Influence of melting heat transfer and chemical reaction on the flow of non-Newtonian nanofluid with Brownian motion: Advancement in mechanical engineering

Abstract

The aim of the present study is to investigate the melting heat and mass transport characteristics on the stagnation point flow of Powell–Eyring nanofluid over a stretchable surface because melting is so important in many processes, such as Permafrost melting, magma solidification, and thawing of frozen grounds, are all examples of soil melting and freezing around the heat exchanger coils of a ground-based pump. The developing mathematical model under the boundary layer flow in terms of differential equations is solved through a numerical algorithm using a boundary value problem solver bvp4c/shooting technique with the help of MATLAB software. The impact of emerging parameters on the velocity profile, temperature profile, and concentration profile is elaborated graphically. The profile and boundary-layer width rate for the value stretching parameter less than one rises when A enhances while the thickness of boundary layer velocity profile for the value stretching parameter greater than one decreases as A. The velocity function shows a decrement response for M, while the opposite behavior is seen against the concentration field. Furthermore, the numeric data for the friction factor and Nusselt number are demonstrated in tabular form, and the result shows a remarkable agreement with the previously published data.

Introduction

Heat transfer over a stretchable surface is an important phenomenon because it has a wide range of applications in engineering and industrial operations, including the extrusion of polymer sheets and cooling metallic plates and glass fibers. Crane ¹ was the pioneer who investigated two-dimensional flow over a stretchable surface in a still fluid, and since then, much literature has been produced on the subject.^2,3 Khan et al. ⁴ investigated the influence of heat transfer on second-grade fluid due to a porous stretched sheet. The impact of thermal radiation and Brownian motion over second-grade fluid through a stretched sheet with mass and heat transfer was discovered by Khan et al.,^5–7 Safdar et al., ⁸ Jawad et al., ⁹ Safdar et al., ¹⁰ and Jawad et al. ¹¹

Nanofluids have recently attracted much interest from researchers due to their improved thermal characteristics. Nanofluids have been offered to enhance the efficacy of heat transfer liquids that are currently unavailable, such as water, toluene, oil, and an ethylene–glycol combination. Choi ¹² may have coined the word “nanofluid” to describe a combination of nanoparticles and a base fluid. The literature on nanofluid flow, in many respects, is rather rich. Buongiorno ¹³ developed a mathematical model for convective nanofluid flow, including Brownian diffusion and thermophoresis effects. The importance of thermophoresis and Brownian motion effects in nanoparticle/base-fluid slide processes cannot be overstated. Kuznetsov and Nield ¹⁴ proposed an analytical solution for the natural convective boundary layer flow of a fluid across a vertical plate in the presence of nanoparticles. They utilized the influence of thermophoresis and Brownian motion to combine the nanofluid model. Khan and Pop ¹⁵ executed the nanoparticles’ effects on boundary layer flow and heat transfer past a linearly stretching surface. In the presence of nanoparticles, Nadeem et al. ¹⁶ investigated Jeffrey's liquid flow model across a stretched surface. Under the impact of a magnetic field and an internal heat source/sink, Ramesh et al. ¹⁷ expanded on the same experiment. The influence of particle concentration on the dusty fluid flow model via a porous surface was investigated by Gireesha et al. ¹⁸ Ibrahim et al. ¹⁹ examined magnetohydrodynamic (MHD) stagnation point flow and heat transfer of nanoliquid over a stretchable surface. Madhu and Kishan ²⁰ studied the impact of nanoparticles in a Power-law non-Newtonian fluid on a stretched sheet with heat source/sink and thermal radiation. Kumari and Rama Subba ²¹ presented a theoretical analysis for warm nanoliquid flow to a horizontal melting plate.

In the advanced technology process, melting and solidification phenomena are critical. The melting phenomena of solid–liquid phase shift have many applications, including welding, crystal formation, thermal protection, heat transfer, permafrost melting, and semiconductor material preparation. In the beginning, Robert ²² reported the melting of ice slabs in a hot air stream. Hayat et al. ²³ examined Maxwell's liquid flow of stagnation point over a stretchable surface with the melting phenomena. The melting process in MHD fluid flow toward a movable surface with thermal radiation was explored by Das. ²⁴ Epstein and Cho ²⁵ conducted a numerical simulation of melting heat transfer in a stationary plate. Kazmierczak et al. ²⁶ explored the impact of melting heat transfer and looked at mixed convective flow towards a porous surface. The melting heat transfer in a mixed convective flow across a vertical plate was predicted by Gorla et al. ²⁷ Bachok et al. ²⁸ considered the flow and heat transfer across a melting surface in a two-dimensional stagnation point. Melting flow and heat transfer across a moving surface were studied by Ishak et al. ²⁹ The influence of melting heat transfer on MHD stagnation point flow of a nanofluid with a heat source/sink was examined by Gireesha et al. ³⁰ Animasaun et al. ³¹ studied the mobility of an electrically conducting fluid when melting heat transfer and stratified concentration were coupled. References^32–38 include some of the research that is relevant to this. Non-Newtonian nanofluid flow over a stretchable surface fascinated many scientists due to its widest applications in manufacturing since there are countless uses. It uses in industrial sectors such as hot rolling, extrusion, glass-fiber production, melting spinning, manufacturing rubber sheets, plastic pieces, and colling a massive plate of metal in a mist wash, which can be used as an electrolyte, etc. Polymer slips and strings are made in industries by constant extrusion of the polymer and wind-up roller, which stands placed at a determinate distance away. Further research shows that the widening surface rate is nearly comparable to the reserve of the hole by Vleggaar. ² Powell and Eyring ³⁹ advised in detail the classical fluid of Eyring–Powell. That one retains, so several benefits concluded the further non-Newtonian fluid simulations by way of simplicity, the prosperity of calculation with the physical strength and it remains realized from the kinetic theory of fluid in its place of the observed relative, so it is contracted to Newtonian performance for both share and special rates. Malik et al. ⁴⁰ observed the flow of Powell–Eyring fluid because of the stretched tube through adjustable viscosity beneath certain margin sheet situations. Hayat et al. ⁴¹ considered Powell–Eyring liquid flow with conductive border circumstances on a shrinking surface. Some communications of an advanced model show the literature must offer the Powell–Eyring fluid. Islam et al. ⁴² discussed the resulting solution of a homotopy perturbation for slider manners lubricants abstaining Powell–Eyring fluids. Makinde et al. ⁴³ discussed the dynamics of Casson fluid subject to Lorentz force applied on stratified melting space.

Motivated by these facts, our main goal is to explore the effect of the nanoparticles on the MHD stagnation point flow of Eyring–Powell fluid with mass and heat transfer. The melting and Brownian effects are considered in the present study. A suitable renovation is employed to convert the nonlinear partial differential equations into simple ordinary differential equations (ODEs), which are then solved numerically. The key parameters involved in the current problem are thermophoresis number Nt, elastic parameter $\beta$ , Lewis number Le, Brownian number Nb, Prandtl number Pr, and M indicates magnetic field parameter, and their influences are discussed pictorially. The calculated values of skin friction and Nusselt number are displayed in Table 2 for some convergence parameters. To validate the enactment of the current study, results are obtained by using bvp4c and ND solver approach, which shows superb accuracy with those obtained by Hayat et al. ²³ and Hiemenz ⁴⁴ can be perceived in Table 1.

Table 1.

Calculation of skin friction coefficient $f^{\mathrm{″}}$ (0) and Nusselt number $-{\theta }^{\mathrm{\prime }}(0)$ for N, Pr, ε, and A when $\delta$  = 0.5.

				Bvp4c		ND solver
N	Pr	ε	A	$(1+\epsilon )f^{\mathrm{″}}(0)-\frac{\epsilon }{3}\delta f^{\mathrm{\prime }{\mathrm{\prime }}^3}(0)$	$-{\theta }^{\mathrm{\prime }}$ (0)
0.0	1.0	0.5		1.1162	0.7088	1.11622	0.70889
0.3				0.7750	0.6910	0.77502	0.69101
0.5				0.5490	0.6953	0.54903	0.69532
0.8				0.2214	0.7116	0.22146	0.71164
	0.8			0.5552	0.7540	0.55521	0.75405
	1.0			0.5652	0.8629	0.56524	0.86296
	1.2			0.5729	0.9630	0.57292	0.96307
		0.0		0.4400	0.8444	0.44003	0.84442
		0.5		0.5652	0.8629	0.56521	0.86293
		1.0		0.6749	0.8766	0.67491	0.87664
		1.2		0.7157	0.8813	0.71572	0.88135
			0.0	0.7037	0.7496	0.70373	0.74961
			0.2	0.5652	0.8629	0.56522	0.86292
			0.5	0.2750	1.0463	0.27501	1.04633

Table 2.

Comparison of skin friction coefficient $f^{\mathrm{″}}(0)$ when Pr = 0.7, Nb = 0.1, while the other parameters are considered to be zero.

A	Mahapatra and Gupta 50	Hayat et al. 23	Bvp4c	ND solver
0.10	−0.9694	−0.96802	−0.969385598777646	−0.969385598777642
0.20	−0.9181	−0.91692	−0.918106431760606	−0.918106431760601
0.50	−0.6673	−0.66722	−0.667263520517571	−0.667263520517565
2.00	2.0175	2.0175	2.017502801675610	2.017502801675607
3.00	4.7293	4.7291	4.729282356533137	4.729282356533134

Mathematical modeling and problem description

Let us consider the two-dimensional, incompressible stagnation point of Eyring–Powell nanofluid flow with melting heat transport against a stretched surface at $y=0$ , which can be seen in Figure 1. Here $(u,v)$ are the velocity components of fluids flow along $(x,y)$ directions. Fluid is embedded in a porous medium with chemical reaction and Brownian motion. The governing equations of temperature and concentration are defined as follows ⁴⁵ :

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0}}$$

(1)

$$u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}=\left(\nu +{\displaystyle \frac{1}{\rho Bc}}\right){\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}-{\displaystyle \frac{1}{2\rho B{C}^3}{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+u_e{\displaystyle \frac{\mathrm{\partial }{u}_e}{\mathrm{\partial }x}-{\displaystyle \frac{\sigma B_O^2}{\rho }(u-u_e)}}}}}}}$$

(2)

$$\begin{array}{rl}u{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }x}}& +v{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}=\alpha \left({\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}}\right)}\\ & +\tau \left\{D_B\left({\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }y}{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}}}\right)+\left({\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}}\right)\left[{\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }X}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}}\right)}^2\right]\right\}\\ & +4\mu \left(1+{\displaystyle \frac{1}{\mu bc}}\right){\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}\right)}^2\end{array}$$

(3)

$$\begin{array}{rl}u{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}+v{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }y}=}}& D_B\left({\displaystyle \frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{y}^2}}\right)+\left({\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}}\right)\left({\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}}\right)\\ & -k(c-c_{\mathrm{\infty }})\end{array}$$

(4)

Figure 1.

Geometry of the flow problem.

Subjected to the following boundary conditions ⁴⁶ :

$$u=u_w(x)=cx,T=T_m,c=cw,k{\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}}\right)}_{y=0}=\rho [\lambda +c_s(T_m-T_0)]v(x,0),\mathrm{at}y=0$$

(5)

$$u=u_e(x)=ax,C=C_{\mathrm{\infty }},T=T_{\mathrm{\infty }},\mathrm{as}y\to \mathrm{\infty }$$

(6)

Here the velocity component along the x-direction is u and along the y-direction is v, $\rho$ is the density of the fluid, T is fluid temperature, ν is the kinematic viscosity, k is the thermal conductivity, $c_p$ is the specific heat, $\lambda$ is the latent heat, $c_s$ is the heat capacity, and $T_0$ is the melting surface temperature.

Similarity approach

By applying similarity transformations, ⁴⁷ we have

$$u=cx{f}^{\mathrm{\prime }}(\eta ),v=-\sqrt{cv}f(\eta ),\eta =y\sqrt{{\displaystyle \frac{c}{v}}},\theta (\eta )={\displaystyle \frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}},\varphi (\eta )={\displaystyle \frac{C-C_{\mathrm{\infty }}}{C_w-C_{\mathrm{\infty }}}}}$$

(6)

Using equation (6), equations (1) to (3) are reduced in the following form:

$$(1+ϵ)f^{\mathrm{″}\mathrm{\prime }}+f{f}^{\mathrm{″}}-f^{{\mathrm{\prime }}^2}-\in \delta f^{\mathrm{\prime }{\mathrm{\prime }}^2}{f}^{\mathrm{″}\mathrm{\prime }}+A^2-M{f}^{\mathrm{\prime }}=0$$

(7)

$${\displaystyle \frac{1}{Pr}{\theta }^{\mathrm{″}}+f{\theta }^{\mathrm{\prime }}+Nb{\phi }^{\mathrm{\prime }}{\theta }^{\mathrm{\prime }}+Nt{\theta }^{{\mathrm{\prime }}^2}+\delta f^{{\mathrm{\prime }}^2}=0}$$

(8)

$${\phi }^{\mathrm{″}}+Lef{\phi }^{\mathrm{\prime }}+{\displaystyle \frac{Nt}{Nb}{\theta }^{\mathrm{″}}-k_0\phi =0}$$

(9)

The corresponding equations (4) and (5) are

$$f^{\mathrm{\prime }}(0)=1,\theta (0)=0,\phi (0)=1,Prf(0)+N{\theta }^{\mathrm{\prime }}(\eta )=0$$

(10)

$$f^{\mathrm{\prime }}(\mathrm{\infty })=A,\theta (\mathrm{\infty })=1,\phi (\mathrm{\infty })=0$$

(11)

Here $Pr=\upsilon /\alpha$ is the Prandtl number, $A=a/c$ is the ratio of free stream velocity to stretching velocity, N is the non-dimensional melting parameter, given as $N=c_{f(T_{\mathrm{\infty }}-T_m)}/\lambda +c_s(T_m-T_{\mathrm{\infty }})$ , where $c_{f(T_{\mathrm{\infty }}-T_m)}/\lambda$ is the Stefan number and $c_s(T_m-T_0)/\lambda ,$ is the solid phase for the given liquid. Here the main point is that if we take N = 0, then equation (9) takes the classical equation form, which was initially improved by Hayat et al. ²³

The skin fraction $C_f$ along with the Local Nusselt number $N{u}_x$ is given as follows^48,49:

$$C_f={\displaystyle \frac{{\tau }_w}{\rho u_w^2},N{u}_x={\displaystyle \frac{x{q}_w}{k(T_w-T_{\mathrm{\infty }})},Sh={\displaystyle \frac{x{q}_m}{k(C_w-C_{\mathrm{\infty }})}}}}$$

(12)

$$C_{f}Re{x}^{1/2}=(1+\epsilon )f^{\mathrm{″}}(0)-{\displaystyle \frac{\epsilon }{3}\delta f^{\mathrm{\prime }{\mathrm{\prime }}^3}(0),N{u}_{x}Re{x}^{-1/2}=-{\theta }^{\mathrm{\prime }}(0)}$$

(13)

Methodology

The analytic solution of the system of equations with corresponding boundary conditions (8) to (12) cannot be found because they are highly nonlinear and coupled. So we use a numerical technique, that is, a shooting technique in MATLAB using the built-in command bvp4c. To solve the system of ODEs (8) to (10) with boundary conditions (11) and (12) using the shooting method, we have to first convert these systems of equations into a system of first-order differential equations. Numerical calculations are achieved by using the MATLAB algorithm in the above ODEs, which can also be seen through the flowchart (see Figure 2) as follows.

Figure 2.

Illustration of the numerical scheme.

Graphical illustration and discussion

In this segment, equations (3) to (11) depict the effects of different physical factors on velocity, temperature, and concentration profiles graphically. $Pr=1$ ; $ϵ=0.1$ ; $e=0.1$ ; $\delta =0.1$ ; $A=0.01$ ; $N=0.1$ ; $Nb=0.1$ ; $Nt=0.3$ ; $Ec=0.1$ ; $Le=0.5$ ; $Ko=4$ ; $M=0.1$ ; Figure 3 illustrates the sketch of the velocity field $f^{\mathrm{\prime }}(\eta )$ and temperature profile $\theta (\eta )$ versus the melting parameter $N$ . From the figure, it is concluded that the velocity field $f^{\mathrm{\prime }}(\eta )$ varies directly against $N$ the melting parameter which improves boundary layer thickness. Hence heat increases, then the difference between the temperature and melting surface and ambient also increases, which improves the velocity field. Similarly, the behavior of temperature profile $\theta (\eta )$ for some values of N. The graph shows decreasing performance for N, because the temperature difference between the melting surface and the ambient surface increases as a consequent reduction occurs in fluid temperature also in thermal boundary layer thickness increases against N. Figure 4 shows that there is an inverse variation between the nanoparticle convection distribution $\varphi (\eta )$ and the melting parameter N. Figure 5 shows the relation between the parameter $\epsilon$ and the velocity field $f^{\mathrm{\prime }}(\eta )$ and temperature profile $\theta (\eta )$ . Here it is observed that both the velocity field and boundary layer thickness enhance as we increase the value of parameter $\epsilon$ , which shows that the fluid parameter has a relation with the thickness of the fluid for the greater value of $ϵ$ stickiness of the given fluid decrease, this improves the velocity field. An opposite relation between the fluid parameter $\epsilon$ and $\theta (\eta )$ is shown, a greater value of $\epsilon$ enhances the temperature profile but the thermal boundary layer thickness decreases. Figure 6 displays a deflection of the concentration profile $\varphi (\eta )$ against the fluid parameter $\epsilon$ . The highest value of $\epsilon$ enhances the temperature profile. A relation between the viscous dissipation parameter $\delta$ on the temperature field and the concentration field is given in Figure 7. Here, we show that an increase in $\delta$ , the viscous dissipation parameter, has an opposite impact on the temperature field $\theta (\eta )$ ; however, improvement occurs in the thickness of the thermal boundary layer. It is found that the concentration decrease as the value of the viscous dissipation parameter $\delta$ increases. Figure 8 shows the relation between the parameter $\epsilon$ and the velocity field $f^{\mathrm{\prime }}(\eta )$ and temperature profile $\theta (\eta )$ . Its shows that the temperature profile enhances for higher values of A and clarifies that concentration profile φ decreases for higher values of A. The influence of the Brownian parameter $Nb$ on the temperature and concentration field is given in Figure 9. The migration of nanoparticles from the hot surface to the cold surface happens when the Brownian motion Nb parameter is raised. The magnetic parameter (M) variation on temperature field, velocity variable, and concentration distribution is depicted in Figures 10 and 11. When M is increased, the speed of fluid decreases, whereas concentration distribution and temperature distribution have a reverse impact. From a physical point of view, it is due to an opposing force named Lorentz force is produced against the speed of the fluid.

Figure 3.

Influence of N on $f^{\mathrm{\prime }}(\eta )$ and $\theta (\eta )$ .

Figure 4.

Influence of N on $\varphi (\eta )$ .

Figure 5.

Influence of $ϵ$ on $f^{\mathrm{\prime }}(\eta )$ and $\theta (\eta )$ .

Figure 6.

Influence of $ϵ$ on $\varphi (\eta )$ .

Figure 7.

Influence of $\delta$ on $\theta (\eta )$ and $\varphi (\eta )$ .

Figure 8.

Influence of A on $f^{\mathrm{\prime }}(\eta )$ and $\theta (\eta )$ .

Figure 9.

Influence of $Nb$ on $f^{\mathrm{\prime }}(\eta )$ and $\theta (\eta )$ .

Figure 10.

Influence of M on $f^{\mathrm{\prime }}(\eta )$ and $\theta (\eta )$ .

Figure 11.

Influence of M on $\varphi (\eta )$ .

Final remarks

In this portion, the effects of melting heat transfer phenomena have been elaborated on MHD stagnation point flow with mass transport of Powell–Eyring nanoliquid toward a stretchable medium with a chemical reaction. The findings drawn from the above analysis are summarized as follows:

The profile and boundary-layer width rate for the value A < 1 rises when A enhances while the thickness of boundary layer velocity profile for the value A > 1 decreases as A.

The velocity shape and concentration field show opposite variations for the value of M.

The performance of the Prandtl number on the velocity field and temperature profile is quite dissimilar.

The rate of material fluid factor ε increases and the velocity field and temperature profile also increase.

The influence of material parameter $\delta$ on both the temperature and velocity field is qualitatively alike.