Dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet

An analysis is carried out for dual solutions of the boundary layer flow of Maxwell fluid over a permeable shrinking sheet. In the investigation, a constant wall mass transfer is considered. With the help of similarity transformations, the governing partial differential equations（PDEs） are converted into a nonlinear self-similar ordinary differential equation（ODE）. For the numerical solution of transformed self-similar ODE, the shooting method is applied. The study reveals that the steady flow of Maxwell fluid is possible with a smaller amount of imposed mass suction compared with the viscous fluid flow. Dual solutions for the velocity distribution are obtained. Also, the increase of Deborah number reduces the boundary layer thickness for both solutions.

DTM-BF method and dual solutions for unsteady MHD flow over permeable shrinking sheet with velocity slip

An unsteady magnetohydrodynamic （MHD） boundary layer flow over a shrinking permeable sheet embedded in a moving viscous electrically conducting fluid is investigated both analytically and numerically. The velocity slip at the solid surface is taken into account in the boundary conditions. A novel analytical method named DTM- BF is proposed and used to get the approximate analytical solutions to the nonlinear governing equation along with the boundary conditions at infinity. All analytical results are compared with those obtained by a numerical method. The comparison shows good agreement, which validates the accuracy of the DTM-BF method. Moreover, the existence ranges of the dual solutions and the unique solution for various parameters are obtained. The effects of the velocity slip parameter, the unsteadiness parameter, the magnetic parameter, the suction/injection parameter, and the velocity ratio parameter on the skin friction, the unique velocity, and the dual velocity profiles are explored, respectively.

Dual solutions in mixed convection with variable physical properties

Assisting and opposing flows in a mixed convection boundary layer flow over an isothermal vertical plate are studied for the case of variable physical properties and uniform free stream.Fluid viscosity and thermal conductivity are assumed to be linear functions of temperature.Using local similarity the flow and heat transfer quantities are found to be functions of four parameters,i.e. Richardson number,Prandtl number,a viscosity variation parameter and a thermal conductivity variation parameter.Numerical solutions are obtained by two methods,a shooting technique and Nachtsheim-Swigert technique,for selected values of parameters appropriate for the fluids considered and specific temperatures of the plate and ambient fluid.For assisting flows,there exist solutions for all values of Richardson number while for opposing flows solutions exist only for a finite set of its values and,in addition,there also exist dual solutions.Important flow and heat transfer quantities of practical interest are determined and the influence of different parameters is discussed.

Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet

The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.

Dual solutions of time-dependent magnetohydrodynamic stagnation point boundary layer micropolar nanofluid flow over shrinking/stretching surface

Time-dependent,two-dimensional(2 D)magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered.Mass and heat transfer characteristics are incorporated in the problem.A model of the partial differential expressions is altered into the forms of the ordinary differential equations via similarity transformations.The obtained equations are numerically solved by a shooting scheme in the MAPLE software.Dual solutions are observed at different values of the specified physical parameters.The stability of first and second solutions is examined through the stability analysis process.This analysis interprets that the first solution is stabilized and physically feasible while the second one is un-stable and not feasible.Furthermore,the natures of various physical factors on the drag force,skin-friction factor,and rate of mass and heat transfer are determined and interpreted.The micropolar nanofluid velocity declines with a rise in the suction and magnetic parameters,whereas it increases by increasing the unsteadiness parameter.The temperature of the micropolar nanofluid rises with increase in the Brownian motion,radiation,thermophoresis,unsteady and magnetic parameters,but it decreases against an increment in the thermal slip constraint and Prandtl number.The concentration of nanoparticles reduces against the augmented Schmidt number and Brownian movement values but rises for incremented thermophoresis parameter values.

Dual Solutions of MHD Boundary Layer Flow of a Micropolar Fluid with Weak Concentration over a Stretching/Shrinking Sheet

We investigate the dual solutions for the MHD flow of micropolar fluid over a stretching/shrinking sheet with heat transfer. Suitable relations transform the partial differential equations into the ordinary differential equations.Closed forms solutions are also obtained in terms of confluent hypergeometric function. This is the first attempt to determine the exact solutions for the non-linear equations of MHD micropolar fluid model. It is demonstrated that the microrotation parameter helps in increasing Nusselt number and the dual solutions exist for all fluid flow parameters under consideration. The dual behavior of dimensionless velocity, temperature, microrotation, skin-friction coefficient,local Nusselt number is displayed on graphs and examined.

Unsteady Flow and Heat Transfer of a Casson Micropolar Nanofluid over a Curved Stretching/Shrinking Surface

We present the results of an investigation into the behavior of the unsteady flow of a Casson Micropolar nanofluid over a shrinking/stretching curved surface,together with a heat transfer analysis of the same problem.The body force acting perpendicular to the surface wall is in charge of regulating the fluid flow rate.Curvilinear coordinates are used to account for the considered curved geometry and a set of balance equations for mass,momentum,energy and concentration is obtained accordingly.These are turned into ordinary differential equations using a similarity transformation.We show that these equations have dual solutions for a number of different combinations of various parameters.The stability of such solutions is investigated by applying perturbations on the steady states.It is found that high values of the Micropolar and Casson parameters cause the flow to move more slowly.However,when compared to a shrunken surface,a stretched surface produces a greater Micro-rotation flux.

Unsteady MHD flow and heat transfer near stagnation point over a stretching/shrinking sheet in porous medium filled with a nanofluid

In this article, the unsteady magnetohydrodynamic （MHD） stagnation point flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated numerically. The similarity solution is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the fourth-order Runge-Kutta method with shooting technique. The ambient fluid velocity, stretching/shrinking velocity of sheet, and the wall temperature are assumed to vary linearly with the distance from the stagnation point. To investigate the influence of various pertinent parameters, graphical results for the local Nusselt number, the skin friction coefficient, velocity profile, and temperature profile are presented for different values of the governing parameters for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid. It is found that the dual solution exists for the decelerating flow. Numerical results show that the extent of the dual solution domain increases with the increases of velocity ratio, magnetic parameter, and permeability parameter whereas it remains constant as the value of solid volume fraction of nanoparticles changes. Also, it is found that permeability parameter has a greater effect on the flow and heat transfer of a nanofluid than the magnetic parameter.

Boundary layer flow and heat transfer of a micropolar fluid near the stagnation point on a stretching vertical surface with prescribed skin friction

The steady laminar mixed convection boundary layer flow and heat transfer of a micropolar fluid near the stagnation point on a stretched vertical surface with prescribed skin friction were considered.The governing partial differential equations were transformed into a system of ordinary differential equations,which were then solved numerically using the shooting method.Results for the stretching velocity,the local Nusselt number,the temperature,and the velocity profiles are presented for various values of the mixed convection parameter λ and material parameter K when the Prandtl number is equal to 1.Both assisting（heated plate） and opposing（cooled plate） flow regions are considered.It is found that dual solutions exist for both assisting and opposing flows.

Stagnation Point Flow Over a Permeable Stretching/Shrinking Sheet with Chemical Reaction and Heat Source/Sink

The present study considers the magnetohydrodynamic(MHD)stagnation point flow with chemical reaction effect over a permeable stretching/shrinking sheet.The partial differential equations are reduced to a set of ordinary differential equations using a similarity transformation.The transformed equations are then solved numerically by employing the bvp4c function available in the MATLAB software.The numerical results illustrate the effects of several parameters on the skin friction coefficient,local Nusselt number and the local Sherwood number.Dual solutions are obtained for a certain range of parameters.The temporal stability analysis is carried out to determine which one of these solutions is stable and thus physically reliable in a long run.

Temporal Stability Analysis of Magnetized Hybrid Nanofluid Propagating through an Unsteady Shrinking Sheet: Partial Slip Conditions

The unsteady magnetohydrodynamic(MHD)flow on a horizontal preamble surface with hybrid nanoparticles in the presence of the first order velocity and thermal slip conditions are investigated.Alumina(Al_(2)O_(3))and copper(Cu)are considered as hybrid nanoparticles that have been dispersed in water in order to make hybrid nanofluid(Cu-Al_(2)O_(3)/water).The system of similarity equations is derived from the system of partial differential equations(PDEs)by using variables of similarity,and their solutions are gotten with shooting method in the Maple software.In certain ranges of unsteadiness and magnetic parameters,the presence of dual solutions can be found.Further,it is examined that layer separation is deferred due to the effect of the hybrid nanoparticles.Moreover,the capacity of the thermal enhancement of Cu-Al_(2)O_(3)/water hybrid nanofluid is higher as compared to Al_(2)O_(3)/water based nanofluid and enhancements inCu are caused to rise the fluid temperature in both solutions.In the last,solutions stability analyzes were also carried out and the first solution was found to be stable.

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.

Hybrid Nanofluid Flow with Homogeneous-Heterogeneous Reactions

This study examines the stagnation point flow over a stretching/shrinking sheet in a hybrid nanofluid with homogeneous-heterogeneous reactions.The hybrid nanofluid consists of copper(Cu)and alumina(Al2O3)nanoparticles which are added into water to form Cu-Al2O3/water hybrid nanofluid.The similarity equations are obtained using a similarity transformation.Then,the function bvp4c in MATLAB is utilised to obtain the numerical results.The dual solutions are found for limited values of the stretching/shrinking parameter.Also,the turning point arises in the shrinking region(λ<0).Besides,the presence of hybrid nanoparticles enhances the heat transfer rate,skin friction coefficient,and the concentration gradient.In addition,the concentration gradient is intensified with the heterogeneous reaction but the effect is opposite for the homogeneous reaction.Furthermore,the velocity and the concentration increase,whereas the temperature decreases for higher compositions of hybrid nanoparticles.Moreover,the concentration decreases for larger values of homogeneous and heterogeneous reactions.It is consistent with the fact that higher reaction rate cause a reduction in the rate of diffusion.However,the velocity and the temperature are not affected by these parameters.From these observations,it can be concluded that the effect of the homogeneous and heterogeneous reactions is dominant on the concentration profiles.Two solutions are obtained for a single value of parameter.The temporal stability analysis shows that only one of these solutions is stable and thus physically reliable over time.

Influence of various shapes of nanoparticles on unsteady stagnation-point flow of Cu-H_(2)O nanofluid on a flat surface in a porous medium:A stability analysis

The nanofluid and porous medium together are able to fulfill the requirement of high cooling rate in many engineering problems.So,here the impact of various shapes of nanoparticles on unsteady stagnation-point flow of Cu-H_(2)O nanofluid on a flat surface in a porous medium is examined.Moreover,the thermal radiation and viscous dissipation effects are considered.The problem governing partial differential equations are converted into self-similar coupled ordinary differential equations and those are numerically solved by the shooting method.The computed results can reveal many vital findings of practical importance.Firstly,dual solutions exist for decelerating unsteady flow and for accelerating unsteady and steady flows,the solution is unique.The presence of nanoparticles affects the existence of dual solution in decelerating unsteady flow only when the medium of the flow is a porous medium.But different shapes of nanoparticles are not disturbing the dual solution existence range,though it has a considerable impact on thermal conductivity of the mixture.Different shapes of nanoparticles act differently to enhance the heat transfer characteristics of the base fluid,i.e.,the water here.On the other hand,the existence range of dual solutions becomes wider for a larger permeability parameter related to the porous medium.Regarding the cooling rate of the heated surface,it rises with the permeability parameter,shape factor(related to various shapes of Cu-nanoparticles),and radiation parameter.The surface drag force becomes stronger with the permeability parameter.Also,with growing values of nanoparticle volume fraction,the boundary layer thickness(BLT)increases and the thermal BLT becomes thicker with larger values of shape factor.For decelerating unsteady flow,the nanofluid velocity rises with permeability parameter in the case of upper branch solution and an opposite trend for the lower branch is witnessed.The thermal BLT is thicker with radiation parameter.Due to the existence of dual solutions,a linear stability analysis is made and it is concluded that the upper branch and unique solutions are stable solutions.

Boundary layer flow over a moving surface in a nanofluid with suction or injection

An analysis is performed to study the heat transfer characteristics of steady two-dimensional boundary layer flow past a moving permeable flat plate in a nanofluid. The effects of uniform suction and injection on the flow field and heat transfer characteristics are numerically studied by using an implicit finite difference method. It is found that dual solutions exist when the plate and the free stream move in the opposite directions. The results indicate that suction delays the boundary layer separation, while injection accelerates it.

MHD flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge

The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted to the similarity equations by techniques of the similarity transformation. The bvp4c function that is available in MATLAB software is utilized for solving the similarity equations numerically. The numerical results are obtained for selected different values of parameters. The results discover that two solutions exist, up to a certain value of the stretching/shrinking and suction strengths. The critical value in which the solution is in existence decreases as nanoparticle volume fractions for copper and wedge angle parameter increase. It is also found that the hybrid nanofluid enhances the heat transfer rate compared with the regular nanofluid. The reduction of the heat transfer rate is observed with the increase in radiation parameter. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.

Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid

An analysis of unsteady boundary layer flow and heat transfer over an exponentially shrinking porous sheet filled with a copper-water nanofluid is presented.Water is treated as a base fluid.In the investigation,non-uniform mass suction through the porous sheet is considered.Using Keller-box method the transformed equations are solved numerically.The results of skin friction coefficient,the local Nusselt number as well as the velocity and temperature profiles are presented for different flow parameters.The results showed that the dual non-similar solutions exist only when certain amount of mass suction is applied through the porous sheet for various unsteady parameters and nanoparticle volume fractions.The ranges of suction where dual non-similar solution exists,become larger when values of unsteady parameter as well as nanoparticle volume fraction increase.So,due to unsteadiness of flow dynamics and the presence of nanoparticles in flow field,the requirement of mass suction for existence of solution of boundary layer flow past an exponentially shrinking sheet is less.Furthermore,the velocity boundary layer thickness decreases and thermal boundary layer thickness increases with increasing of nanoparticle volume fraction in both non-similar solutions.Whereas,for stronger mass suction,the velocity boundary layer thickness becomes thinner for the first solution and the effect is opposite in the case of second solution.The temperature inside the boundary layer increases with nanoparticle volume fraction and decreases with mass suction.So,for the unsteadiness and for the presence of nanoparticles,the flow separation is delayed to some extent.

Mixed convection boundary layer flow near stagnation-point on vertical surface with slip

This paper considers the steady mixed convection boundary layer flow of a viscous and incompressible fluid near the stagnation-point on a vertical surface with the slip effect at the boundary. The temperature of the sheet and the velocity of the external flow are assumed to vary linearly with the distance from the stagnation-point. The governing partial differential equations are first transformed into a system of ordinary differential equations, which are then solved numerically by a shooting method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. The results indicate that for the opposing flow, the dual solutions exist in a certain range of the buoyancy parameter, while for the assisting flow, the solution is unique. In general, the velocity slip increases the heat transfer rate at the surface, while the thermal slip decreases it.

Stability analysis and modeling for the three-dimensional Darcy-Forchheimer stagnation point nanofluid flow towards a moving surface

In this research,the three-dimensional(3D)steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed.The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation.The slip for viscous fluid is considered.The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate.The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method.Dual solutions are obtained for the velocity,skin friction coefficient,temperature,and Nusselt number subject to sundry flow parameters,magnetic parameter,Darcy-Forchheimer number,thermal radiation parameter,suction parameter,and dimensionless slip parameter.In this research,the main consideration is given to the engineering interest like skin friction coefficient(velocity gradient or surface drag force)and Nusselt number(temperature gradient or heat transfer rate)and discussed numerically through tables.In conclusion,it is noticed from the stability results that the upper branch solution(UBS)is more reliable and physically stable than the lower branch solution(LBS).

Flow and heat transfer of nanofluid past stretching/shrinking sheet with partial slip boundary conditions

The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.