Melting heat transfer in the boundary layer flow of a couple stress fluid over a stretching surface is investigated. The developed differential equations are solved for homotopic solutions. It is observed that the vel...Melting heat transfer in the boundary layer flow of a couple stress fluid over a stretching surface is investigated. The developed differential equations are solved for homotopic solutions. It is observed that the velocity and the boundary layer thickness are decreasing functions of the couple stress fluid parameter. However, the temperature and surface heat transfer increase when the values of the couple stress fluid parameter increase. The velocity and temperature fields increase with an increase in the melting process of the stretching sheet.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 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
The characteristics of heat transfer in the three-dimensional stagnationpoint flow past a stretching/shrinking surface of the Al2O3-Cu/H2O hybrid nanofluid with anisotropic slip are investigated.The partial differenti...The characteristics of heat transfer in the three-dimensional stagnationpoint flow past a stretching/shrinking surface of the Al2O3-Cu/H2O hybrid nanofluid with anisotropic slip are investigated.The partial differential equations are converted into a system of ordinary differential equations by valid similarity transformations.The simplified mathematical model is solved computationally by the bvp4c approach in the MATLAB operating system.This solving method is capable of generating more than one solutions when suitable initial guesses are proposed.The results are proven to have dual solutions,which consequently lead to the application of a stability analysis that verifies the achievability of the first solution.The findings reveal infinite values of the dual solutions at several measured parameters causing the non-appearance of the turning points and the critical values.The skin friction increases with the addition of nanoparticles,while the escalation of the anisotropic slip effect causes a reduction in the heat transfer rate.展开更多
The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic...The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.展开更多
The study of non-axisymmetric Homann stagnation-point flow of Walter’s B nanofluid along with magnetohydrodynamic(MHD) and non-linear Rosseland thermal radiation over a cylindrical disk in the existence of the time-i...The study of non-axisymmetric Homann stagnation-point flow of Walter’s B nanofluid along with magnetohydrodynamic(MHD) and non-linear Rosseland thermal radiation over a cylindrical disk in the existence of the time-independent free stream is considered. Moreover, the notable impacts of thermophoresis and Brownian motion are analyzed by Buongiorno’s model. The momentum, energy, and concentration equations are converted into the dimensionless coupled ordinary differential equations via similarity transformations, which are later numerically solved by altering the values of the pertinent parameters. The numerical and asymptotic solutions for the large shear-to-strain rate ratio γ =a/bfor the parameters of the displacement thicknesses and the wall-shear stress are computed by perturbative expansion and analyzed. Furthermore, the technique bvp4c in MATLAB is deployed as an efficient method to analyze the calculations for the non-dimensional velocities, temperature, displacement thickness, and concentration profiles. It is observed that the two-dimensional displacement thickness parameters α andβ are reduced due to the viscoelasticity and magnetic field effects. Moreover, when the shear-to-strain rate ratio approaches infinity, α is closer to its asymptotic value, while βand the three-dimensional displacement thickness parameter δ1 show the opposite trend.The outcomes of the viscoelasticity and the magnetic field on the skin friction are also determined. It is concluded that ■ reaches its asymptotic behavior when the shearto-strain rate ratio approaches infinity. Meanwhile, ■ shows different results.展开更多
In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types： direct pow...In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types： direct power-law variation with the distance along the sheet and inverse power-law variation with the distance. The governing partial differential equations （PDEs） are transformed into non linear self-similar ordinary differential equations （ODEs） by similarity transformations, and then those are solved using very efficient shooting method. The direct variation and inverse variation of heat flux along the sheet have completely different effects on the temperature distribution. Moreover, the heat transfer characteristics in the presence of non-uniform heat flux for several values of physical parameters are also found to be interesting.展开更多
The problem of steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface is reexamined. Here the surface is stretched with a velocity proportional to the dist...The problem of steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface is reexamined. Here the surface is stretched with a velocity proportional to the distance from a fixed point. Previous studies on this problem are reviewed and the errors in the boundary conditions at infinity are rectified. It is found that for a very small value of shear in the free stream, the flow has a boundary layer structure when , where and are the free stream stagnation-point velocity and the stretching velocity of the sheet, respectively, being the distance along the surface from the stagnation-point. On the other hand, the flow has an inverted boundary layer structure when . It is also observed that for given values of and free stream shear, the horizontal velocity at a point decreases with increase in the pressure gradient parameter.展开更多
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...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.展开更多
In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a syste...In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into self-similar ordinary differential equations before they are solved numerically using the shooting method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the velocity slip parameter (α), the thermal slip parameter (β), the curvature parameter (γ) and the velocity ratio parameter (c/a). The physical quantities of interest are the skin friction coefficient and the local Nusselt number measured by f’’(0) and –θ’(0), respectively. The numerical results show that the velocity slip parameter α increases the heat transfer rate at the surface, while the thermal slip parameter β decreases it. On the other hand, increasing the velocity slip parameter α causes the decrease in the flow velocity. Further, it is found that the solutions for a shrinking cylinder (c/ac/a>0) case. Finally, it is also found that the values of f’’(0) and –θ’(0) increase as the curvature parameter γ increases.展开更多
Exact solution of the steady Navier-Stokes equations has been obtained for the thermal stagnation-point flow at the leading edge of a turbine blade under the assumptions of constant nose radius and external vorticity,...Exact solution of the steady Navier-Stokes equations has been obtained for the thermal stagnation-point flow at the leading edge of a turbine blade under the assumptions of constant nose radius and external vorticity, and fluid properties independent of temperature. The solutions reveal that curvature affects local heat transfer and skin friction while external vorticity does not. The effect of external vorticity is to shift the zero skin friction point away from the stagnation point. This solution is valid for all Reynolds number, external vorticity, and nose radius. In the limit of nose radius going to infinity and external vorticity, going to zero, the exact solution for two-dimensional plane stagnation-point flow is recovered identically. In addition, it can be shown that the velocity field around the stagnation point of a rotating curved surface is the same as that around the stagnation point of a stationary curved surface with an external vorticity which equals to twice of the rotational speed. This realization renders the present solution equally valid for thermal stagnation point flow at the leading edge of centrifugal impeller blades.展开更多
The effect of a uniform transverse magnetic field on two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a stretching surface is investigated when the surface is stret...The effect of a uniform transverse magnetic field on two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. This magnetohydrodynamic (MHD) flow problem is governed by the parameter b representing the ratio of the strain rate of the stagnation-point flow to that of the stretching sheet and the magnetic field parameter M. It is known from a previous paper [9] that if b > 1, the steady solution to the problem is monotonic increasing and the solution is also unique. But when 0 0.23919, the non-monotonic solution cannot exist and so in this case, the only solution is monotonic decreasing. A stability analysis reveals that when 0 < b < bc, the solutions along the upper branch corresponding to the monotonic solution are linearly stable while those along the lower branch for the non-monotonic solution are linearly unstable. It is also shown that the decay rate of a disturbance increases with increasing M for the stable solution but the growth rate of instability for the non-monotonic solution decreases with increasing M.展开更多
A numerical study of partial slip boundary condition is investigated. The stagnation-point flow problem involving some physio-chemical parameters has been elucidated. The process involves developing a multivariate mat...A numerical study of partial slip boundary condition is investigated. The stagnation-point flow problem involving some physio-chemical parameters has been elucidated. The process involves developing a multivariate mathematical model for the flow and transforming it into a coupled univariate equation. Key parameters of interest in the study are the buoyancy force, the surface stretching, the unsteadiness, the radiation, the dissipation effects, the slip effects, the species reaction and the magnetic field parameters. It is concluded that the impact of physio-chemical factors significantly alters the kinematics of the flow in order to optimally achieve desired product characteristics.展开更多
A numerical investigation is presented to show the effects of thermo-diffusion on the hydromagnetic stagnation-point flow of Sisko nanofluid over a linearly stretching sheet by considering viscous dissipation and Joul...A numerical investigation is presented to show the effects of thermo-diffusion on the hydromagnetic stagnation-point flow of Sisko nanofluid over a linearly stretching sheet by considering viscous dissipation and Joule heating with suction/injection.A suitable set of similarity transformations is considered in order to convert the basic partial differential equations into a set of coupled nonlinear ordinary differential equations.An efficient numerical method along with shooting technique is involved in older to solve the reduced governing basic equations.The influences of several emerging physical parameters of Sisko nanofluid on the profiles of velocity,temperature,solutal concentration,nanoparticle volume fraction,skin-friction coefficient,Nusselt number,and Sherwood number have been studied and analyzed in detail through graphs and tables.It is found that the skin friction coefficient decays more rapidly against the material parameter of Sisko fluid.ALso,it is noticed that the Brownian motion and thermophoresis parameter have the reverse effects on Sisko nanofluid Sherwood number.It is analyzed that the Nusselt number decreases with an increase in the values of thermophoresis parameter.Brownian motion parameter,and Eckert number,while its value increases with material parameter and Dufour solutal Lewis number.It is observed that the Sherwood number has ascending behavior for thermophoresis and.Brownian motion parameters,whereas nanofluid Sherwood number gets amplified with a hike in the nano Lewis number parameter for all the values of Brownian motion parameter.展开更多
An investigation is made to study the heat transfer in boundary layer stagnationpoint flow over a non-isothermal permeable shrinking sheet with suction/injection.In this study,power-law variation of sheet temperature...An investigation is made to study the heat transfer in boundary layer stagnationpoint flow over a non-isothermal permeable shrinking sheet with suction/injection.In this study,power-law variation of sheet temperature is considered.By similarity transformation,the governing equations with the boundary conditions are transformed to self-similar nonlinear ordinary differential equations and then those are solved numerically by shooting method.In presence of variable sheet temperature,the variation of temperature is analysed.For larger shrinking rate compared to that of straining rate,dual solutions for velocity and temperature are obtained.It is found that for positive value of power-law exponent of variable sheet temperature heat transfer at the sheet as well as heat absorption at the sheet with temperature overshoot near the sheet occur and for negative value heat transfer from the sheet occurs though there is overshoot away from the sheet.With increasing positive power-law exponent heat transfer reduces for first solution and heat absorption enhances for second solution.Whereas,with increasing magnitude of negative power-law exponent heat transfer increases for second solution and for first solution the heat transfer increases for larger shrinking rate and it decreases for smaller shrinking rate.Due to suction heat transfer/absorption increases in all cases and for injection heat transfer/absorption increases for first solution and decreases for second solution.Also,interesting effects of suction/injection and Prandtl number on temperature distribution are observed when the sheet temperature varies(directly/inversely)along the sheet.展开更多
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 provides mathematical modeling for melting heat and thermal radiation in stagnationpoint flow of carbon nanotubes towards a nonlinear stretchable surface of variable thickness. The process of homog...The present article provides mathematical modeling for melting heat and thermal radiation in stagnationpoint flow of carbon nanotubes towards a nonlinear stretchable surface of variable thickness. The process of homogeneousheterogeneous reactions is considered. Diffusion coefficients are considered equal for both reactant and autocatalyst.Water and gasoline oil are taken as base fluids. The conversion of partial differential system to ordinary differential system is done by suitable transformations. Optimal homotopy technique is employed for the solutions development of velocity, temperature, concentration, skin friction and local Nusselt number. Graphical results for various values of pertinent parameters are displayed and discussed. Our results indicate that the skin friction coefficient and local Nusselt number are enhanced for larger values of nanoparticles volume fraction.展开更多
For high Re number flows,a new computational fluid dynamics （CFD） validation method,namely,wall-surface compatibility criteria method and also called Gao＇s criteria,is proposed to verify near wall flows.As examples...For high Re number flows,a new computational fluid dynamics （CFD） validation method,namely,wall-surface compatibility criteria method and also called Gao＇s criteria,is proposed to verify near wall flows.As examples,Gao＇s criteria method and other two CFD verification methods （analytic solutions method and grid convergence analysis） are used to verify FLUENT＇s near wall flows solutions for two-dimensional incompressible laminar stagnation-point flows.The exact solution of the Navier-Stokes equations for laminar stagnation-point flows satisfy accurately all Gao＇s criteria,numerical simulations of laminar stagnation-point flows show that when physical parameters （the pressure coefficient and the Stanton number at wall surface） achieve convergence with more and more fine-screen grid,a part of Gao＇s criteria are not satisfied.展开更多
In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transfo...In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold's model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically.展开更多
The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrical...The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.展开更多
基金supported by the Deanship of Scientific Research(DSR),King Abdulaziz University,Jeddah,Saudi Arabia
文摘Melting heat transfer in the boundary layer flow of a couple stress fluid over a stretching surface is investigated. The developed differential equations are solved for homotopic solutions. It is observed that the velocity and the boundary layer thickness are decreasing functions of the couple stress fluid parameter. However, the temperature and surface heat transfer increase when the values of the couple stress fluid parameter increase. The velocity and temperature fields increase with an increase in the melting process of the stretching sheet.
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.
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.
文摘The characteristics of heat transfer in the three-dimensional stagnationpoint flow past a stretching/shrinking surface of the Al2O3-Cu/H2O hybrid nanofluid with anisotropic slip are investigated.The partial differential equations are converted into a system of ordinary differential equations by valid similarity transformations.The simplified mathematical model is solved computationally by the bvp4c approach in the MATLAB operating system.This solving method is capable of generating more than one solutions when suitable initial guesses are proposed.The results are proven to have dual solutions,which consequently lead to the application of a stability analysis that verifies the achievability of the first solution.The findings reveal infinite values of the dual solutions at several measured parameters causing the non-appearance of the turning points and the critical values.The skin friction increases with the addition of nanoparticles,while the escalation of the anisotropic slip effect causes a reduction in the heat transfer rate.
基金supported by the Fundamental Research Grant Scheme(FRGS)from the Ministry of HigherEducation in Malaysia(No.5524295)the Research University Grant from the Universiti Kebangsaan Malaysia(No.GUP-2013-040)
文摘The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.
文摘The study of non-axisymmetric Homann stagnation-point flow of Walter’s B nanofluid along with magnetohydrodynamic(MHD) and non-linear Rosseland thermal radiation over a cylindrical disk in the existence of the time-independent free stream is considered. Moreover, the notable impacts of thermophoresis and Brownian motion are analyzed by Buongiorno’s model. The momentum, energy, and concentration equations are converted into the dimensionless coupled ordinary differential equations via similarity transformations, which are later numerically solved by altering the values of the pertinent parameters. The numerical and asymptotic solutions for the large shear-to-strain rate ratio γ =a/bfor the parameters of the displacement thicknesses and the wall-shear stress are computed by perturbative expansion and analyzed. Furthermore, the technique bvp4c in MATLAB is deployed as an efficient method to analyze the calculations for the non-dimensional velocities, temperature, displacement thickness, and concentration profiles. It is observed that the two-dimensional displacement thickness parameters α andβ are reduced due to the viscoelasticity and magnetic field effects. Moreover, when the shear-to-strain rate ratio approaches infinity, α is closer to its asymptotic value, while βand the three-dimensional displacement thickness parameter δ1 show the opposite trend.The outcomes of the viscoelasticity and the magnetic field on the skin friction are also determined. It is concluded that ■ reaches its asymptotic behavior when the shearto-strain rate ratio approaches infinity. Meanwhile, ■ shows different results.
基金the National Board for Higher Mathematics(NBHM),DAE,Mumbai,India
文摘In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types： direct power-law variation with the distance along the sheet and inverse power-law variation with the distance. The governing partial differential equations （PDEs） are transformed into non linear self-similar ordinary differential equations （ODEs） by similarity transformations, and then those are solved using very efficient shooting method. The direct variation and inverse variation of heat flux along the sheet have completely different effects on the temperature distribution. Moreover, the heat transfer characteristics in the presence of non-uniform heat flux for several values of physical parameters are also found to be interesting.
文摘The problem of steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface is reexamined. Here the surface is stretched with a velocity proportional to the distance from a fixed point. Previous studies on this problem are reviewed and the errors in the boundary conditions at infinity are rectified. It is found that for a very small value of shear in the free stream, the flow has a boundary layer structure when , where and are the free stream stagnation-point velocity and the stretching velocity of the sheet, respectively, being the distance along the surface from the stagnation-point. On the other hand, the flow has an inverted boundary layer structure when . It is also observed that for given values of and free stream shear, the horizontal velocity at a point decreases with increase in the pressure gradient parameter.
文摘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.
文摘In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into self-similar ordinary differential equations before they are solved numerically using the shooting method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the velocity slip parameter (α), the thermal slip parameter (β), the curvature parameter (γ) and the velocity ratio parameter (c/a). The physical quantities of interest are the skin friction coefficient and the local Nusselt number measured by f’’(0) and –θ’(0), respectively. The numerical results show that the velocity slip parameter α increases the heat transfer rate at the surface, while the thermal slip parameter β decreases it. On the other hand, increasing the velocity slip parameter α causes the decrease in the flow velocity. Further, it is found that the solutions for a shrinking cylinder (c/ac/a>0) case. Finally, it is also found that the values of f’’(0) and –θ’(0) increase as the curvature parameter γ increases.
文摘Exact solution of the steady Navier-Stokes equations has been obtained for the thermal stagnation-point flow at the leading edge of a turbine blade under the assumptions of constant nose radius and external vorticity, and fluid properties independent of temperature. The solutions reveal that curvature affects local heat transfer and skin friction while external vorticity does not. The effect of external vorticity is to shift the zero skin friction point away from the stagnation point. This solution is valid for all Reynolds number, external vorticity, and nose radius. In the limit of nose radius going to infinity and external vorticity, going to zero, the exact solution for two-dimensional plane stagnation-point flow is recovered identically. In addition, it can be shown that the velocity field around the stagnation point of a rotating curved surface is the same as that around the stagnation point of a stationary curved surface with an external vorticity which equals to twice of the rotational speed. This realization renders the present solution equally valid for thermal stagnation point flow at the leading edge of centrifugal impeller blades.
文摘The effect of a uniform transverse magnetic field on two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. This magnetohydrodynamic (MHD) flow problem is governed by the parameter b representing the ratio of the strain rate of the stagnation-point flow to that of the stretching sheet and the magnetic field parameter M. It is known from a previous paper [9] that if b > 1, the steady solution to the problem is monotonic increasing and the solution is also unique. But when 0 0.23919, the non-monotonic solution cannot exist and so in this case, the only solution is monotonic decreasing. A stability analysis reveals that when 0 < b < bc, the solutions along the upper branch corresponding to the monotonic solution are linearly stable while those along the lower branch for the non-monotonic solution are linearly unstable. It is also shown that the decay rate of a disturbance increases with increasing M for the stable solution but the growth rate of instability for the non-monotonic solution decreases with increasing M.
文摘A numerical study of partial slip boundary condition is investigated. The stagnation-point flow problem involving some physio-chemical parameters has been elucidated. The process involves developing a multivariate mathematical model for the flow and transforming it into a coupled univariate equation. Key parameters of interest in the study are the buoyancy force, the surface stretching, the unsteadiness, the radiation, the dissipation effects, the slip effects, the species reaction and the magnetic field parameters. It is concluded that the impact of physio-chemical factors significantly alters the kinematics of the flow in order to optimally achieve desired product characteristics.
文摘A numerical investigation is presented to show the effects of thermo-diffusion on the hydromagnetic stagnation-point flow of Sisko nanofluid over a linearly stretching sheet by considering viscous dissipation and Joule heating with suction/injection.A suitable set of similarity transformations is considered in order to convert the basic partial differential equations into a set of coupled nonlinear ordinary differential equations.An efficient numerical method along with shooting technique is involved in older to solve the reduced governing basic equations.The influences of several emerging physical parameters of Sisko nanofluid on the profiles of velocity,temperature,solutal concentration,nanoparticle volume fraction,skin-friction coefficient,Nusselt number,and Sherwood number have been studied and analyzed in detail through graphs and tables.It is found that the skin friction coefficient decays more rapidly against the material parameter of Sisko fluid.ALso,it is noticed that the Brownian motion and thermophoresis parameter have the reverse effects on Sisko nanofluid Sherwood number.It is analyzed that the Nusselt number decreases with an increase in the values of thermophoresis parameter.Brownian motion parameter,and Eckert number,while its value increases with material parameter and Dufour solutal Lewis number.It is observed that the Sherwood number has ascending behavior for thermophoresis and.Brownian motion parameters,whereas nanofluid Sherwood number gets amplified with a hike in the nano Lewis number parameter for all the values of Brownian motion parameter.
文摘An investigation is made to study the heat transfer in boundary layer stagnationpoint flow over a non-isothermal permeable shrinking sheet with suction/injection.In this study,power-law variation of sheet temperature is considered.By similarity transformation,the governing equations with the boundary conditions are transformed to self-similar nonlinear ordinary differential equations and then those are solved numerically by shooting method.In presence of variable sheet temperature,the variation of temperature is analysed.For larger shrinking rate compared to that of straining rate,dual solutions for velocity and temperature are obtained.It is found that for positive value of power-law exponent of variable sheet temperature heat transfer at the sheet as well as heat absorption at the sheet with temperature overshoot near the sheet occur and for negative value heat transfer from the sheet occurs though there is overshoot away from the sheet.With increasing positive power-law exponent heat transfer reduces for first solution and heat absorption enhances for second solution.Whereas,with increasing magnitude of negative power-law exponent heat transfer increases for second solution and for first solution the heat transfer increases for larger shrinking rate and it decreases for smaller shrinking rate.Due to suction heat transfer/absorption increases in all cases and for injection heat transfer/absorption increases for first solution and decreases for second solution.Also,interesting effects of suction/injection and Prandtl number on temperature distribution are observed when the sheet temperature varies(directly/inversely)along the sheet.
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 provides mathematical modeling for melting heat and thermal radiation in stagnationpoint flow of carbon nanotubes towards a nonlinear stretchable surface of variable thickness. The process of homogeneousheterogeneous reactions is considered. Diffusion coefficients are considered equal for both reactant and autocatalyst.Water and gasoline oil are taken as base fluids. The conversion of partial differential system to ordinary differential system is done by suitable transformations. Optimal homotopy technique is employed for the solutions development of velocity, temperature, concentration, skin friction and local Nusselt number. Graphical results for various values of pertinent parameters are displayed and discussed. Our results indicate that the skin friction coefficient and local Nusselt number are enhanced for larger values of nanoparticles volume fraction.
基金Sponsored by the National Natural Science Foundation of China(10702009)
文摘For high Re number flows,a new computational fluid dynamics （CFD） validation method,namely,wall-surface compatibility criteria method and also called Gao＇s criteria,is proposed to verify near wall flows.As examples,Gao＇s criteria method and other two CFD verification methods （analytic solutions method and grid convergence analysis） are used to verify FLUENT＇s near wall flows solutions for two-dimensional incompressible laminar stagnation-point flows.The exact solution of the Navier-Stokes equations for laminar stagnation-point flows satisfy accurately all Gao＇s criteria,numerical simulations of laminar stagnation-point flows show that when physical parameters （the pressure coefficient and the Stanton number at wall surface） achieve convergence with more and more fine-screen grid,a part of Gao＇s criteria are not satisfied.
文摘In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold's model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically.
基金supported by the C.S.I.R.,India in the form of Junior Research Fellowship(JRF)(Grant No.09/149(0593)/2011-EMR-I)
文摘The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.
