Flow induced by a continuously stretching sheet in a quiescent fluid have applications in manufacturing processes of plastic films, hot rolling, glass fiber, wire drawing, paper production, extrusion, spinning of laments, production of synthetic sheets, glass fiber and paper production, extrusion, continuous casting, spinning of laments, production of synthetic sheets etc [1–10]. In convection, there is a fluid flowing relative to surface and a temperature difference between fluid and surface. In these heat convection processes, energy is transferred from a surface to fluid flowing over it as a result of temperature difference between fluid and surface. The flow over a surface is also induced by thermal buoyancy and these thermal buoyancy effects are more prominent if the surface is vertical. When the sheet is being stretched vertically with a forced velocity then buoyance forces cannot be neglected and flow in this case is mixed convective flow [11–19].

Boundary layer flow over an exponentially stretching surface is firstly investigated by Magyari and Keller [20]. Elbashbeshy [21] performed heat transfer analysis of the flow induced by a permeable exponentially stretching sheet. Sajid and Hayat [22] computed the series solution of the self-similar equations that governs the flow over an exponentially stretching sheet in the presence of linear thermal radiation by the homotopy analysis method. The effect of thermal radiation in the presence of viscous dissipation on the flow over an exponentially stretching sheet is examined by Bidin and Nazar [23]. Partha et al [24] provided the similarity solution of mixed convection flow over an exponentially stretching sheet in the presence of frictional heating. Bhattacharyya and Layek [25] investigated boundary layer flow over an exponentially stretching surface in an exponentially moving parallel free stream. Dual solutions of MHD boundary layer flow induced by an exponentially stretching sheet with non-uniform heat sink/source are analysed by Raju et al [26]. Sulochana and Sandeep [27] examined the stagnation point flow of nanofluid over an exponentially shrinking/stretching cylinder. Three-dimensional flow of viscous fluid over an exponentially stretching sheet is firstly studied by Liu et al [28]. Afridi and Qasim [29] inspected the entropy generation in three-dimensional boundary flow over an exponentially stretching surface in the presence of viscous dissipation.

All studies cited above related to flow induced by the exponentially stretching sheet with constant fluid properties. It is now well established that the physical properties of fluids may vary with temperature [30–34]. The magnitude of variation in properties varies from one fluid to another and depends on temperature limits of ambient fluid. It can be observed that the variable viscosity and thermal conductivity are quite significant for most of the fluids as compared to the variations in other properties. It is also believed that the difference between the theoretical and experimental results is because the properties of fluid are assumed to be constant and not temperature-dependent.

To best of our knowledge no one analysed the three-dimensional boundary layer flow over an exponentially stretching sheet with temperature-dependent thermal conductivity. Keeping this in mind, our aim is to analyse three-dimensional mixed convection flow induced by exponentially stretching sheet in the presence of frictional heating. The thermal conductivity of fluid is assumed to vary with temperature. Governing equations are modelled and simplified under boundary layer approximations. Modelled partial differential equations are then transformed into non-linear ordinary differential equations by utilizing similarity transformations. Special forms of stretching velocity and surface temperature varying exponentially are chosen for accurate similarity transformations. The novelty of the present study is the self-similar solution of three-dimensional boundary-layer flow in the presence of mixed convection, viscous dissipation and temperature-dependent thermal conductivity. To check the validation of our numerical code of the shooting method obtained results are also compared with Matlab boundary value solver bvp4c [35]. Effects of physical parameters of interest on velocity and temperature profile are analyzed.

Three-dimensional mixed convection boundary layer flow over a sheet that is being stretched exponentially has been considered for investigation. The sheet is being stretched in both $x-$ and $y-$ directions with velocities ${u}_{w}={u}_{0}{{\rm{e}}}^{\left(\tfrac{x+y}{L}\right)}$ and ${v}_{w}={v}_{0}{{\rm{e}}}^{\left(\tfrac{x+y}{L}\right)}$ respectively. The temperature of the sheet is ${T}_{s}={T}_{a}+A{{\rm{e}}}^{2\left(\tfrac{x+y}{L}\right)}$ (A is the constant characteristic temperature) and the temperature of the ambient fluid is ${T}_{a}.$ In the presence of frictional heating the boundary layer equations for the flow and heat transfer are [28, 29]: along with physical boundary conditions [28, 29] where $u,v$ and w are the components of velocity along $x-\,,\,y-$ and $z-$ axes, respectively, u₀ and v₀ are stretching parameters, μ is the dynamic viscosity, ρ is the density of fluid, g is the acceleration due to gravity, ${\alpha }_{T}$ is the thermal expansion coefficient, S_p denotes the specific heat, $k\left(T\right)$ is the thermal conductivity of the fluid, which is assumed to be temperature-dependent and take the following form [30–34]:here δ is the variable thermal conductivity parameter.

Introducing similarity transformations With similarity variable ξ, ${F}^{{\prime} }$ and ${G}^{{\prime} }$ being the dimensionless axial and transverse velocities, respectively, Θ is the dimensionless temperature and prime denotes the differentiation with respect to ξ. Equations $\left(1\right)\,-\,\left(4\right)$ along with boundary conditions (5) and (6) after using transformations (8) are transformed to self-similar equations with boundary conditionsHere $\varepsilon =\tfrac{{u}_{0}}{{v}_{0}}$ is velocity ratio parameter, ${Ec}=\tfrac{{u}_{0}^{2}}{{S}_{p}A}$ is Eckert number, and ${\Pr }=\tfrac{\mu {S}_{p}}{{k}_{a}}$ is Prandtl number and $\lambda =\tfrac{{{Gr}}_{x}}{{\mathrm{Re}}_{x}^{2}}=\tfrac{{gA}{\alpha }_{T}L}{{u}_{0}^{2}}$ is the mixed convection parameter.

The effects of mixed convection parameter λ, Prandtl number $\Pr $ Eckert number Ec, velocity ratio parameter ϵ and variable thermal conductivity parameter δ on velocity components ${F}^{{\prime} }\left(\xi \right),$ ${G}^{{\prime} }\left(\xi \right)$ and temperature ${\rm{\Theta }}\left(\xi \right)$ are investigated. This purpose is achieved by plotting figures 1–9. The effect of the mixed convection parameter on velocity ${F}^{{\prime} }\left(\xi \right)$ is seen in figure 1. Velocity ${F}^{{\prime} }\left(\xi \right)$ increases by increasing $\lambda .$ For large λ, there is a stronger buoyancy force due to which momentum boundary layer increases. The influence of Prandtl number $\Pr $ on velocity ${F}^{{\prime} }\left(\xi \right)$ is displayed in figure 2. The large values of Prandtl number $\Pr $ correspond to stronger momentum and weaker thermal diffusivity, which therefore reduces the thermal boundary layer thickness. Figure 3 explicates the effect of Eckert number Ec on the velocity As expected velocity increases by increasing ${Ec}.$ Figure 4 illustrates that velocity ${F}^{{\prime} }\left(\xi \right)$ decreases by increasing velocity ratio parameter ϵ whereas velocity ${G}^{{\prime} }\left(\xi \right)$ increases by increasing velocity ratio parameter ϵ (see figure 5). Temperature and thermal boundary layer decreases upon increasing mixed convection parameter λ (see figure 6). Increasing the buoyancy parameter specifies the higher temperature from the surface relative to ambient, therefore thermal boundary layer decreases. The influence of Prandtl number $\Pr $ on the temperature field is illustrated in figure 7. The increase in the Prandtl number $Pr$ indicates a reduction in the thermal conductivity of the fluid and because of this reduction, the penetration depth of the thermal energy decreases, therefore temperature of the fluid decreases. Figure 8 shows that temperature distribution and thermal boundary layer thickness increases with an increase in Eckert number ${Ec}.$ Physically, by increasing friction between the adjacent layers of fluid increases and the kinetic energy of fluid is converted into thermal energy, which in turn rises the fluid temperature. The influence of the thermal conductivity parameter $\delta \ $ on the temperature profile is portrayed in figure 9. It is observed that the temperature of the fluid in the boundary layer increases by increasing the value of the variable thermal conductivity parameter δ. Physically, as the thermal conductivity parameter increases, the thermal conductivity of the fluid increases and, as a result, the thermal energy of fluid increases, resulting in an increase in temperature. Table 1 shows that skin friction coefficients along $-{F}^{{\prime\prime} }\left(0\right)$ transverse direction decreases by increasing mixed convection parameter λ, whereas $-{G}^{{\prime\prime} }\left(0\right)$ and local Nusselt number $-{{\rm{\Theta }}}^{{\prime} }\left(0\right)$ increases with increasing mixed convection parameter $\lambda .$ Skin friction coefficients and local Nusselt number decreases by increasing the variable thermal conductivity parameter. $-{F}^{{\prime\prime} }\left(0\right)$ and $-{{\rm{\Theta }}}^{{\prime} }\left(0\right)$ are decreasing functions of Eckert number Ec whereas increasing functions of Prandtl number ${\Pr }.$

From the current analysis, the following specific conclusions are deduced: