Impacts of various shaped Cu-nanomaterial on the heat transfer over a bilateral stretchable surface: Numerical investigation

Abstract

The heat transport in the nanofluids attained much interest of the researchers and engineers due to broad uses in medical sciences, paint industries, aerodynamics, wheel alignment and manufacturing of aircraft parts. Therefore, keeping in mind the paramount significance of the heat transfer, the study of Cu-nanomaterials based nanofluid is conducted. The governing nanofluid model transformed in dimensionless version via similarity transformations. For numerical simulation of the dynamics of Cu-H₂O, RK technique with shooting algorithm is employed and presented behavior of the fluid motion, temperature, wall shear stresses and local thermal performance rate via graphical aid. It is noted that the heat transfer augmented promptly by increasing $\lambda$ and volumetric fraction of Cu nanomaterial. Further, graphical and tabular comparison is also provided under certain assumptions which authenticate the study.

Introduction

The fundamental fluid mechanics problem in two dimensions due to a stretching slit, firstly explored by Crane. ¹ Applications of such type of problems comprised in paper production, plastic films drawing, and polymer extrusion etc. Due to versatile uses, researcher’s community focused to analyze such type of problems under certain conditions. These are mixed convection, mass transfer, convective condition, variable slit temperature, suction and injection etc. The flow characteristics of Newtonian fluid in three dimensions past a bi-laterally stretchable sheet in three dimensions reported by Wang ² in 1988. In 2014, Sheikholeslami and Ganji ³ reported the variations in the nanofluid characteristics in rotating Cartesian coordinates system. For mathematical investigation of the model, they implemented numerical technique and examined the variations of flow quantities in the flow field. Implementation of two-phase model for nanofluid is also part of his work. In 2010, Rashidi and Keimanesh ⁴ inspected magnetohydrodynamic flow of laminar liquid film which placed horizontally. The sheet is capable to stretching. They adopted analytical technique for mathematical analysis. The effects of Lorentz force on the nanofluid flow explored in Sheikholeslami et al. ⁵ MHD flow of $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ -H₂O examined in Sheikholeslami et al. ⁶ They focused on the flow characteristics due to fluctuating various flow quantities.

A numerical investigation of thermal transport in the boundary layer flow of the nanofluid over a stretchable surface was conducted by Rana and Bhargava. ⁷ They used the model comprising the thermophoretic and Brownian motion effects. The results for the flow regimes against the parameters were drawn by implementing variational finite element technique. In 2015, Das ⁸ presented the analysis of nanofluid dynamics over a permeable surface. To find fascinating results of the study, they incorporated slip effects over the surface and tackled the resultant model numerically. The study was interesting in the context of the topic but, the more novel results could be achieved by ingrained the effects of thermal radiations, Lorentz forces and resistive heating.

The dynamics of non-Newtonian nanofluid under convective flow condition is a paramount research topic in the area of Fluid Dynamics. Keeping in view the significance of such flow, Sandeep et al. ⁹ analyzed the flow characteristics over a stretching surface with suction/blowing properties of the surface. They observed that thermal performance in Oldroyd-B nanofluid is very effective compared to other fluid like Maxwell and Jeffery nanofluids. In order to make the results more effective, they plugged the magnetic field phenomena in the constitutive model. The heat transfer inspection in radiative and dissipative nanofluid over a stretchable geometry by taking the effects of Lorentz forces into account were found in Pal et al. ¹⁰ The influences of suction and injection, Lorentz forces and Grashof number in the dynamics of fluid were core outcomes of the study. Another significant heat transfer investigation in the nanofluid over an inclined stretchable surface is conducted in Thumma et al. ¹¹ The Keller’s box technique is implemented for mathematical analysis and displayed the results against the flow quantities. They validated the study by comparing the results with existing literature.

The effects of Lorentz forces on the dynamics of fluid are significant and broadly applicable in the accomplishment of various industrial products. Due to these applications, a heat transfer investigation in the nanofluid by taking the impacts of Lorentz forces into account over a stretching surface were examined in Ibrahim et al. ¹² They found that the imposed magnetic field resists the fluid motion due to which maximum heat transfer at the surface is observed. Further, they validate the study by making a comparative analysis. The investigation of second law analysis in Jeffery nanofluid under the impacts of nonlinear thermal radiation and activation energy is presented in Hayat et al. ¹³ They observed reverse variations in the temperature and concentration regimes against Deborah number. Further, the escalations in the entropy generation are inspected for stronger Lorentz forces and Bejan number. A 3D heat transport mechanism in Carreau fluid under thermal radiation and heat generation/absorption was determined in Khan et al. ¹⁴ They imposed convective condition on the surface and explored the interesting results for the heat transport.

The nonlinear flow model in the presence of various shaped nanoparticles of copper between nonparallel walls was reported in Khan et al. ¹⁵ They also discussed the model analytically and explored the velocity and thermal fields graphically. Recently, heat transfer in magneto-nanofluid with natural convection discussed in Sheikholeslami et al. ¹⁶ They used Lattice Boltzmann technique for solution purpose. The effects of Ag-nanoparticles on non-Newtonian in the existence of Lorentz forces past a wavering surface studied in Mohd Zin et al. ¹⁷ They considered the appropriate geometry of the vertically placed plate.

In 2015, Gul et al. ¹⁸ examined the effects of various shaped nanoparticles in porous media. Ahmed et al. ¹⁹ examined H₂O flow composed by SWCNT bounded between two Riga plates. They discussed the velocity and thermal profiles graphically. The flow of ferro-fluids over a plate positioned vertically described in Gul et al. ²⁰ They also examined the stimulus of convective boundary condition. The characteristics of rotating nanofluid by encountering the influences of magnetic field and radiative heat flux presented in Sheikholeslami et al. ²¹ Furthermore, they focused on numerical computation of the model. The effects of variable magnetic field and Cu-H₂O cavity examined in Sheikholeslami and Vajravelu ²² and Sheikholeslami, ²³ respectively. The study comprised in Mohyud-Din et al. ²⁴ confined to H₂O flow composed by Cu nanoparticles in stretchable opening/narrowing channel. Further, they examined the impacts of Lorentz force on the flow properties. The flow of magneto-casson fluid past a stretchable sheet by considering of slip parameter discussed in Ullah et al. ²⁵ For numerical and analytical investigation of Newtonian model between two oblique walls described in Adnan et al.^26,27 respectively. Rashidi et al. ²⁸ discussed rotating nanofluid model between porous disks. Further, 3D rotating nanofluid squeezed flow discussed in Khan et al. ²⁹ For the flow of nanofluids in various geometries under certain assumptions and boundary conditions, we can study^30–36 and references comprised therein.

From the inspection of literature, it is pointed that the heat transfer in the nanofluids under various shape effects over a bidirectional stretchable sheet is not discussed so far. Therefore, the analysis is conducted to fill this research opening. The nanofluid model is formulated via similarity transforms and then handled numerically. At the end, the variations in the velocity and temperature of the nanofluids against various plugged parameters are explored and discussed.

Formulation of the model

Considered the nanofluid flow over a plane stretchable sheet which is capable to stretching in opposite directions. The sheet is placed at $z=0.$ The flow over the sheet is due to stretching of the sheet. The velocity in the directions of x and y axes are described by U_w = a/(x + y)⁻ⁿ and V_w = b/(x + y)⁻ⁿ, respectively. Here, invariables are a, b, and n. The variable temperature at the sheet is described by $T_w(x,y)=T_{\infty }+A{(x+y)}^n$ , in which $A$ is constant. Figure 1 demonstrated the flow configuration:

Figure 1.

Flow configuration of the nanofluid.

In the light of above imposed restriction, the specific set of PDE’s representing the flow of nanofluid flow over a bi-directionally stretchable surface are ³⁷ :

$$\frac{\partial \tilde{u}}{\partial x}+\frac{\partial \tilde{v}}{\partial y}+\frac{\partial \tilde{w}}{\partial z}=0,$$

(1)

$$\tilde{u}\frac{\partial \tilde{u}}{\partial x}+\tilde{v}\frac{\partial \tilde{u}}{\partial y}+\tilde{w}\frac{\partial \tilde{u}}{\partial z}=\frac{{\mu }_{\mathrm{nf}}^{*}}{{\rho }_{\mathrm{nf}}^{*}}\left(\frac{{\partial }^2\tilde{u}}{\partial z^2}\right),$$

(2)

$$\tilde{u}\frac{\partial \tilde{v}}{\partial x}+\tilde{v}\frac{\partial \tilde{v}}{\partial y}+\tilde{w}\frac{\partial \tilde{v}}{\partial z}=\frac{{\mu }_{\mathrm{nf}}^{*}}{{\rho }_{\mathrm{nf}}^{*}}\left(\frac{{\partial }^2\tilde{v}}{\partial z^2}\right),$$

(3)

$$\tilde{u}\frac{\partial \tilde{T}}{\partial x}+\tilde{v}\frac{\partial \tilde{T}}{\partial y}+\tilde{w}\frac{\partial \tilde{T}}{\partial z}=\frac{k_{\mathrm{nf}}^{*}}{{{\left(\rho C_p\right)}^{*}}_{\mathrm{nf}}}\left(\frac{{\partial }^2\tilde{T}}{\partial z^2}\right),$$

(4)

Equation (1) presenting the continuity equation in three dimensions which gratify identically. The set of equations (2)–(4) described momentum and energy equations, respectively. Furthermore, thermal conductivity is $k_{\mathrm{nf}}^{*}$ , dynamic viscosity and density of the nanofluid denoted by ${\mu }_{\mathrm{nf}}^{*},\mathrm{and}{\rho }_{\mathrm{nf}}^{*}$ , respectively. The following theoretical models ¹⁸ are used for thermo-physical characteristics of nanofluid:

$$\begin{array}{c}\multicolumn{1}{c}{{\mu }_{\mathrm{nf}}^{*}={\mu }_f^{*}(1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2),}\\ \multicolumn{1}{c}{k_{\mathrm{nf}}^{*}=k_f^{*}\left[\frac{k_s^{*}+(n-1)k_f^{*}+(n-1)(k_s^{*}-k_f^{*}){\varphi }^{*}}{k_s^{*}+(n-1)k_f^{*}-(k_s^{*}-k_f^{*}){\varphi }^{*}}\right],}\\ \multicolumn{1}{c}{{\rho }_{\mathrm{nf}}^{*}=\frac{{\rho }_f^{*}}{{(1-{\varphi }^{*})}^{-1}}+{\varphi }^{*}{\rho }_s^{*},}\\ \multicolumn{1}{c}{{{\left(\rho C_p\right)}^{*}}_{\mathrm{nf}}=[(1-{\varphi }^{*})+\frac{{\varphi }^{*}{{\left(\rho C_p\right)}^{*}}_s}{{{\left(\rho C_p\right)}^{*}}_f}]{{\left(\rho C_p\right)}^{*}}_f,}\end{array}$$

In the above models, $a^{*}\mathrm{and}{b}^{*}$ are constants. The empirical shape factor parameter is denoted by $n$ , ${\varphi }^{*}$ denotes the volumetric fraction of the nanofluid, densities of carrier fluid and nanofluid are ${\rho }_f^{*}$ and ${\rho }_s^{*}$ , respectively. The empirical shapes factor of the nanoparticles, Sphericity, thermal and physical properties of the nanofluid comprised in. ¹⁸

The dimensional boundary conditions for the specific model are as under ³⁷ :

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{At}z=0:}\\ \multicolumn{1}{c}{\begin{array}{c}\tilde{u}=U_w=\frac{a}{{(x+y)}^{-n}}\hfill \\ \tilde{v}=V_w=\frac{b}{{(x+y)}^{-n}}\hfill \\ w=0\hfill \\ \tilde{T}=T_w=T_{\infty }+\frac{A}{{(x+y)}^{-n}}\hfill \end{array}\},}\\ \multicolumn{1}{c}{\mathrm{At}z\to \infty :}\\ \multicolumn{1}{c}{\begin{array}{c}\tilde{u}\to 0\hfill \\ \tilde{v}\to 0\hfill \\ \tilde{T}\to T_{\infty }\hfill \end{array}\},}\end{array}$$

The similarity variables are ³⁷ :

$$\begin{array}{c}\tilde{u}=\frac{F^{\prime }\left(\eta \right)a}{{(x+y)}^{-n}}\hfill \\ \tilde{v}=\frac{G^{\prime \left(\eta \right)}a}{{(x+y)}^{-1}}\hfill \\ \tilde{w}=-\sqrt{a{\nu }_f}[\frac{\left(\frac{n+1}{2}\right)(F\left(\eta \right)+G\left(\eta \right))}{{(x+y)}^{\frac{1-n}{2}}}+\frac{\left(\frac{n-1}{2}\right)\eta (F^{\prime }\left(\eta \right)+G^{\prime }\left(\eta \right))}{{(x+y)}^{\frac{1-n}{2}}}]\hfill \\ \beta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}\hfill \\ \eta =\sqrt{\frac{a}{{\nu }_f}}z{(x+y)}^{\frac{n-1}{2}}\hfill \end{array}\},$$

By entreating the suitable partial derivatives and above defined self-similar transformations in equations (1)–(4), the following is attained:

$$\begin{array}{c}\multicolumn{1}{c}{F″\prime -\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{\rho }_s^{*}}{{\rho }_f^{*}}\right]}{(1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2)}}\\ (n{F}^{\prime }(F^{\prime }+G^{\prime })+0.5(n+1)F″(F+G))0,\hfill \end{array}$$

(5)

$$\begin{array}{c}\multicolumn{1}{c}{G″\prime +\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{\rho }_s^{*}}{{\rho }_f^{*}}\right]}{(1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2)}}\\ (0.5(n+1)G″(F+G)-\mathrm{nG}\prime (F^{\prime }+G\prime ))=0,\hfill \end{array}$$

(6)

$$\begin{array}{c}\beta ″+\mathrm{Pr}\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{{\left(\rho C_p\right)}^{*}}_s}{{{\left(\rho C_p\right)}^{*}}_f}\right]}{\left[\frac{k_s^{*}+(n-1)k_f^{*}+(n-1)(k_s^{*}-k_f^{*}){\varphi }^{*}}{k_s^{*}+(n-1)k_f^{*}-(k_s^{*}-k_f^{*}){\varphi }^{*}}\right]}\hfill \\ (0.5(n+1){\beta }^{\prime }(F+G)-n\beta (F^{\prime }+G^{\prime }))=0,\hfill \end{array}$$

(7)

The supporting self-similar boundary conditions are:

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}\multicolumn{1}{c}{F\left(\eta \right)=G\left(\eta \right)=0}\\ \multicolumn{1}{c}{F^{\prime }\left(\eta \right)=1}\\ \multicolumn{1}{c}{G^{\prime }\left(\eta \right)=\lambda }\\ \multicolumn{1}{c}{\beta \left(\eta \right)=1}\end{array}\}\mathrm{at}\eta =0,}\\ \multicolumn{1}{c}{\begin{array}{c}\multicolumn{1}{c}{F^{\prime }\left(\eta \right)\to 0}\\ \multicolumn{1}{c}{\beta \left(\eta \right)\to 0}\\ \multicolumn{1}{c}{G\prime (\eta )\to 0}\end{array}\}\mathrm{at}\eta \to \infty ,}\end{array}$$

Here, $\mathrm{Pr}=\frac{{\mu }_f^{*}{{\left(c_p\right)}^{*}}_f}{k_f^{*}}$ .

Further, dimensional formulas for shear stresses and local Nusselt number are as follows:

$$C_{\mathrm{Fx}}=\frac{{\tau }_{\mathrm{zx}}}{{\rho }_{\mathrm{nf}}{U}_w^2},C_{\mathrm{Fy}}=\frac{{\tau }_{\mathrm{zy}}}{{\rho }_{\mathrm{nf}}{V}_w^2}\mathrm{and}N{u}_x=\frac{q_w(x+y)}{k(T_w-T_{\infty })},$$

Where, ${\tau }_{\mathrm{zx}},{\tau }_{\mathrm{zy}}$ and radiative heat flux $q_w$ described as under:

$$\begin{array}{c}{\tau }_{\mathrm{zx}}={\mu }_{\mathrm{nf}}(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}){\downarrow }_{z=0},\hfill \\ {\tau }_{\mathrm{zy}}={\mu }_{\mathrm{nf}}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}){\downarrow }_{z=0}\mathrm{and}{q}_w=-k_{\mathrm{nf}}\left(\frac{\partial T}{\partial z}\right){\downarrow }_{z=0},\hfill \end{array}$$

After simplification, transformed version is given as:

$$\begin{array}{c}\multicolumn{1}{c}{C_{\mathrm{Fx}}\sqrt{R{e}_x}=\frac{{\stackrel{⌣}{A}}_2}{{\stackrel{⌣}{A}}_1}F″(\eta ){\downarrow }_{\eta =0},}\\ \multicolumn{1}{c}{C_{\mathrm{Fy}}\sqrt{R{e}_y}{\lambda }^{\frac{3}{2}}=\frac{{\stackrel{⌣}{A}}_2}{{\stackrel{⌣}{A}}_1}G″(\eta ){\downarrow }_{\eta =0}\mathrm{and}}\\ \multicolumn{1}{c}{N{u}_x{\left(R{e}_x\right)}^{-\frac{1}{2}}=-\stackrel{\beta }{{\stackrel{⌣}{A}}_4}\prime (\eta ){\downarrow }_{\eta =0}}\\ \multicolumn{1}{c}{{\stackrel{⌣}{A}}_1=\frac{1}{{(1-{\varphi }^{*})}^{-1}}+{\varphi }^{*}\left[\frac{{\rho }_s^{*}}{{\rho }_f^{*}}\right],{\stackrel{⌣}{A}}_2=1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2,}\end{array}$$

• $\stackrel{=}{{\stackrel{⌣}{A}}_4}\left[\frac{k_s^{*}+{\mathrm{gk}}_f^{*}+g(k_s^{*}-k_f^{*}){\varphi }^{*}}{k_s^{*}+{\mathrm{gk}}_f^{*}-(k_s^{*}-k_f^{*}){\varphi }^{*}}\right]$ , where, $g=n-1$ .

Numerical analysis of the model

It is a paramount fact that the coupled nonlinear mathematical models fail to tackle in the form of closed solution. Therefore, numerical solutions are credible for such nature of the flow models. The formulated model is highly nonlinear and coupled in nature so, closed form solution does not hold. Thus, the model is then treated numerically with RK technique ³⁷ with coupling of shooting technique. In this scheme, firstly the nonlinear mathematical model transformed into a system of first order initial value problem and then solved using the algorithm. Most importantly, ${\eta }_{\infty }$ is chosen in such a way that the velocity and thermal fields satisfy the asymptotic behavior far from the surface. In order to obtain the desired first order model, the following transformation are made for the velocity and thermal equations:

$$\begin{array}{c}{\stackrel{⌣}{s}}_1=F,{\stackrel{⌣}{s}}_2=F^{\prime },{\stackrel{⌣}{s}}_3=F″,{\stackrel{⌣}{s}}_4=G,{\stackrel{⌣}{s}}_5=G^{\prime },{\stackrel{⌣}{s}}_6=G″,\hfill \\ {\stackrel{⌣}{s}}_7=\beta \mathrm{and}{\stackrel{⌣}{s}}_8=\beta \prime .\hfill \end{array}$$

In the light of these substitutions, the nonlinear coupled model transformed into the following version:

$$\left[\begin{array}{c}\stackrel{⌣}{s}{\prime }_1\hfill \\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_2}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_3}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_4}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_5}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_6}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_7}\\ \multicolumn{1}{c}{\stackrel{⌣}{s}{\prime }_8}\end{array}\right]=\left[\begin{array}{c}{\stackrel{⌣}{s}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_3}\\ \multicolumn{1}{c}{\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{\rho }_s^{*}}{{\rho }_f^{*}}\right]}{(1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2)}(\frac{n+1}{2}({\stackrel{⌣}{s}}_1+{\stackrel{⌣}{s}}_4){\stackrel{⌣}{s}}_3+n({\stackrel{⌣}{s}}_2+{\stackrel{⌣}{s}}_5){\stackrel{⌣}{s}}_2)}\\ \multicolumn{1}{c}{s_5}\\ \multicolumn{1}{c}{s_6}\\ \multicolumn{1}{c}{-\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{\rho }_s^{*}}{{\rho }_f^{*}}\right]}{(1+a^{*}{\varphi }^{*}+b^{*}{{\varphi }^{*}}^2)}(\frac{n+1}{2}({\stackrel{⌣}{s}}_1+{\stackrel{⌣}{s}}_4){\stackrel{⌣}{s}}_6+n({\stackrel{⌣}{s}}_2+{\stackrel{⌣}{s}}_5){\stackrel{⌣}{s}}_5)}\\ \multicolumn{1}{c}{s_8}\\ \multicolumn{1}{c}{-\frac{(1-{\varphi }^{*})+{\varphi }^{*}\left[\frac{{{\left(\rho C_p\right)}^{*}}_s}{{{\left(\rho C_p\right)}^{*}}_f}\right]}{\left[\frac{k_s^{*}+(n-1)k_f^{*}+(n-1)(k_s^{*}-k_f^{*}){\varphi }^{*}}{k_s^{*}+(n-1)k_f^{*}-(k_s^{*}-k_f^{*}){\varphi }^{*}}\right]}\mathrm{Pr}(\frac{n+1}{2}({\stackrel{⌣}{s}}_1+{\stackrel{⌣}{s}}_4){\stackrel{⌣}{s}}_8-n{s}_2({\stackrel{⌣}{s}}_2+{\stackrel{⌣}{s}}_5){\stackrel{⌣}{s}}_7}\end{array}\right],$$

The appropriate initial conditions are:

$$\left[\begin{array}{c}{\stackrel{⌣}{s}}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_2}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_3}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_4}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_5}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_6}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_7}\\ \multicolumn{1}{c}{{\stackrel{⌣}{s}}_8}\end{array}\right]=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{\stackrel{⌣}{g_1}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\lambda }\\ \multicolumn{1}{c}{\stackrel{⌣}{g_2}}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{\stackrel{⌣}{g_3}}\end{array}\right]$$

Where, $\stackrel{⌣}{g_1}=F″\left(0\right),\stackrel{⌣}{g_2}=G″\left(0\right)$ and $\stackrel{⌣}{g_3}=\beta \prime (0)$ . After this, integration is carried out and Mathematica 10.0 is used for further computation of the results.

Results and discussion

This section deals with the graphical results and their comprehensive discussion. The results are presented against various flow parameters over the domain of interest. these results are displayed in Figures 2 to 10. The results are displayed for the velocities, temperature, shear stresses and local thermal performance of the nanofluids against the pertinent flow parameters.

Figure 2.

Effects of $\lambda$ on $F\prime (\eta )$ .

Figure 3.

Effects of $\lambda$ on $G\prime (\eta )$ .

Figure 4.

Effects of $\lambda$ on $\beta (\eta )$ .

Figure 5.

Effects of $\lambda$ on the shear stresses for (a) $n=1$ and (b) $n=2$ in $x$ -direction.

Figure 6.

Effects of $\varphi$ on the shear stresses for (a) $n=1$ and (b) $n=2$ in $x$ -direction.

Figure 7.

Effects of $\lambda$ on the shear stresses (a) $n=1$ and (b) $n=2$ in $y$ -direction.

Figure 8.

Effects of $\varphi$ on the shear stresses (a) $n=1$ and (b) $n=2$ in $y$ -direction.

Figure 9.

Effects of $\lambda$ on the heat transport rate for (a) $n=1$ and (b) $n=2$ .

Figure 10.

Effects of $\varphi$ on the heat transport rate for (a) $n=1$ and (b) $n=2$ .

Velocity and thermal distribution

The trends in horizontal velocity component $F\prime (\eta )$ against arising values of the parameter $\lambda$ are displayed in Figure 2. The velocity distribution is sketched for platelets, blades, cylinders, and bricks shaped based nanofluid. It is examined that the fluid velocity in the horizontal direction drops for higher values of $\lambda$ . The motion of the nanofluid comprising platelets nanomaterial is slow in comparison with rest of the nanofluids. The physical reason behind these trends is the thermophysical characteristics of the nanomaterial. The effective density of platelets nanomaterial is less compared to rest of the nanomaterials. Due to low dense material, the internal molecular forces become weaker which allows the fluid to move freely. On the other side, rest of the nanofluids become more denser due to high density of bricks, blades and cylinders nanomaterial. Ultimately internal fluid forces become dominant which opposes the fluid motion.

The vertical fluid motion against $\lambda$ for various nanofluids is examined in Figure 3. It is observed that the vertical velocity $G\prime (\eta )$ augmented for increasing values of $\lambda$ . The velocity rises slowly for the nanofluids composed by blades and bricks nanomaterial. The physical reason behind these trends is the effective density. Due to higher density of these materials, the resultant fluid becomes more denser which leads to slow increment in the vertical component of the velocity. Near the stretchable surface abrupt fluid motion is examined and its gradually slow down away from the it. The velocity of the nanofluids asymptotically vanishes at ambient level of the surface.

Thermal behavior of the nanofluids for the multiple ranges of the parameter $\lambda$ are portrayed in Figure 4. It is obvious that $\lambda$ opposes the fluid temperature under the particular scenario. The temperature promptly declines in the region $0.2<\eta <0.8$ . Beyond the region, fluid temperature asymptotically vanishes and obeys the ambient temperature condition. The thermal boundary layer region strated beyond $\eta >1.2$ . A slow decrement in the temperature is observed for the nanofluid composed by blades nanomaterial in which thermophysical characteristics playing dominant role especially thermal conductivity.

Trends in shear stresses

The trends in stresses along horizontal and vertical directions due to varying flow parameters are displayed in Figures 5 to 8, respectively. The stresses along horizontal direction against $\varphi$ versus $\lambda$ are depicted in Figure 5. It is examined that the shear stresses drop inconsequently as the volumetric fraction tends to zero and abruptly drops for higher volumetric fraction. The physical reason for negligible decrement in the shear stresses for smaller volumetric fraction is that the fluid under goes to its conventional liquid state for zero $\varphi$ due to which the decrement is negligible. The more decreasing trends in the shear stresses are examined for stronger volumetric fraction of the nanomaterials. These trends are elucidated in Figure 5(a) and (b), respectively.

Figure 6 portraying the trends in shear stresses against $\varphi$ versus $\lambda$ . The volumetric fraction of the nanomaterials varies curve wise while $\lambda$ varies horizontally. The shear stresses promptly decline for higher $\lambda$ . These trends controlled by increasing the index parameter $n$ in the flow model. In both the cases, abrupt decrement is examined for the nanofluid composed by platelets shaped nanomaterial in which the effective density playing its significant role.

The behavior of vertical shear stresses by increasing the values of $\lambda$ and volumetirc fraction of the nanomaterial is presented in Figures 7 and 8, respectively. From Figure 7, it is observed that the vertical stresses drops very rapidly by increasing the parameter $\lambda$ and this decrement becomes almost similar as the volumetric fraction tends to zero. For the nanofluid containing bricks shaped nanomaterial, the vertical trends in the shear stresses becomes slow to rest of the nanofluids. Further, it is obvious that by increasing index parameter value, the vertical stresses declines abruptly. In Figure 8, the higher volumetric fraction causes the decrement in the shear stresses but it is slow compared to the trends displayed in Figure 7. For higher $\lambda$ and $\varphi$ , prompt decrement is examined for vertical shear stresses in Figure 8.

Trends in local thermal performance

Figures 9 and 10 elucidating the behavior of local thermal performance in the nanofluids under varying $\lambda$ and the volumetric fraction $\varphi$ , respectively. Form Figures 9 and 10, it is examined that the local thermal performance rate escalated for higher $\lambda$ and $\varphi$ . However, slow increment is observed as the volume fraction and $\lambda$ tends to zero. The maximum rate of thermal transport is investigated for the nanofluid suspended by blades shaped nanomaterial.

Authentication of the study

The under consideration nanofluid model is modified version of Khan et al. ³⁷ In order to verify the presented results with existing science literature under the same boundary conditions and the same flow parameters, we need to impose some restriction ( $\varphi =0$ ) on our model mentioned in Figure 11(a) and (b), respectively. From Figure 11(a) and (b), it is evident that the presented study has an excellent agreement with the previous literature Khan et al. ³⁷ which validate the study.

Figure 11.

Comparison of present study with previous work Khan et al. ³⁷ for (a) F′(η) and (b) β(η)

Moreover, a comparative analysis for $F″(0)$ and $G″(0)$ is provided for zero volumetric fraction of the nanomaterial. The results are computed and given in Table 1. From the table, the present results are in very good agreement with existing science literature by taking ${\eta }_{\infty }=15$ . This proves the validation of the analysis.

Table 1.

Authentication of the study for $F″(0)$ and $G″(0)$ , $\varphi =0$ .

$n$	$\lambda$	Present		Khan et al. 37
		$F″(0)$	$G″(0)$	$F″(0)$	$G″(0)$
$1.0$	$0.0$	$-1.0$	$0$	$-1.0$	$0$
	$0.5$	$-1.22474$	$-0.612372$	$-1.224745$	$-0.612372$
	$1.0$	$-1.41421$	$-1.41421$	$-1.414214$	$-1.414214$

Conclusions

The thermal transport mechanism is examined under various shaped nanomaterial based nanofluids. The colloidal model is treated numerically and displayed the results against the flow quantities over the feasible region. It is observed that the vertical motion of the nanofluids significantly escalated by increasing $\lambda$ . The temperature of the nanofluids containing bricks, blades, cylinders and platelets shaped nanomaterial drop for higher $\lambda$ . Further, the shear stresses along horizontal direction declines for higher volumetric fraction and it could be controlled by increasing the value of index parameter $n$ . The local thermal performance of the nanofluids improved against higher $\varphi$ and significant contribution of blades nanomaterial is examined regarding thermal rate analysis. Finally, the comparative investigation is provided which showing good agreement with existing literature.