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

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

Boundary Layer Flow and Heat Transfer over an Exponentially Shrinking Sheet

An analysis is made to study boundary layer flow and heat transfer over an exponentially shrinking sheet.Using similarity transformations in exponential form,the governing boundary layer equations are transformed into self-similar nonlinear ordinary differential equations,which are then solved numerically using a very efficient shooting method.The analysis reveals the conditions for the existence of steady boundary layer flow due to exponential shrinking of the sheet and it is found that when the mass suction parameter exceeds a certain critical value,steady flow is possible.The dual solutions for velocity and temperature distributions are obtained.With increasing values of the mass suction parameter,the skin friction coefficient increases for the first solution and decreases for the second solution.

Slip Effects on an Unsteady Boundary Layer Stagnation-Point Flow and Heat Transfer towards a Stretching Sheet

An analysis is presented for an unsteady boundary layer stagnation-point flow of a Newtonian fluid and the heat transfer towards a stretching sheet taking non-conventional partial slip conditions at the sheet.The self-similar equations are obtained using similarity transformations and solved numerically by the shooting method.Effects of the parameters involved in the equations,especially velocity slip and thermal slip parameters on the velocity and temperature profiles,are analyzed extensively.It is revealed that due to the velocity and thermal slip parameters,the rate of heat transfer from the sheet and the wall skin friction change significantly.

MHD Boundary Layer Slip Flow and Heat Transfer over a Flat Plate

An analysis of magnetohydrodynamic(MHD)boundary layer flow and heat transfer over a flat plate with slip condition at the boundary is presented.A complete self-similar set of equations are obtained from the governing equations using similarity transformations and are solved by a shooting method.In the boundary slip condition no local similarity occurs.Velocity and temperature distributions within the boundary layer are presented.Our analysis reveals that the increase of magnetic and slip parameters reduce the boundary layer thickness and also enhance the heat transfer from the plate.

Analytic solution for magnetohydrodynamic boundary layer flow of Casson fluid over a stretching/shrinking sheet with wall mass transfer

In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing equations are converted to an ordinary differential equation and then solved analytically.The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case.The similarity solution is always unique in the stretching case,and in the shrinking case the solution shows dual nature for certain values of the parameters.For stronger magnetic field,the similarity solution for the shrinking sheet case becomes unique.

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

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

Heat transfer in boundary layer stagnation-point flow towards a shrinking sheet with non-uniform heat flux

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.

Dual Solutions in Unsteady Stagnation-Point Flow over a Shrinking Sheet

An analysis is made to study the dual nature of solution of unsteady stagnation-point flow due to a shrinking sheet.Using similarity transformations,the governing boundary layer equations are transformed into the self-similar nonlinear ordinary differential equations.The transformed equations are solved numerically using a very efficient shooting method.The study reveals the conditions of existence,uniqueness and non-existence of unsteady similarity solution.The dual solutions for velocity distribution exist for certain values of velocity ratio parameter(c/a),and the increment in the unsteadiness parameter A increases the range of c/a where solution exists.Also,with increasing A,the skin friction coefficient increases for the first solution and decreases for the second.

Heat transfer for boundary layers with cross flow

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

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

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

Exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects

The effects of transpiration on forced convection boundary layer non-Newtonian fluid flow and heat transfer toward a linearly stretching surface are reported. The flow is caused solely by the stretching of the sheet in its own plane with a velocity varying linearly with the distance from a fixed point. The constitutive relationship for the Casson fluid is used. The governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations by using similarity transformations. Exact solutions of the resulting ordinary differential equations are obtained. The effect of increasing Casson parameter, i.e., with decreasing yield stress （the fluid behaves as a Newtonian fluid as the Casson parameter becomes large）, is to suppress the velocity field. However, the temperature is enhanced as the Casson parameter increases. It is observed that the effect of transpiration is to decrease the fluid velocity as well as the temperature. The skin-friction coefficient is found to increase as the transpiration parameter increases.

MHD Boundary Layer Flow of Dilatant Fluid in a Divergent Channel with Suction or Blowing

An analysis is carried out to study a steady magnetohydrodynamic(MHD)boundary layer flow of an electrically conducting incompressible power-law non-Newtonian fluid through a divergent channel.The channel walls are porous and subjected to either suction or blowing of equal magnitude of the same kind of fluid on both walls.The fluid is permeated by a magnetic field produced by electric current along the line of intersection of the channel walls.The governing partial differential equation is transformed into a self-similar nonlinear ordinary differential equation using similarity transformations.The possibility of boundary layer flow in a divergent channel is analyzed with the power-law fluid model.The analysis reveals that the boundary layer flow(without separation)is possible for the case of the dilatant fluid model subjected to suitable suction velocity applied through its porous walls,even in the absence of a magnetic field.Further,it is found that the boundary layer flow is possible even in the presence of blowing for a suitable value of the magnetic parameter.It is found that the velocity increases with increasing values of the power-law index for the case of dilatant fluid.The effects of suction/blowing and magnetic field on the velocity are shown graphically and discussed physically.

Insight into the relationship between non-linear mixed convection and thermal radiation:The case of Newtonian fluid flow due to non-linear stretching

The current research focuses the light on the characterization of buoyancy-driven non-linear mixed convection and non-linear radiation in a Newtonian flow over a nonlinearly stretching vertical sheet,and this type of flow has useful applications in many industrial processes,such as the paper and pulp industry,polymer industry,electronic device cooling,solar collectors,gas turbine plants,and nuclear power.Using appropriate transformations,governing PDEs for non-linear mixed convection are reduced to higher-order non-linear ODEs and those are numerically solved.Along with tabular presentations of computed results,the graphical representations are generated to elucidate the effects of involved parameters on convection transport properties and their inter-relations.It demonstrates that flow velocity increases near the surface and decreases away from the surface as the non-linear convection parameter increases.Furthermore,increments in the thermal buoyancy,temperature ratio and non-linear radiation parameters result in the boost of velocity.The temperature decreases as linear and non-linear buoyancy-related parameters(non-linear convection and thermal buoyancy parameters)are of higher levels.In contrast,the temperature rises with two non-linear thermal radiation-related parameters(thermal ratio and non-linear radiation parameters).For greater values of the non-linear stretching related parameter,a lower velocity and a higher temperature are witnessed.The non-linear convection,thermal buoyancy,thermal ratio and non-linear radiation parameters contribute toward the reduction of the magnitude of surface-drag force and growth of the surface cooling rate.But,with the non-linearity in surface stretching there are significant percentage hikes of surface-drag force magnitude and surface cooling rate.

Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection

In this paper,an investigation is made to study the effects of radiation and heat source/sink on the unsteady boundary layerflow and heat transfer past a shrinking sheet with suction/injection.Theflow is permeated by an externally applied magneticfield normal to the plane offlow.The self-similar equations correspond-ing to the velocity and temperaturefields are obtained,and then solved numerically byfinite difference method using quasilinearization technique.The study reveals that the momentum boundary layer thickness increases with increasing unsteadiness and decreases with magneticfield.The thermal boundary layer thickness decreases with Prandtl number,radiation parameter and heat sink parameter,but it increases with heat source parameter.Moreover,increasing unsteadiness,magneticfield strength,radiation and heat sink strength boost the heat transfer.

Thermal boundary layer in stagnation-point flow past a permeable shrinking sheet with variable surface 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.

Micropolar fluid flow and heat transfer over exponentially permeable shrinking sheet

The importance of boundary layer flow of micropolar fluid and heat transfer over an exponentially penneable shrinking sheet is analysed.The similarity approach is adopted and self-similar ordinary differential equations are obtained and then those are solved numerically using very efficient shooting method.Similar to that of Newtonian fluid flow case,here also dual similarity solutions for velocity,microrotation and temperature are obtained when certain amount of mass suction is applied through the porous sheet.For steady flow of micropolar fluid over exponentially shrinking porous sheet the mass suction need to be stronger compared to the Newtonian fluid flow.From dual velocity,microrotation,and temperature profiles it is found that the velocity decreases with material parameter(related to micropolar fluid)for first solution and it increases for second,whereas the effects are opposite for fluid temperature.On the other hand,for larger material parameter microrotation profile reduces for both types of solutions.But it significant that the skin friction coefficient,the couple stress coefficient and the heat transfer coefficient show similar variation with increasing material parameter,all those physical quantities decrease for first solution and increase for second solution.