip condition is considered at the surfaces because of the non-adherence of the fluid-solid interface and the microscopic roughness in microchannels. Design/methodology/approach – Employing the Brinkman model for the flow in the porous medium and the “clear fluid compatible” model as a viscous dissipation model, the conservative partial differential equations have been transformed into a system of ordinary ones via the similarity variables. Closed form exact solutions are obtained analytically based on dimensionless parameters of velocity, temperature and species concentration. Findings – Results show that the addition of Cu-nanoparticles to the fluid has a significant influence on decreasing concentration, temperature distribution at the both walls and velocity profile along the microchannel. In addition, total heat transfer in microchannel increases as nanoparticles add to the fluid. Slip parameter and Hartmann number have the decreasing effects on concentration and temperature distributions. Slip parameter leads to increase velocity profiles, while Hartmann number has an opposite trend in velocity profiles. These two parameters increase the total heat transfer rate significantly. Originality/value – In the present study, a comprehensive analytical solution has been obtained for convective heat and mass transfer characteristics of Cu-water nanofluid inside a porous microchannel in the presence of a uniform magnetic field. Finally, the effects of several parameters such as Darcy number, nanoparticle volume fraction, slip parameter, Hartmann number, Brinkman number, asymmetric heat flux parameter, Soret and Damkohler numbers on total heat transfer rate and fluid flow profiles are studied in more detail. To the best of author’s knowledge, no study has been conducted to this subject and the results are original.

Computational solutions for non-isothermal, nonlinear magneto-convection in porous media with hall/ionslip currents and ohmic dissipation

A theoretical and numerical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flow and heat transfer in a non-Darcian, isotropic, homogenous porous medium, in the presence of Hall currents, Ionslip currents, viscous heating and Joule heating. A power-law variation is used for the temperature at the wall. The governing nonlinear coupled partial differential equations for momentum conservation in the x and z directions and heat conservation, in the flow regime are transformed from an (x, y, z) coordinate system to a (ξ,η) coordinate system in terms of dimensionless x-direction velocity (∂F/∂η) and z-direction velocity (G) and dimensionless temperature function (H) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also local Nusselt number. Excellent correlation is achieved with a Nakamura tridiagonal finite difference scheme (NTM). The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Basic cerebrospinal fluid flow patterns in ventricular catheters prototypes.

A previous study by computational fluid dynamics (CFD) of the three-dimensional (3-D) flow in ventricular catheters (VC) disclosed that most of the total fluid mass flows through the catheter's most proximal holes in commercially available VC. The aim of the present study is to investigate basic flow patterns in VC prototypes. The general procedure for the development of a CFD model calls for transforming the physical dimensions of the system to be studied into a virtual wire-frame model which provides the coordinates for the virtual space of a CFD mesh, in this case, a VC. The incompressible Navier-Stokes equations, a system of strongly coupled, nonlinear, partial differential conservation equations governing the motion of the flow field, are then solved numerically. New designs of VC, e.g., with novel hole configurations, can then be readily modeled, and the corresponding flow pattern computed in an automated way. Specially modified VCs were used for benchmark experimental testing. Three distinct types of flow pattern in prototype models of VC were obtained by varying specific parameters of the catheter design, like the number of holes in the drainage segments and the distance between them. Specifically, we show how to equalize and reverse the flow pattern through the different VC drainage segments by choosing appropriate parameters. The flow pattern in prototype catheters is determined by the number of holes, the hole diameter, the ratio hole/segment, and the distance between hole segments. The application of basic design principles of VC may help to develop new catheters with better flow circulation, thus reducing the possibility of becoming occluded.

Effect of radiation on heat and mass transfer of fluid flow across a horizontal cylinder

The free convective boundary layer flow, heat and mass transfer across an isothermal cylinder in the diffusing species is investigated. The partial differential conservation equations are non-dimensionalized and solved numerically by using an implicit iterative tridiagonal finite-difference method. The effects of radiation parameters on dimensionless velocity, temperature and species concentration distributions are studied, for the case of water (Pr = 7.0), concentration is slightly affected by radiation parameter; Temperatures are however increased by an increase in radiation parameter. Furthermore, the local skin friction, local Nusselt number and local Sherwood number have been analyzed for various parametric conditions. It is found that as the thermal radiation parameter increases the skin-friction coefficient increases and the local Nusselt number decreases. Excellent agreement is obtained with the earlier studies for the purely fluid.

Analysis and Multi-Dimensional Modeling of Lithium-Air Batteries

This study contributes to: 1) a multi-dimensional model framework of lithium-air (Li-air) battery, 2) incorporation of mechanisms of insoluble precipitates’ impacts, and 3) analysis and discussion on oxygen supply channel for Li-air battery. The model consists of a set of partial differential equations of species and charge conservation, in conjunction with the electrochemical reaction kinetics, and takes into account the two major mechanisms of voltage loss caused by insoluble discharge products: namely, electrode passivation and increased oxygen transport resistance. Two-dimensional (2-D) simulation indicates that the pore space in the cathode electrode is not fully utilized for Li compounds storage, particularly under high discharging current. For selected battery designs, considerable variation of quantities is observed only in the thickness direction. Through analysis, we evaluate the oxygen concentration drop along an oxygen supply channel and relate it to the Damköhler (Da) number, and further explore potential cases that yield oxygen starvation.

Effect of magnetic fields on heat convection inside a concentric annulus filled with Al2O3–water nanofluid

In the current study, forced convective heat transfer of an MHD fully developed laminar nanofluid between two concentric horizontal cylinders is investigated in the presence of a radial magnetic field. In contrast to a conventional no-slip condition at the surfaces, the Navier’s slip condition is considered at the surface to represent the non-equilibrium region near the surfaces. Employing the modified Buongiorno model, the conservative partial differential equations have been collapsed to two-point ordinary boundary value differential equations before being numerically solved. To consider the effects of thermal boundary condition on nanoparticle migration, two distinctive cases including constant heat flux at the outer wall and adiabatic inner wall (Case A) and constant heat flux at the inner wall with adiabatic outer wall (Case B) have been considered. Our results indicate that due to thermophoresis force, the distribution of nanoparticles was denser at the adiabatic wall for the case A which affects the local and the universal fluid flow and heat transfer characteristics. Moreover, inducing a radial magnetic field on the system, heat transfer rate was increased for the case A which had a decreasing effect on the case B. Finally, slip velocity at the walls enhances heat transfer rate for both cases.

A thermodynamically compatible splitting procedure in hyperelasticity

A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations.In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.

A general framework for finding energy dissipative/conservative $$H^1$$ H 1 -Galerkin schemes and their underlying $$H^1$$ H 1 -weak forms for nonlinear evolution equations

A general framework for constructing energy dissipative or conservative Galerkin schemes for time-dependent partial differential equations (PDEs) is presented. The framework embraces a variety of dissipative and conservative PDEs with variational structure. An advantageous feature is that implementation of the resulting scheme requires only P1 elements. The schemes and their underlying \(H^1\)-weak forms are derived from the concept of formal weak form and an \(L^2\)-projection technique.

BIOCONVECTION IN A NON-DARCY POROUS MEDIUM SATURATED WITH A NANOFLUID AND OXYTACTIC MICRO-ORGANISMS

The aim of this paper is to present a continuum model for bioconvection of oxytactic micro-organisms in a non-Darcy porous medium and to investigate the effects of bioconvection and mixed convection on the steady boundary layer flow past a horizontal plate embedded in a porous medium filled with a water-based nanofluid. The governing partial differential equations for momentum, heat, oxygen and micro-organism conservation are reduced to a set of nonlinear ordinary differential equations using similarity transformations that are numerically solved using a built-in MATLAB ODE solver. The effects of the bioconvection parameters on the nanofluid fluid properties, nanoparticle concentration and the density of the micro-organism are analyzed. A comparative analysis of our results with those previously reported in the literature is given. Among the significant findings in this study is that bioconvection parameters highly influence heat, mass and motile micro-organism transfer rates.

Analysis and Three-Dimensional Modeling of Vanadium Flow Batteries

This study presents 1.) a multi-dimensional model of vanadium Redox Flow Batteries (RFB); 2.) rigorous explanation of pore-level transport resistance, dilute solution assumption, and pumping power; and 3.) analysis of time constants of heat and mass transfer and dimensionless parameter. The model, describing the dynamic system of a RFB, consists of a set of partial differential equations of mass, momentum, species, charges, and energy conservation, in conjunction with the electrode's electrochemical reaction kinetics. The governing equations are successfully implemented into three-dimensional numerical simulation of charging, idling, and discharging operations. The model, validated against experimental data, predicts fluid flow, concentration increase/decrease, temperature contours and local reaction rate. The prediction indicates a large variation in local reaction rate across electrodes and the time constants for reactant variation and temperature evolution, which are consistent with theoretical analysis.

Unsteady Free Convection in a Walter’s-B Viscoelastic Flow past a Semi-Infinite Vertical Plate with Radiation and Chemical Reaction

A numerical solution for the free convective, unsteady, laminar convective heat and mass transfer in a viscoelastic fluid along a semi-infinite vertical plate with radiation and chemical reaction is presented. The Walters-B liquid model is employed to simulate medical creams and other rheological liquids encountered in biotechnology and chemical engineering. This rheological model introduces supplementary terms into the momentum conservation equation. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Prandtl number, viscoelasticity parameter, Schmidt number, radiation parameter, chemical reaction parameter and buoyancy parameters. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed graphically. It is observed that, when the viscoelasticity parameter increases, the velocity increases close to the plate surface. An increase in Schmidt number is observed to significantly decrease both velocity and concentration.

Network and Nakamura tridiagonal computational simulation of electrically-conducting biopolymer micro-morphic transport phenomena

Magnetic fields have been shown to achieve excellent fabrication control and manipulation of conductive bio-polymer characteristics. To simulate magnetohydrodynamic effects on non-Newtonian electro-conductive bio-polymers (ECBPs) we present herein a theoretical and numerical simulation of free convection magneto-micropolar biopolymer flow over a horizontal circular cylinder (an “enrobing” problem). Eringen's robust micropolar model (a special case of the more general micro-morphic or “microfluid” model) is implemented. The transformed partial differential conservation equations are solved numerically with a powerful and new code based on NSM (Network Simulation Method) i.e. PSPICE. An extensive range of Hartmann numbers, Grashof numbers, micropolar parameters and Prandtl numbers are considered. Excellent validation is also achieved with earlier non-magnetic studies. Furthermore the present PSPICE code is also benchmarked with an implicit tridiagonal solver based on Nakamura's method (BIONAK) again achieving close correlation. The study highlights the excellent potential of both numerical methods described in simulating nonlinear biopolymer micro-structural flows.