Stability analysis on dual solutions of MHD Casson fluid flow with thermal and chemical reaction over a permeable elongating sheet

The endeavor of this study is to explore the nature of dual solutions (steady and unsteady) for the Casson fluid flow with the simultaneous consequences of both thermal and mass transmissions. The flow passes above an absorbent elongating sheet in the existence of a constant magnetic field. The supported leading equations are remodeled into a set of solvable forms with the assist of suitable similarity variables and hence deciphered utilizing the “MATLAB routine bvp4c scheme.” Due to the sudden changes in the surface with time, the temperature and flow behavior over the sheet also change, and hence dual-type flow solutions exist. Stability scrutiny is implemented to examine the less (more) stable and visually achievable solutions. From this study, we have achieved many interesting facts, among them, we can use magnetic and Casson fluid parameters to control the motion of the fluid and to enlarge of thermal transmission of the fluid. This flow model has many important applications in different physical fields, such as engineering sciences, medical sciences, and different industrial processes. One of the most important results, which has been achieved from this investigation, is that the Prandtl number enriches the heat transfer rate of the fluid at the surface during the time-independent case under the suction environment. Also, the chemical reaction parameter helps to enhance the mass accumulation rate of the fluid in both cases.     

Casson fluid flow and heat transfer past a symmetric wedge

Boundary‐layer forced convection flow of a Casson fluid past a symmetric wedge is investigated. Similarity transformations are used to convert the governing partial differential equations to ordinary ones and the reduced equations are then solved numerically with the help of the shooting method. Comparisons with various previously published works on special cases are performed and the results are found to be in excellent agreement. A representative set of graphical results is obtained and illustrated graphically. The velocity is found to increase with an increasing Falkner–Skan exponent whereas the temperature decreases. With the rise of the Casson fluid parameter, the fluid velocity increases but the temperature is found to decrease in this case. The skin friction decreases with increasing values of the Casson fluid parameter. It is found that the temperature decreases as the Prandtl number increases and thermal boundary layer thickness decreases with increasing values of the Prandtl number. A significant finding of this investigation is that flow separation can be controlled by increasing the value of the Casson fluid parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(8): 665–675, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21065     

Effects of chemical reaction,Soret and Lorentz force on Casson fluid flow past an exponentially accelerated vertical plate: A comprehensive analysis

The aim of the current study is to explore the effects of heat and mass transfer on unsteady chemically reacted Casson liquid flow over an exponentially accelerated vertical plate in a porous medium. It is assumed that the bounding plate has varying temperatures as well as concentrations in a porous medium under a uniform magnetic field. This phenomenon is modeled in the form of a system of partial differential equations (PDEs) with boundary conditions. The governing dimensionless PDEs are solved using Laplace transform method for velocity, temperature, and concentration. The impact of nondimensional parameters, which are controlling the flow like Casson parameter, Soret number, magnetic parameter, heat generation parameter, Prandtl number, radiation parameter, and Schmidt number is analyzed through graphs. The incremental values of the Casson fluid parameter lead to a reduction in velocity and discovered that for large values of the Casson parameter, the fluid is near to the Newtonian fluid. Also, the Sherwood number increases with enhancing dissimilar estimators of the Schmidt and Soret numbers. A comparison has been made with the published work (Kataria et al.) for a particular case, which was in good agreement.     

Effects of the viscous dissipation and chemical reaction on Casson nanofluid flow over the permeable stretching/shrinking sheet

A steady two‐dimensional Casson nanofluid flow over the permeable stretching/shrinking sheet along the viscous dissipation and the chemical reaction is studied in this article. The convective boundary condition is incorporated in energy equation. Similarity variables are applied to convert the governing partial differential equations into ordinary differential equations. The numerical solutions of the equations are obtained by using the shooting method with Maple implementation. The numerical findings indicate occurrence of the dual solutions for a certain range of stretching/shrinking and suction parameters. Therefore, a stability analysis is done to find the solution that is stable and physically realizable. The effects of the pertinent physical parameters on velocity, temperature, and concentration profiles are investigated graphically. Numerical results of various parameters involved for skin friction coefficient, the local Nusselt as well as Sherwood numbers are determined and also discussed in detail. The Casson and suction parameters decrease the velocity in the first solution, whereas they increase it in the second solution. The rate of heat transfer increases in both solutions with an increment in Eckert number, Biot number, thermophoresis, and Brownian motion parameters. Thermophoresis and Brownian motion parameters show opposite behavior in the nanoparticle's concentration. The nanoparticle concentration decreases in both solutions with increment in Schmidt number, Brownian motion, and chemical reaction parameters.     

Heat and mass transfer in MHD Casson nanofluid flow past a stretching sheet with thermophoresis and Brownian motion

The foremost objective of the current article is to explore the impact of Brownian motion on magnetohydrodynamic Casson nanofluid flow toward a stretching sheet in the attendance of nonlinear thermal radiation. The combined heat and mass transfer characteristics are investigated. The influence of chemical reaction, nonuniform heat source/sink, Soret, and Dufour is deemed. The convective boundary condition is taken. The appropriate transformations are utilized to transform the flow regulating partial differential equations into dimensionless ordinary differential equations (coupled). The numerical outcomes of the converted nonlinear system are solved by the Runge-Kutta based Shooting procedure. Results indicate that the temperature is an increasing function of both thermophoresis and Brownian motion parameters. The concentration of the fluid and the corresponding boundary layer thickness reduces with an enhancement in Lewis number.     

Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet

In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet. The governing equations are transformed into a set of self-similar ordinary differential equations by similarity transformations. The transformed equations are solved numerically using very efficient shooting method. The study reveals that the dual solutions of velocity and concentration exist for certain values of velocity ratio parameter (the ratio of stretching/shrinking rate and straining rate). The concentration boundary layer thickness decreases with increasing values of Schmidt number and reaction-rate parameter for both solutions.     

MHD stagnation point flow of a Maxwell nanofluid over a shrinking sheet (multiple solution)

In the current study, a realistic approach is used to investigate the MHD stagnation point flow of a Maxwell nanofluid past a shrinking sheet with a chemical reaction. First, the flow model is made non dimensionalized via an appropriate transformation. The non dimensionalized equations are numerically tackled by adopting the bvp4c technique. It is also analyzed that the dual solutions are obtained for a particular choice of shrinking parameter. A detailed analysis of the impact of several parameters on the velocity field, temperature distribution, and concentration distribution is carried out graphically. The computed result shows that the first solution significantly increases for higher values of the magnetic parameter, whereas the second solution decreases. Furthermore, it is noted that the first and second solutions decreases for the relaxation parameter. The physical quantities are observed graphically. It is exhibited that the Nusselt number shows a decreasing behavior for the both solutions via relaxation parameter.     

The effect of Joule heating and viscous dissipation on the boundary layer flow of a magnetohydrodynamics micropolar-nanofluid over a stretching vertical Riga plate

Joule heating and viscous dissipation effects on the behavior of the boundary layer flow of a micropolar nanofluid over a stretching vertical Riga plate (electro magnetize plate) are considered. The flow is disturbed by an external electric magnetic field. The problem is formulated mathematically by nonlinear system of partial differential equations (PDEs). By using suitable variables transformations, this system is transformed onto a system of nonlinear ordinary differential equations (ODEs). The Parametric NDsolve package of the commercial software Mathematica is used to solve the obtained ODEs as well as the considered numerical results for different physical parameters with appropriate boundary conditions. Novel results are obtained by studying the stream lines flow around the plate in two and three dimensions. Moreover, the effects of the pertinent parameters on the skin friction coefficient, couple stress, local Nusselt, and Sherwood number are discussed. Special cases of the obtained results show excellent agreements with previous works. The results showed that as the magnetic field parameter increases the velocity of the boundary layer adjacent to the stretching sheet decreases. Also, for a productive chemical reaction near the sheet surface, the angular velocity decreases but opposite trend is observed far from the sheet surface. The importance of this study comes from its significant applications in many scientific fields, such as nuclear reactors, industry, medicine, and geophysics.     

Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet

The effect of radiation on the boundary layer flow and heat transfer of a viscous fluid over an exponentially stretching sheet is studied. The homotopy analysis method (HAM) is employed to determine the convergent series expressions of velocity and temperature. The physical interpretation to these expressions is assigned through graphs. It is found that the effects of Prandtl and radiation numbers on the temperature are opposite.     

Linear stability analysis of MHD flow of micropolar fluid with thermal radiation and convective boundary condition: Exact solution

Magnetohydrodynamic (MHD) flow of micropolar fluid by including the thermal radiation and convective condition on a shrinking surface in the presence of mass suction effects has been investigated. The momentum, angular momentum and energy equations, and the solutions of these equations are valid for whole Navier stokes, and microrotational and energy equations have been solved exactly. We obtain the solution in the form of an incomplete γ function for the energy equation. The results reveal that dual solutions exist for certain domains of different physical parameters. Furthermore, high suction produces the high effect of drag force, and as a result, coefficient of skin friction increases in the first solution. Stability analysis has been performed and determined that the first solution is more stable.     

MHD Double‐diffusive thermosolutal Marangoni convection non‐Newtonian Casson fluid flow over a permeable stretchable sheet

In the present paper, we have discussed the thermosolutal Marangoni force acting on the electrically conducting Casson fluid flow over a permeable horizontal stretching surface. It is presumed that the condition at the interfaces is influenced by the surface tension, which is proportional to the temperature and concentration profiles. At the interface, both concentration and temperature are heated in such a way that they are quadratic functions in $x.$ Furthermore, we have introduced the magnetic field in the transverse direction of the fluid flow along with heat generation/absorption, thermal radiation, viscous dissipation, and first‐order chemical effect with heat and mass flux into the present system. Similarity transformations have been used to convert the system of the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The reduced ODEs are then solved using the MATLAB program bvp4c, which is based on the fourth‐order Runge‐Kutta and shooting method. The impact of various pertinent flow parameters on the flow field, temperature, and species concentration has been studied through graphs. To know the characteristics of shear stress, heat and mass rate near the boundary, numerical values of them are also calculated and given in the tabular form. The results show that the momentum boundary layer's thickness is getting thicker with an increase in solutal surface tension ratio, while its opposite trends have been observed in the thermal boundary layer region, this is due to the Marangoni effect. This Marangoni effect is very much important in the field of melting metals, crystal growth, welding, and electron beam.     

Nonsimilar boundary layer flow of Cross fluid induced by a heated stretched sheet

This paper deals with the nonisothermal boundary layer flow of Cross fluid due to a stretching sheet. Unlike previous studies on boundary layer flow of Cross fluid, a nonsimilar formulation is adopted to transform the boundary layer equations into nondimensional form. The problem is characterized by three dimensionless parameters, namely, the Deborah number, the Prandtl number, and dimensionless distance along the sheet. The transformed equations are simulated by a numerical scheme with the help of MAPLE software. The velocity and temperature profiles inside the boundary layer are calculated and shown graphically. The skin friction coefficient and Nusselt number at various axial stations are also tabulated for several values of Deborah number and Prandtl number.     

Steady MHD Casson Ohmic heating and viscous dissipative fluid flow past an infinite vertical porous plate in the presence of Soret,Hall, and ion‐slip current

In the present study, the influence of Hall and ion‐slip current on steady magnetohydrodynamics mixed convective, Ohmic heating, and viscous dissipative Casson fluid flow over an infinite vertical porous plate in the presence of Soret effect and chemical reaction are investigated. The modeling equations are transformed into dimensionless equations and then solved analytically through the multiple regular perturbation law. Computations are performed graphically to analyze the behavior of fluid velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number on the vertical plate with the difference of emerging physical parameters. This study reflects that the incremental values of Casson fluid parameter and Schmidt number lead to reduction in velocity. However, fluid velocity rises due to enhancement of ion‐slip parameter but an opposite effect is observed in case of Hall parameter. In addition, the Sherwood number declines with enhancing dissimilar estimators of the chemical reaction, Schmidt number, as well as Soret number.     

Study of chemically reactive and thermally radiative Casson nanofluid flow past a stretching sheet with a heat source

Nanoparticle (NP) delivery is an exciting and rapidly developing field that adequately takes care of thermal radiation in blood flow and is likely to have bearing on the therapeutic procedure of hyperthermia, blood flow, and heat transfer in capillaries. The NP parameters such as size, shape, and surface characteristics can be regulated to improve nano-drug delivery efficiency in biological systems. The NPs outperform traditional drug delivery processes in drug carrying capacity and controlled release. The current article investigates the boundary layer flow and heat transfer of thermally radiative Casson nanofluid (NF) over a stretching sheet with chemical reaction and internal heat source. In our study, Cu and Al₂O₃ are taken as NPs in a suitable base fluid. The problem is analyzed by using similarity transformations and is solved with MATLAB's built-in solver bvp4c. The effects of pertinent parameters characterizing the flow model are presented through graphs and tables. The important findings of the investigation are noted as: the use of metallic oxide is more beneficial to attain higher temperature within a few layers close to the bounding surface; the appearance of convexity and concavity in the concentration profile attributed to flow instability, and the constructive and destructive heterogeneous reactions at the bounding surface have distinct roles to modify the NF flow in the boundary layer.     

Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface

The present study deals with new similarity solution of steady mixed convection boundary layer flow over a permeable surface for convective boundary condition. It has been shown that a self similar solution is possible when the mass transfer velocity at the surface of the plate varies like x^−1/2, where x is the distance from the leading edge of the solid surface. Two point boundary value problem governed by non-linear coupled ordinary differential equations have been solved numerically using implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile and the Prandtl number strongly affects the thermal boundary layer thickness including the surface heat transfer rate.     

Natural convection boundary layer flow of a micropolar fluid over a vertical permeable cone with variable wall temperature

This work studies the natural convection boundary layer flow of a micropolar fluid near a vertical permeable cone with variable wall temperature. The transformed boundary layer governing equations are solved by the cubic spline collocation method. The local Nusselt numbers are presented as functions of suction variables for different values of vortex viscosity parameter, surface temperature exponent, and Prandtl number. Results show that the heat transfer rates of the permeable cones with higher suction variables are higher than those with lower suction variables. Moreover, the heat transfer rate from a vertical permeable cone in Newtonian fluids is higher than that in micropolar fluids.     

Heat and mass transfer in MHD boundary layer flow of a second-grade fluid past an infinite vertical permeable surface

Heat and mass transfer of non-Newtonian fluids is increasingly being studied by researchers due to its applications in many branches of science and engineering, such as metallurgical processes, polymer extrusion, glass blowing, crystal growing, and so forth. The present work is mainly concerned with the unsteady laminar magnetohydrodynamic flow of a heat-generating or absorbing second-grade fluid past an infinite vertical porous plate. The nondimensional governing equations are solved for the best analytical solution. Results for various flow characteristics are presented through graphs and tables delineating the effect of various parameters characterizing the flow. For engineering interest, the shear stresses, Nusselt number, and Sherwood number are computed and exchanged of views with reference to the important parameters. Our analysis explored that the influences of a chemical reaction and fluid oscillations reduced the concentration distribution in the entire liquid region. The rotation effect decreases the shear stress, whereas it is augmented through an increase in the permeability of porous medium and second-grade fluid parameters' impact.     

Melting heat transfer analysis of electrically conducting nanofluid flow over an exponentially shrinking/stretching porous sheet with radiative heat flux under a magnetic field

Modern magnetic nanomaterial processing operations are progressing rapidly and require increasingly sophisticated mathematical models for their optimization. Stimulated by such developments, in this paper, a theoretical and computational study of a steady magnetohydrodynamic nanofluid over an exponentially stretching/shrinking permeable sheet with melting (phase change) and radiative heat transfer is presented. Besides, wall transpiration, that is, suction and blowing (injection), is included. This study deploys Buongiorno's nanofluid model, which simulates the effects of the Brownian motion and thermophoresis. The transport equations and boundary conditions are normalized via similarity transformations and appropriate variables, and the similarity solutions are shown to depend on the transpiration parameter. The emerging dimensionless nonlinear coupled ordinary differential boundary value problem is solved numerically with the Newton-Fehlberg iteration technique. Validation with special cases from the literature is included. The increase in the magnetic field, that is, the Hartmann number, is observed to elevate nanoparticle concentration and temperature, whereas it dampens the velocity. Higher values of the melting parameter consistently decelerate the boundary layer flow and suppress temperature and nanoparticle concentration. A higher radiative parameter strongly increases temperature (and thermal boundary layer thickness) and weakly accelerates the flow. The increase in the Brownian motion reduces nanoparticle concentrations, whereas a greater thermophoretic body force strongly enhances them. The Nusselt number and Sherwood number are observed to be decreased with an increasing Hartmann number, whereas they are elevated with a stronger wall suction and melting parameter.     

Nonlinear convective boundary layer flow of micropolar‐couple stress nanofluids past permeable stretching sheet using Cattaneo‐Christov heat and mass flux model

The present work aims to examine the effects of viscous dissipation and unsteadiness parameters on nonlinear convective laminar boundary layer flow of micropolar‐couple stress nanofluid past a permeable stretching sheet with non‐Fourier heat flux model in the presence of suction/injection variable. The unsteadiness in the flow, temperature, and concentration profile is caused by the time‐dependence of the stretching velocity, surface temperature, and surface concentration of the boundary layer flow. Similarity transformation is applied to transform the time‐dependent boundary layer flow equations into the corresponding highly nonlinear coupled ordinary differential equations with appropriate boundary conditions. The robust numerical technique called Galerkin finite element method is used to solve the obtained dimensionless governing equations of the flow. The effects of Eckert number, unsteadiness parameter, suction/injection parameter, mixed convection parameter, material parameter, Schmidt number, and couple stress parameter on linear velocity, angular velocity, temperature, concentration, local skin friction coefficient, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed with the help of graphical and tabular form. Under special conditions, the present result is compared with the existing literature and revealed good agreement. Our result shows that as unsteadiness parameter boost, both heat and mass transfer rate rises. The present study has a significant application in material processing technology.     

Symmetry analysis of MHD Casson fluid flow for heat and mass transfer near a stagnation point over a linearly stretching sheet with variable viscosity and thermal conductivity

In this article, the impacts of variable viscosity and thermal conductivity on magnetohydrodynamic, heat transfer, and mass transfer flow of a Casson fluid are analyzed on a linearly stretching sheet inserted in a permeable medium along with heat source/sink and viscous dissipation. To reduce the ascendant partial differential equations into ordinary differential equations, Lie group transformation is utilized. Further, the fourth-order Runge–Kutta strategy is utilized to solve the ordinary differential equations numerically. The numerical results obtained for various parameters by employing coding in MATLAB programming are investigated and considered through graphical representation and tables. We anatomize the impacts of distinctive parameters on velocity, temperature, and concentration distributions.