Double Stratification Research Articles

Discussion: “Double Stratification in Flow by Curved Stretching Sheet With Thermal Radiation and Joule Heating” (Hayat, T., Qayyum, S., Imtiaz, M., and Alsaedi, A., 2018, J. Therm. Sci. Eng. Appl., 10(2), 021010)

Abstract The present discussion concerns some doubtful results included in the above paper.

Rheology of mixed convective Casson nanofluid in a convectively heated stratified medium

A mathematical framework for mixed convective non-Newtonian nanoliquid subjected to hydromagnetic characteristics is addressed in this attempt. The well-known Buongiorno nanoliquid model which elaborates thermophoretic and Brownian diffusions aspect is opted for formulation. In addition, heat generation, double stratification and convective conditions are retained. Apposite variables are introduced for dimensionless procedure. Homotopy scheme is employed to attain convergent expressions. Graphs are presented for description of sundry variables effect versus dimensionless quantities.

The study of double stratification and viscous dissipation effects on chemically radiative MHD free convective flow of micropolar fluid embedded in non-Darcian porous medium

Numerical expression for steady 2D natural convection magnetohydrodynamics flow of micropolar fluid past along a vertical surface embedded through non-Darcian porous medium in the presence of heat source, chemical reaction, viscous dissipation and thermal radiation is obtained. Variable wall temperature and concentration in a doubly stratified are taken into account. The initial governing boundary layer equations are transformed to a system of ODEs, which are then solved numerically by using the Runge– Kutta–Fehlberg fourth–fifth order method along shooting technique. A parametric study is conducted and so that numerical result are obtained for the velocity, micro-rotation, temperature and concentration as well as the local friction factor coefficient, local couple stress, local Nusselt and Sherwood numbers for different values of the governing parameters, namely, the magnetic parameter, material parameter, Prandtl number, radiation parameter, chemical reaction parameter and viscous dissipation and discussed in detail. Comparison in limiting case is also part of present study to validate the obtained results.

Magnetohydrodynamics (MHD) stagnation point flow past a shrinking/stretching surface with double stratification effect in a porous medium

The present work emphasizes the MHD mixed convective stagnation point flow over a shrinking/stretching surface saturated in a porous medium. The double stratification with heat source effects are also considered while the magnetic field is imposed normal to the sheet. The governing model (partial differential equations) is converted into a system of ordinary (similarity) differential equations using similarity transformations. The boundary value problem solver (bvp4c) in the MATLAB software is utilized for the numerical computations. Numerical results are graphically illustrated in the form of velocity, temperature and concentration profiles for several values of buoyancy, magnetic, thermal and solutal stratification parameters. The graphs of skin friction coefficient, local Nusselt and Sherwood numbers portray that the dual solutions are achievable within a certain range of the buoyancy and velocity ratio parameters. Both assisting and opposing flow cases can generate two solutions, whereas the forced convective flow only produces a unique solution. The execution of stability analysis affirms the reliability of the first solution. Both heat and mass transfer rates intensify with the increment of the velocity ratio parameter for all type of convective flows. The fluid temperature and concentration decrease with the increment of the thermal and solutal stratification parameters, respectively, whereas the magnetic and buoyancy parameters reduce both temperature and concentration profiles.

Effects of double stratification on MHD chemically reacting second‐grade fluid through porous medium between two parallel plates

AbstractThe present study focuses on the study of heat transfer in a magnetohydrodynamic double stratification fluid flow through porous channel. In getting the solution of Navier‐Stokes equations, we have adapted the similarity transformation technique to convert the partial differential equations into a coupled system of nonlinear ordinary differential equations. We further applied the fourth‐order Runge‐Kutta method with shooting technique to solve the problem numerically. The results are exhibited via graphs, and the analysis has been addressed for various flow characteristics with respect to the parameters concerned. It is found that thermal stratification enhances both temperature and heat transfer rates, and solutal stratification enhances concentration and mass transfer rates. Concentration decreases with increasing chemical reaction and it increases with stratification. A significant variation is observed between concentration profiles in the presence of stratification.

Significance of activation energy in stratified flow of tangent hyperbolic fluid

Purpose The purpose of this paper is to address double stratification and activation energy in flow of tangent hyperbolic fluid. Flow is induced by non-linear stretching sheet of variable thickness. Heat flux by Cattaneo–Christov theory is implemented. Design/methodology/approach Non-linear system is computed for the convergent solutions. Attention is particularly focused to the velocity, temperature and concentration. Findings It is found that temperature and thermal layer thickness are decreased for larger stratification. Originality/value In view of aforementioned communication, the aim of the present study is fourfold: First, to inspect stagnation point flow of tangent hyperbolic liquid by a stretched sheet; second, to discuss effect of non-Fourier heat flux and double stratification; third, to investigate activation energy; and fourth, to examine variable thickness effect.

Transportation of activation energy in the Oldroyd-B nanofluid by considering double stratification over a surface with variable thickness

In this communication, the impact of activation energy on the nonlinear binary chemically reactive flow of an Oldroyd-B nanofluid has been examined. Buongiorno’s nanofluid model is used in mathematical modelling. The flow behaviour is discussed over a nonlinear stretchable surface with variable thickness. Nonlinear mixed convection is considered. The energy equation is modelled subject to a heat source / sink and radiative flux. Furthermore, double stratification at the boundary of the sheet is considered for the heat and mass transfers. Important slip mechanisms such as Brownian and thermophoresis diffusions are accounted. The obtained flow expressions are analytically solved by using the optimal homotopy asymptotic method (OHAM). Computational analysis for concentration, temperature and velocity is obtained and discussed using plots. Nusselt and Sherwood numbers are discussed using a tabulated form. Total squared residual error is calculated for velocity, temperature and concentration. The obtained results show that for increased values of Hartmann (magnetic parameter) and Deborah numbers, the fluid velocity decreases. The temperature field shows an increasing impact in the presence of larger radiative parameters. Sherwood and Nusselt numbers increase with higher values of thermophoresis and solutal stratified parameters.

Double stratified stagnation-point flow of Williamson nanomaterial with entropy generation through a porous medium

PurposeThis paper aims to address double-stratified stagnation-point flow of Williamson nanomaterial with entropy generation. Flow through porous medium is discussed. Energy equation is modeled in existence of viscous dissipation, Brownian motion and thermophoresis. Furthermore, convective boundary conditions are considered. Total entropy rate is presented.Design/methodology/approachThe non-linear flow expressions are converted to ordinary ones by implementation of suitable transformations. The obtained ordinary system is tackled for series solutions via homotopy analysis method.FindingsTill date no one has considered the irreversibility analysis in stagnation-point flow of Williamson nanomaterial with double stratification, porous medium and convective conditions. The basic objective of present research is to investigate the convective stagnation point flow of Williamson liquid with entropy concept and porous medium.Originality/valueAs per the authors’ knowledge, no such work is yet present in the literature.

Magnetohydrodynamics (MHD) Flow and Heat Transfer of a Doubly Stratified Nanofluid Using Cattaneo-Christov Model

The present study utilized Cattaneo-Christov heat flux model for solving nanofluid flow and heat transfer towards a vertical stretching sheet with the presence of magnetic field and double stratification. Thermal and solutal buoyancy forces are also examined to deal with the double stratification effects. Buongiorno's model of nanofluid is used to incorporate the effects of Brownian motion and thermophoresis. The boundary layer with non-Fourier energy equations are reduced into a system of nonlinear ordinary (similarity) differential equations using suitable transformations and then numerically solved using bvp4c solver in MATLAB software. The local Nusselt and Sherwood numbers of few limited cases are tabulated and compared with the earlier published works. It showed that a positive agreement was found with the previous study and thus, validated the present method. Numerical solutions are graphically demonstrated for several parameters namely magnetic, thermal relaxation, stratifications (thermal and solutal), thermophoresis and Brownian motion on the velocity, temperature and nanoparticles volume fraction profiles. An upsurge of the heat transfer rate was observed with the imposition of the thermal relaxation parameter (Cattaneo-Christov model) whereas the accretion of thermal and solutal stratification parameters reduced the temperature and nanoparticles concentration profiles, respectively.

Stratified electromagnetohydrodynamic flow of nanofluid supporting convective role

This study numerically examined unsteady double stratified EMHD mixed convection flow of nanofluid via permeable stretching sheet. It also looked at the convective heat and mass boundary conditions as well as the Navier velocity slip. In the thermal field, the effects of radiative heat transfer, heat generation/absorption, viscous dissipation, together with Ohmic heating (both magnetic and electric fields) were considered. The concentration field accounts for the chemical reaction. These show the physical behavior of electromagnetohydrodynamic flow associated with the problem formulation. The characteristics in regard to convective heat and mass, Navier slips conditions, as well as double stratification, were imposed. Such structure arises in energy efficiency and performance, which is achievable without higher pumping power, serves in the extrusion manufacturing process involving the thermal system for efficient devices particularly in polymeric, paper production, and food processing. The governing equations, which are nonlinear partial differential equations, were modelled by ordinary differential equations using suitable transformations. The ODEs were solved numerically, using implicit finite difference method (Keller box method). The physical implications deliberated on the behavior via the velocity, thermal energy, and concentration fields as well as the skin friction coefficient; the Nusselt and Sherwood numbers were scrutinized in relation to several parameters via mathematical model. The analysis shows that thermal and concentration stratifications decrease the distributions adjacent to the sheet surface, indicating decrease in the concentration nanoparticles and reduction in thermal energy. Augmentation occurs with convective heat and mass Biot numbers with the fields. The electric and magnetic parameters exhibit opposite flow behavior to the velocity and temperature. Chemical reaction and viscous dissipation weaken the concentration profile. Numerical results were compared with the published data available in the literature for limiting cases, and good agreement was noticed.

The combined effects of induced magnetic field, thermophoresis and Brownian motion on double stratified nonlinear convective-radiative Jeffrey nanofluid flow with heat source/sink

This communication explores the theoretical investigation of induced magnetic field, thermophoresis and Brownian motion on an unsteady, laminar and incompressible nonlinear convective flow of Jeffrey nanofluid through a rectangular channel in presence of thermal radiation and heat source/sink. Further, we also considered double stratification and heat and mass flux conditions at the boundaries. The governing flow field equations are shortened into coupled non-dimensional system ordinary differential equation using an appropriate transformation, then carried out a numerical solution using fifth-order Runge–Kutta–Fehlberg scheme along with shooting technique. The effect of various non dimensional form of the characteristic function of flow, heat and mass transfers, induced magnetic field and skin friction are plotted with respect to the prominent physical parameters and calculated heat and mass transfer rates at the plates. The results so obtained are compared with the existing published work for viscous case which leads a considerable agreement.

Activation energy and dual stratification effects for Walter-B fluid flow in view of Cattaneo-Christov double diffusionon

The purpose of the present article is to explore the novel aspects of activation energy in nonlinearly convective flow of Walter-B nanofluid in view of Cattaneo-Christov double diffusion model over a permeable stretched sheet. Generalized forms of Fourier's and Fick's law are utilized through Cattaneo-Christov double diffusion. Walter-B nanomaterial model is used that describes the significant slip mechanism namely Brownian and thermophoresis diffusions. Double stratification, heat generation/absorption and chemical reaction are considered. Modified Arrhenius formula for activation energy is implemented. The acquired nonlinear system is cracked through homotopic analysis method. Effects of emanating variables are examined through graphs and tables. It is evident that temperature distribution and thermal boundary layer thickness is monotonically dwindling function of thermal stratification parameter(P1), thermal relaxation time (δ2) and upsurge for the thermophoresis parameter (Nt), heat generation/absorption parameter (B₁), Brownian motion parameter (Nb). Nanoparticle concentration is directly comparable to the activation energy of reaction and impact of thermal relaxation time is qualitatively opposite to that of thermophoretic force.

Dual Stratified Nanofluid Flow Past a Permeable Shrinking/Stretching Sheet Using a Non-Fourier Energy Model

The present study emphasizes the combined effects of double stratification and buoyancy forces on nanofluid flow past a shrinking/stretching surface. A permeable sheet is used to give way for possible wall fluid suction while the magnetic field is imposed normal to the sheet. The governing boundary layer with non-Fourier energy equations (partial differential equations (PDEs)) are converted into a set of nonlinear ordinary differential equations (ODEs) using similarity transformations. The approximate relative error between present results (using the boundary value problem with fourth order accuracy (bvp4c) function) and previous studies in few limiting cases is sufficiently small (0% to 0.3694%). Numerical solutions are graphically displayed for several physical parameters namely suction, magnetic, thermal relaxation, thermal and solutal stratifications on the velocity, temperature and nanoparticles volume fraction profiles. The non-Fourier energy equation gives a different estimation of heat and mass transfer rates as compared to the classical energy equation. The heat transfer rate approximately elevates 5.83% to 12.13% when the thermal relaxation parameter is added for both shrinking and stretching cases. Adversely, the mass transfer rate declines within the range of 1.02% to 2.42%. It is also evident in the present work that the augmentation of suitable wall mass suction will generate dual solutions. The existence of two solutions (first and second) are noticed in all the profiles as well as the local skin friction, Nusselt number and Sherwood number graphs within the considerable range of parameters. The implementation of stability analysis asserts that the first solution is the real solution.

Effects of double stratification and heat flux damping on convective flows over a vertical cylinder

Unsteady free convection flows of viscous fluids over a vertical circular cylinder are investigated by taking in consideration thermal and mass stratification and the thermal memory effects. The mathematical model of thermal transport is based on the fractional generalized Fourier's law for thermal flux with the kernel of power-law kind. In this model the histories of the temperature gradient influence the thermal and mass transport process and the fluid motion. On the cylinder's surface the temperature (or the thermal flux) and solute concentration are constant. Solutions in the transformed domain for the perturbation temperature and concentration and fluid velocity are determined using the Laplace transform coupled with the classical method for the ordinary non-homogeneous differential equations. The inverse Laplace transforms are obtained numerically by employing the Stehfest's algorithm. Solutions for the case corresponding to classical Fourier's law are obtained as particular case of general solutions by taking the memory parameter equal to zero. The influence of the thermal memory and of thermal and mass stratifications is numerically and graphically analyzed by using the software Mathcad 15.

Comment on the paper “Mechanisms of double stratification and magnetic field in flow of third grade fluid over a slendering stretching surface with variable thermal conductivity, Tasawar Hayat, Sajid Qayyum, Ahmed Alsaedi, Bashir Ahmad, Results in Physics 8 (2018) 819–828”

Comment on the paper “Mechanisms of double stratification and magnetic field in flow of third grade fluid over a slendering stretching surface with variable thermal conductivity, Tasawar Hayat, Sajid Qayyum, Ahmed Alsaedi, Bashir Ahmad, Results in Physics 8 (2018) 819–828”

Consequences of activation energy and chemical reaction in radiative flow of tangent hyperbolic nanoliquid

Mixed convection flow of tangent hyperbolic liquid over stretching sheet is explored. Joule heating, double stratification, non-linear thermal radiation, Brownian motion and thermophoresis are present. Phenomenon of mass transfer is examined by activation energy along with binary chemical. Computations of convergent solutions are carried out for the nonlinear mathematical system. Graphical representation is employed for outcome of sundry variables on velocity profile, temperature field and concentration of nanoparticles. Moreover, Nusselt number, coefficient of drag force and mass transfer rate are examined. It is observed that velocity decays for larger Weissenberg number. Concentration of fluid enhances for higher activation energy parameter.

Numerical simulation for activation energy impact in Darcy–Forchheimer nanofluid flow by impermeable cylinder with thermal radiation

Here, the mixed convection and activation energy characteristics in Darcy–Forchheimer nanomaterial flow by impermeable cylinder are addressed. The novel chemical species model which elaborates activation energy impact is introduced. Formulation is based on significant slip mechanisms, namely the Brownian and thermophoretic diffusions. Besides, thermal radiation, double stratification, heat generation and convective conditions are also taken into account. The formulated expressions are converted into non-dimensional quantities. Shooting scheme is opted to tackle governing nonlinear mathematical problems. Plots and tabular results are presented regarding the importance of physical constraints. It is found that the inertia coefficient and porosity variable yield lower velocity. Also, temperature and Sherwood number are increasing functions of activation energy factor.

Influence of viscous dissipation and double stratification on MHD Oldroyd-B fluid over a stretching sheet with uniform heat source

The present report explores the discussion on 3D flow of dual stratified Oldroyd-B fluid in the direction of thermal radiation, viscous dissipation and heat source/sink. Mathematical modeling is formulated with an applied magnetic field through a stretching surface. The present flow governing system has been transformed as nonlinear ODE via suitable transformations and then concluded by using bvp4c. The graphs are described and illustrated for various non-dimensional parameters. Thermal stratification and Prandtl number parameters reduce the temperature, whereas thermal radiation and heat source parameters show reverse behavior. Numerical results are used to obtain the values of frictional drag, rate of heat and mass transfers. Finally, numerical results are compared with previous established work in a limiting case.

Attributes of Activation Energy and Exponential Based Heat Source in Flow of Carreau Fluid with Cross-Diffusion Effects

Abstract In the present study we aim to model and examine the impact of melting and double stratification on MHD flow of Carreau liquid over a stretching sheet. Incompressible non-Newtonian liquid is electrically conducting and exposed to a constant magnetic field. The exponential space dependent internal heat source effect is incorporated in the energy expression. Mass transfer is characterized in terms of activation energy and a binary chemical reaction. In addition, Soret and Dufour phenomena are included. The boundary layer concept is utilized to simplify the governing equations. A system of ordinary differential equations is obtained through proper transformations. The coupled non-linear system is then computed by the NDSolve technique. Plots and tabulated values are arranged for interpretation of different variables. The obtained results show that the flow field is influenced appreciably by melting, the heat source, the reaction rate and the activation energy. Moreover, Nusselt and Sherwood numbers are increased by the activation energy.

Significance of improved Fourier-Fick laws in non-linear convective micropolar material stratified flow with variable properties

Present research article describes the effectiveness of improved Fourier-Fick fluxes and temperature-dependent conductivity on the 2-D, incompressible steady micropolar material flow over a stretchable surface. Non-linear mixed convection, double stratification and heat generation aspects are considered. The considered flow non-linear PDE are converted to ODE via appropriate transformations. Through implementation of homotopy method the obtain system is solved for series solutions. The effects of pertinent parameters are discussed through graphical sketch. Skin friction coefficient (drag force) is calculated. Main findings are pointed out.