An analysis was made to study the steady momentum and heat transfer characteristics of a viscous electrically conducting fluid near a stagnation point due to a stretching/shrinking sheet in the presence of a transvers...An analysis was made to study the steady momentum and heat transfer characteristics of a viscous electrically conducting fluid near a stagnation point due to a stretching/shrinking sheet in the presence of a transverse magnetic field and generalized slip condition. Two flow problems corresponding to the planar and axisymmetric stretching/shrinking sheet were considered. By means of similarity transformations, the obtained resultant nonlinear ordinary differential equations were solved numerically using a shooting method for dual solutions of velocity and temperature profiles. Some important physical features of the flow and heat transfer in terms of the fluid velocity, the temperature distribution, the skin friction coefficient and the local Nusselt number for various values of the controlling governing parameters like velocity slip parameter, critical shear rate, magnetic field, ratio of stretching/shrinking rate to external flow rate and Prandtl number were analyzed and discussed. An increase of the critical shear rate decreases the fluid velocity whereas the local Nusselt number increases. The comparison of the present numerical results with the existing literature in a limiting case is given and found to be in an excellent agreement.展开更多
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 dif...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.展开更多
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 similarit...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.展开更多
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...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.展开更多
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 s...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.展开更多
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 differenti...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.展开更多
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 con...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.展开更多
In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a m...In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a magnetic dipole.To make the results physically realistic,stability analysis is also carried out in this study so that we realized which solution is stable and which is not.The governing partial equations are converted into ordinary differential equations by using similarity transformations and the numerical solution is calculated by applying bvp4c function technique in MATLAB software.The effects of different physical parameters are plotted graphically and discussed according to the outcomes of results.From the present study we observe that ferromagnetic interaction parameter had a great influenced on fluid velocity and temperature distributions.It is also found from the current analysis that the first and second solutions of shrinking cylinder obtained only when we applied particular ranges values of suction parameter.The most important characteristics part of study is to analyze the skin friction coefficient and rate of heat transfer which also covered in this analysis.It reveals that both skin friction coefficient and rate of heat transfer are reduced with rising values of ferromagnetic number.A comparison has also been made to make the solution feasible.展开更多
The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and ...The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and porous velocities and variable thickness of the sheet and they are combined in a relation.Consequently,the new problem reproduces the different available forms of flow motion and heat transfer maintained over a stretching(shrinking)and porous sheet of variable thickness in one go.As a result,the governing equations are embedded in several parameters which can be transformed into classical cases of stretched(shrunk)flows over porous sheets.A set of general,unusual and new variables is formed to simplify the governing partial differential equations and boundary conditions.The final equations are compared with the classical models to get the validity of the current simulations and they are exactly matched with each other for different choices of parameters of the current problem when their values are properly adjusted and manipulated.Moreover,we have recovered the classical results for special and appropriate values of the parameters(δ_(1),δ_(2),δ_(3),c,and B).The individual and combined effects of all inputs from the boundary are seen on flow and heat transfer properties with the help of a numerical method and the results are compared with classical solutions in special cases.It is noteworthy that the problem describes and enhances the behavior of all field quantities in view of the governing parameters.Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter.A stability analysis is accomplished and apprehended in order to establish a criterion for the determinations of linearly stable and physically compatible solutions.The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable(uniform)thickness with variable(uniform)stretching/shrinking and injection/suction velocities.展开更多
This study investigates the heat transfer and flow dynamics of a ternary hybrid nanofluid comprising alumina,copper,and silica/titania nanoparticles dispersed in water.The analysis considers the effects of suction,mag...This study investigates the heat transfer and flow dynamics of a ternary hybrid nanofluid comprising alumina,copper,and silica/titania nanoparticles dispersed in water.The analysis considers the effects of suction,magnetic field,and Joule heating over a permeable shrinking disk.Amathematicalmodel is developed and converted to a systemof differential equations using similarity transformation which then,solved numerically using the bvp4c solver in Matlab software.The study introduces a novel comparative analysis of alumina-copper-silica and alumina-coppertitania nanofluids,revealing distinct thermal conductivity behaviors and identifying critical suction values necessary for flow stabilization.Dual solutions are found within a specific range of parameters such that the minimum required suction values for flow stability,with S_(c)=1.2457 for alumina-copper-silica/water and S_(c)=1.2351 for alumina-coppertitania/water.The results indicate that increasing suction by 1%enhances the skin friction coefficient by up to 4.17%and improves heat transfer efficiency by approximately 1%,highlighting its crucial role in stabilizing the opposing flow induced by the shrinking disk.Additionally,the inclusion of 1%silica nanoparticles reduces both skin friction and heat transfer rate by approximately 0.28%and 0.85%,respectively,while 1%titania concentration increases skin friction by 3.02%but results in a slight heat transfer loss of up to 0.61%.These findings confirm the superior thermal performance of alumina-copper-titania/water,making it a promising candidate for enhanced cooling systems,energy-efficient heat exchangers,and industrial thermal management applications.展开更多
This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity a...This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity and thermal slip boundary conditions are also examined.The considered hybrid nano-fluid contains silver(Ag)and iron oxide(Fe_(3)O_(4))nanoparticles dispersed in the water to prepare the Ag-Fe_(3)O_(4)/water-based hybrid nanofluid.The requisite posited partial differential equations model is converted to ordinary differential equations using similarity transformations.For a numerical solution,the shooting method in Maple is employed.Moreover,the duality in solutions is achieved for both cases of the disk(stretching(λ>0)and shrinking(λ<0)).At the same time,a unique solution is observed for λ=0.No solution is found for them at λ<λ_(c),whereas the solutions are split at the λ=λ_(c).Besides,the value of the λ_(c) is dependent on the φ_(hnf).Meanwhile,the values of f″(0)and -θ′(0)intensified with increasing φ_(hnf).Stability analysis has been applied using bvp4c in MATLAB software due to a dual solution.Furthermore,analysis shows that the first solution is stable and feasible physically.For the slip parameters,an increase in the velocity slip parameter increases the velocity and shear stress profiles while increasing the temperature profile in the first solutions.While the rise in thermal slip parameter reduces the temperature profile nanoparticle volume fractions increase it.展开更多
Model of Casson nanofluid flow over a nonlinear shrinking surface is considered.Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver.The governing nonlinear equations in...Model of Casson nanofluid flow over a nonlinear shrinking surface is considered.Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver.The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems (BVPs) of ordinary differential equations (ODEs) by using appropriate similarity transformations.The resulting equations are converted into initial value problems (IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order.In order to determine the stability of the dual solutions obtained,stability analysis is performed and discovered that the first (second) solution is stable (unstable) and physically realizable (unrealizable).Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter (β) is increased in the second solution.展开更多
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...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.展开更多
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 ...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.展开更多
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 ...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).展开更多
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 s...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.展开更多
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 ...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.展开更多
An analysis is carried out to investigate the steady mixed convection bound- ary layer flow of a water based nanofluid past a vertical semi-infinite flat plate. Using an appropriate similarity transformation, the gove...An analysis is carried out to investigate the steady mixed convection bound- ary layer flow of a water based nanofluid past a vertical semi-infinite flat plate. Using an appropriate similarity transformation, the governing partial differential equations are transformed into the coupled, nonlinear ordinary （similar） differential equations, which are then solved numerically for the Prandtl number Pr = 6.2. The skin friction coeffi- cient, the local Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ and the mixed convection parameter λ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from an assisting flow, it is found that the solutions for an opposing flow are non-unique. In order to establish which solution branch is stable and physically realizable in practice, a stability analysis is performed.展开更多
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 par...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.展开更多
A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation ...A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das （Tiwari I K and Das M K 2007 Int. J. HeatMass Transfer 50 2002）. A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge-Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter ~, solid volume fraction tp, magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.展开更多
文摘An analysis was made to study the steady momentum and heat transfer characteristics of a viscous electrically conducting fluid near a stagnation point due to a stretching/shrinking sheet in the presence of a transverse magnetic field and generalized slip condition. Two flow problems corresponding to the planar and axisymmetric stretching/shrinking sheet were considered. By means of similarity transformations, the obtained resultant nonlinear ordinary differential equations were solved numerically using a shooting method for dual solutions of velocity and temperature profiles. Some important physical features of the flow and heat transfer in terms of the fluid velocity, the temperature distribution, the skin friction coefficient and the local Nusselt number for various values of the controlling governing parameters like velocity slip parameter, critical shear rate, magnetic field, ratio of stretching/shrinking rate to external flow rate and Prandtl number were analyzed and discussed. An increase of the critical shear rate decreases the fluid velocity whereas the local Nusselt number increases. The comparison of the present numerical results with the existing literature in a limiting case is given and found to be in an excellent agreement.
文摘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.
基金the financial support of National Board for Higher Mathematics(NBHM),DAE,Mumbai,Indiapartially supported by Deanship of Scientific Research(DSR),King Abdulaziz University,Jeddah,Saudi Arabia
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 50936003 and 51076012)
文摘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.
文摘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.
文摘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.
文摘In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a magnetic dipole.To make the results physically realistic,stability analysis is also carried out in this study so that we realized which solution is stable and which is not.The governing partial equations are converted into ordinary differential equations by using similarity transformations and the numerical solution is calculated by applying bvp4c function technique in MATLAB software.The effects of different physical parameters are plotted graphically and discussed according to the outcomes of results.From the present study we observe that ferromagnetic interaction parameter had a great influenced on fluid velocity and temperature distributions.It is also found from the current analysis that the first and second solutions of shrinking cylinder obtained only when we applied particular ranges values of suction parameter.The most important characteristics part of study is to analyze the skin friction coefficient and rate of heat transfer which also covered in this analysis.It reveals that both skin friction coefficient and rate of heat transfer are reduced with rising values of ferromagnetic number.A comparison has also been made to make the solution feasible.
文摘The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and porous velocities and variable thickness of the sheet and they are combined in a relation.Consequently,the new problem reproduces the different available forms of flow motion and heat transfer maintained over a stretching(shrinking)and porous sheet of variable thickness in one go.As a result,the governing equations are embedded in several parameters which can be transformed into classical cases of stretched(shrunk)flows over porous sheets.A set of general,unusual and new variables is formed to simplify the governing partial differential equations and boundary conditions.The final equations are compared with the classical models to get the validity of the current simulations and they are exactly matched with each other for different choices of parameters of the current problem when their values are properly adjusted and manipulated.Moreover,we have recovered the classical results for special and appropriate values of the parameters(δ_(1),δ_(2),δ_(3),c,and B).The individual and combined effects of all inputs from the boundary are seen on flow and heat transfer properties with the help of a numerical method and the results are compared with classical solutions in special cases.It is noteworthy that the problem describes and enhances the behavior of all field quantities in view of the governing parameters.Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter.A stability analysis is accomplished and apprehended in order to establish a criterion for the determinations of linearly stable and physically compatible solutions.The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable(uniform)thickness with variable(uniform)stretching/shrinking and injection/suction velocities.
基金funded by Universiti Teknikal Malaysia Melaka,through Fakulti Teknologi dan Kejuruteraan Mekanikal(FTKM)’s publication fund-K23003.
文摘This study investigates the heat transfer and flow dynamics of a ternary hybrid nanofluid comprising alumina,copper,and silica/titania nanoparticles dispersed in water.The analysis considers the effects of suction,magnetic field,and Joule heating over a permeable shrinking disk.Amathematicalmodel is developed and converted to a systemof differential equations using similarity transformation which then,solved numerically using the bvp4c solver in Matlab software.The study introduces a novel comparative analysis of alumina-copper-silica and alumina-coppertitania nanofluids,revealing distinct thermal conductivity behaviors and identifying critical suction values necessary for flow stabilization.Dual solutions are found within a specific range of parameters such that the minimum required suction values for flow stability,with S_(c)=1.2457 for alumina-copper-silica/water and S_(c)=1.2351 for alumina-coppertitania/water.The results indicate that increasing suction by 1%enhances the skin friction coefficient by up to 4.17%and improves heat transfer efficiency by approximately 1%,highlighting its crucial role in stabilizing the opposing flow induced by the shrinking disk.Additionally,the inclusion of 1%silica nanoparticles reduces both skin friction and heat transfer rate by approximately 0.28%and 0.85%,respectively,while 1%titania concentration increases skin friction by 3.02%but results in a slight heat transfer loss of up to 0.61%.These findings confirm the superior thermal performance of alumina-copper-titania/water,making it a promising candidate for enhanced cooling systems,energy-efficient heat exchangers,and industrial thermal management applications.
基金the Researchers Supporting Project number(RSPD2025R997),King Saud University,Riyadh,Saudi Arabia.
文摘This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity and thermal slip boundary conditions are also examined.The considered hybrid nano-fluid contains silver(Ag)and iron oxide(Fe_(3)O_(4))nanoparticles dispersed in the water to prepare the Ag-Fe_(3)O_(4)/water-based hybrid nanofluid.The requisite posited partial differential equations model is converted to ordinary differential equations using similarity transformations.For a numerical solution,the shooting method in Maple is employed.Moreover,the duality in solutions is achieved for both cases of the disk(stretching(λ>0)and shrinking(λ<0)).At the same time,a unique solution is observed for λ=0.No solution is found for them at λ<λ_(c),whereas the solutions are split at the λ=λ_(c).Besides,the value of the λ_(c) is dependent on the φ_(hnf).Meanwhile,the values of f″(0)and -θ′(0)intensified with increasing φ_(hnf).Stability analysis has been applied using bvp4c in MATLAB software due to a dual solution.Furthermore,analysis shows that the first solution is stable and feasible physically.For the slip parameters,an increase in the velocity slip parameter increases the velocity and shear stress profiles while increasing the temperature profile in the first solutions.While the rise in thermal slip parameter reduces the temperature profile nanoparticle volume fractions increase it.
基金Universiti Utara Malaysia (UUM) for the moral and financial support in conducting this research
文摘Model of Casson nanofluid flow over a nonlinear shrinking surface is considered.Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver.The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems (BVPs) of ordinary differential equations (ODEs) by using appropriate similarity transformations.The resulting equations are converted into initial value problems (IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order.In order to determine the stability of the dual solutions obtained,stability analysis is performed and discovered that the first (second) solution is stable (unstable) and physically realizable (unrealizable).Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter (β) is increased in the second solution.
文摘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.
基金the Ministry of Education of Malaysia(No.FRGS/1/2019/STG06/UKM/01/4)Ministry of Science of Romania(No.PN-III-P4-ID-PCE-2016-0036)。
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.11971142,11871202,61673169,11701176,11626101,and 11601485)。
文摘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).
基金supported by Universiti Kebangsaan Malaysia(No.UKM-GUP-2011-202)
文摘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.
基金Project supported by the Ministry of Higher Education in Malaysia(No.FRGS/1/2012/SG04/UKM/2001/1)the Universiti Kebangsaan Malaysia(No.DIP-2012-31)
文摘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.
基金Project supported by the grant of the Romanian National Authority for Scientific Research,CNCSUEFISCDI(No.PN-II-RU-TE-2011-3-0013)
文摘An analysis is carried out to investigate the steady mixed convection bound- ary layer flow of a water based nanofluid past a vertical semi-infinite flat plate. Using an appropriate similarity transformation, the governing partial differential equations are transformed into the coupled, nonlinear ordinary （similar） differential equations, which are then solved numerically for the Prandtl number Pr = 6.2. The skin friction coeffi- cient, the local Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ and the mixed convection parameter λ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from an assisting flow, it is found that the solutions for an opposing flow are non-unique. In order to establish which solution branch is stable and physically realizable in practice, a stability analysis is performed.
基金the financial supports received in the form of fundamental research grant scheme (FRGS)the financial supports received in the form of research university grant (GUP)
文摘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.
文摘A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper （Cu）-water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das （Tiwari I K and Das M K 2007 Int. J. HeatMass Transfer 50 2002）. A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge-Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter ~, solid volume fraction tp, magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.
