Numerical Solution of Natural Convection on a Vertical Stretching Surface with Suction and Blowing

Heat transfer by natural convection frequently occurs in many physical problems and engineering applications, Viscous fluid flow studies are conducted in many practical applications with heat transmission due to stretching surface areas in production, where manufacturing process of plastic and rubber sheet through the non-functional product. Other notable applications of the study are spinning of fibers, Extrusion processes, continuous coating, and glass blowing. This was noted by some of the researchers viz., Surma Devi et al. [1] analyzed on a stretching sheet of an unsteady, 3-D boundary-layer flow. Heat and mass transfer on a stretching sheet with suction or blowing was examined by Gupta and Gupta [2], Chen and Char [3], and Vajravelu [4]. Bhattacharyya [5] shows the heat source/sink impacts of MHD flow and the transfer of heat over a shrink layer in the existence of mass suction. Jat and Chaudhary [6] investigated a stretching sheet with heat transfer on MHD flow. Fathizadeh et al. [7] have considered MHD viscous flow and heat transfer over a stretching sheet with an effective modification of the homotopy perturbation. Heat transfer over a steady stretching surface with suction was analyzed by Siri et al. [8]. Daskalakis [9] then studied free convection effects in the boundary layer along a vertically stretching flat surface. Similarity answer on MHD boundary layer across stretching surface considering thermal flux was studied by Ferdows et al. [10]. Goud and Shekar [11] have studied the effects on unsteady MHD flow past a parabolic started vertical plate with mass diffusion and viscous dissipation with variable temperature by the technique of finite element process. Wang [12] presented free convection on a perpendicular stretching surface. Bestman [13] investigated the natural convection boundary layer in a porous medium with mass transfer and the effect of suction. Acharya et al. [14] have studied the suction and blowing effect on an acceleration surface with heat and mass transfer. An unsteady MHD flow past an impulsive inclined oscillating plate with the impact of viscous dissipation and diffusion thermo in the presence of variable temperature was deliberated by Goud and Rajahekar [15]. Ali [16] investigated the influence of suction/injection on the stretched surface power-law thermal boundary. Free convection boundary layer over a vertical porous flat plate with internal thermal generation in a permeable medium analyzed by Postelnicu et al. [17]. Shateyi [18] presented the buoyancy impacts and thermal radiation on heat & mass transfer through a semi-infinite stretching surface with suction and blowing. Chemically reactive solutions flow across the MHD boundary layer over a permissible stretching sheet with suction or blow have been explored by Bhattacharyya and Layek [19]. Goud and Rajashekar [20] presented an unsteady MHD boundary flow on the semi-infinite vertical with thermal source and suction effects via porous medium outcomes obtained by implementing the finite element method. Some researchers have studied the suction/ injection and other aspects also ref. [21-29].

The current work is about an incompressible, natural convective heat transfer 3-D steady flow past on a stretching sheet involving blowing and suction. Using Boussinesq assumptions, a 3-D similarity result in the Navier-Stokes equations is reported furthermore, the current results are consistent with previously reported outcomes and seem to be a good contract in a defining context Gorla and Sidawi [30]. The governing equations are eased to a couple of non-linear ODEs that are explained using a MATLAB solver.

When similarity solutions are used, the number of free variables is summarized by one or more. This is the most probably used method in fluid mechanics to obtain exact solutions. Ames [31] describes the methods for equations of physical interest in generating similarity transformations. In the field of limiting solutions, asymptotic solutions are used as similarity solutions. Physical insight into these specifics of complex flows of fluid may be obtained by using similarity solutions for complex fluid flows. Most of the characteristics exhibited by these solutions describe the actual problem for the physical, dynamic, and thermal parameters.

$u=b x f^{\prime}(\eta)+A \cos \alpha \cdot \psi(\eta), \quad v=A \sin \alpha \phi(\eta)$,

$w=-\sqrt{b v \varphi}, \theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}$,

$A=\frac{g \beta\left(T_{w}-T_{\infty}\right)}{b}, \eta=\sqrt{\frac{b}{v}}$               (6)

The above Eq. (6) are substituted in (1)-(5) equations, then the following are the resultant equations:

${f}'''={{({f}')}^{2}}-f{f}''$               (7)

${\theta }''+\Pr f{\theta }'=0$                (8)

${\varphi }''+f{\varphi }'+\theta =0$                (9)

${\psi }''+f{\psi }'+\theta =\psi {f}'$                (10)

In Eqns. (7)-(8), prime refers to diff. with resp. to η only. The reduced boundary conditions for flow fields are mentioned by:

$\left.\left.\begin{array}{l}f(\eta)=f_{w}, \\ f^{\prime}(\eta)=1, \\ \theta(\eta)=1, \\ \phi(\eta)=0, \\ \psi(\eta)=0\end{array}\right\}_{\eta =0} ;\quad \begin{array}{r}f^{\prime}(\eta)=0, \\ \theta(\eta)=0, \\ \phi(\eta)=0, \\ \psi(\eta)=0\end{array}\right\}_{\eta \rightarrow \infty}$                     (11)

Here, $f_{w}=\frac{-W}{\sqrt{b v}}$ is the mass transfer parameter changes in a sign for injection and surjection. It is negative for injection and positive for surjection.

The stretching surface undergoes shear stresses and is given by:

${{\tau }_{zx}}={{\left. \mu \frac{\partial u}{\partial z} \right|}_{z=0}}=\rho \sqrt{b\nu }\left[ bx{f}''(0)+A\cos \alpha .{\psi }'(0) \right]$              (12)

${{\tau }_{zx}}={{\left. \mu \frac{\partial v}{\partial z} \right|}_{z=0}}=\rho \sqrt{b\nu }A\sin \alpha .{\varphi }'(0)$                  (13)

In the direction of x and y at any point on the surface, the skin friction factors can be taken as:

$C f x=\frac{2 \tau_{z x}(0)}{\rho(b x)^{2}}=\frac{2}{\sqrt{\operatorname{Re}_{x}^{2}}}\left(-1+\left(\frac{G r_{x}}{\operatorname{Re}_{x}^{2}}\right) \cdot \operatorname{Cos} \alpha \cdot \psi^{\prime}(0)\right)$                   (14)

$C f y=\frac{2 \tau_{z y}(0)}{\rho(b x)^{2}}=\frac{2}{\sqrt{\operatorname{Re}_{x}^{2}}}\left(\frac{G r_{x}}{\operatorname{Re}_{x}^{2}}\right) .\operatorname{Sin} \alpha .\phi^{\prime}(0)$                  (15)

where, ${{\operatorname{Re}}_{x}}=\frac{(bx)x}{\nu },G{{r}_{x}}=\frac{{{x}^{3}}g\beta \left( T{}_{w}-{{T}_{\infty }} \right)}{{{\nu }^{2}}}\,\,\,$. With the help of Fourier’s law, the heat flux may be noted that:

$q_{w}=-K_{f}\left(\frac{\partial T}{\partial z}\right)_{z=0}=-K_{f} \sqrt{\frac{b}{v}}\left(T_{w}-T_{\infty}\right) \theta^{\prime}(0)$.              (16)

At any point on the surface area, the heat transfer coefficient may be noted as:

$h=\frac{{{q}_{w}}}{\left( T{}_{w}-{{T}_{\infty }} \right)}$                    (17)

At any point on the surface Nusselt number be taken as:

$N{{u}_{L}}=\frac{hL}{{{K}_{f}}}==-{\theta }'(0).\sqrt{{{\operatorname{Re}}_{L}}}$                    (18)

In this work, The effects of the surface mass transfer on natural convection flow from a vertical stretching surface. Tables are depicted for the missing values of thermal functions and velocity. Predictions on the rate of heat transfer can be performed using this information. Also, the surface friction factor can be estimated. The temperature field and flow field experience a higher influence by surface mass transfer. Higher entrainment velocity is observed at the surface due to suction. Linearity temperature and velocity profiles are higher with injection.

• The thickness of the thermal boundary layer is increased with an increase of injection values and alternatively reduced by surface suction as well as temperature and velocity profiles decay exponentially.
• With an increase of Pr values the result in heat transfer rate increases.
• With minor Prandtl numbers, the thermal boundary layer’s thickness is found to reduce leading to reduced resistance to heat transfer.