On Magnetohydrodynamic Flow of Viscoelastic Nanofluids with Homogeneous–Heterogeneous Reactions

Abstract

This article explores magnetohydrodynamic stretched flow of viscoelastic nanofluids with heterogeneous–homogeneous reactions. Attention in modeling has been specially focused to constitutive relations of viscoelastic fluids. The heat and mass transport process is explored by thermophoresis and Brownian dispersion. Resulting nonlinear systems are computed for numerical solutions. Findings for temperature, concentration, concentration rate, skin-friction, local Nusselt and Sherwood numbers are analyzed for both second grade and elastico-viscous fluids.

1. Introduction

It is now acknowledged that non-Newtonian fluids in industrial, physiological and technological processes are more significant than viscous fluids. Few examples of such fluids may include silicon oils, printer ink, mud, ice cream, egg yolk, blood at low shear rate, shampoo, gypsum paste, polymer solutions, nail polish, sand in water, ketchup etc. Rheological properties of such fluids are different and thus all these cannot be explained employing one constitutive relationship between shear rate and rate of strain. The modelled expressions for the non-Newtonian liquids are more tedious and of higher order than Navier–Stokes expressions for viscous fluids. Researchers in the field face challenges in modelling, analysis and computations from different quarters. Through different non-Newtonian fluids, the objective here is to explore second grade and elastico-viscous fluids [1,2,3,4,5,6,7,8].

Nanofluids are described by carbon nanotubes (CNTs) [9,10,11], Buongiorno [12] and Tiwari and Das [13] models. Therefore, the information is very significant about flows involving thermophoresis aspects. Impact of slip in flow of copper-water nanoliquid over an extendable surface is examined by Pandey and Kumar [14]. Flow of couple stress nanomaterial bound by an oscillatory stretchable surface is analyzed by Khan et al. [15]. Turkyilmazoglu [16] discussed free and circular jets in view of single phase nanomaterial. Few relevant investigatons for nanoliquids can be seen in studies [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45].

According to previous literature, it is found that magnetohydrodynamic stretched flow of viscoelastic nanofluids with heterogeneous–homogeneous reactions has not been reported yet. Attention in modeling has been specially focused on constitutive relations of viscoelastic fluids. Heat and mass transport process is explored by thermophoresis and Brownian dispersion. Adequate transformations are considered to dimensionless the governing system. Numerical solutions of the resulting system are obtained by employing the shooting method. Contributions of numerous sundary variables on flow fields are interpreted through plots and numerical data.

2. Problem Formulation

Two-dimensional (2D) steady magnetohydrodynamic flow of incompressible viscoelastic nanoliquids by a linear stretchable surface with heterogeneous–homogeneous reactions is analyzed. Second grade and elastico-viscous liquids are considered. Attention in modeling has been specially focused on constitutive relations of viscoelastic fluids. Heat and mass transport process is explored by thermophoresis and Brownian dispersion. Let $u_w\left(x\right)=cx$ denotes wall velocity along x-axis (see Figure 1). Homogeneous-reaction for cubic catalysis is [37]:

$$A+B\to 3B,rate=k_{c}a{b}^2.$$

(1)

At catalyst surface heterogeneous-reaction is [37]:

$$A\to B,rate=k_{s}a.$$

(2)

In above relations rate constants are described by $k_s$ and $k_c$ and chemical species B and A have concentrations b and a separately. Relevant equations for 2D flow satisfy [5,7]:

$$div\mathbf{V}=0,$$

(3)

$$\rho \frac{d\mathbf{V}}{dt}=div\mathsf{\varnothing }+\rho \mathbf{b}.$$

(4)

Cauchy stress tensor of second-order fluid is

$$\mathsf{\varnothing }=-p\mathbf{I}+\mu {\mathbf{A}}_1+{\alpha }_1{\mathbf{A}}_2+{\alpha }_2{\mathbf{A}}_1^2,$$

(5)

in which ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ stand for 1st and 2nd Rivlin-Ericksen tensors respectively i.e.,

$${\mathbf{A}}_1={\left(grad\mathbf{V}\right)}^{*}+\left(grad\mathbf{V}\right),$$

(6)

$${\mathbf{A}}_2=\frac{d{\mathbf{A}}_1}{dt}+{\left(grad\mathbf{V}\right)}^{*}{\mathbf{A}}_1+{\mathbf{A}}_1\left(grad\mathbf{V}\right),$$

(7)

where ${\alpha }_1$ and ${\alpha }_2$ stand for material constants, $\mathbf{b}$ for body force, $\frac{d}{dt}$ for material derivative and p for pressure. Material moduli satisfy following relationships for second grade fluid:

$${\alpha }_1\ge 0,\mu \ge 0,{\alpha }_1+{\alpha }_2=0,$$

(8)

in which * stands for matrix transpose and velocity distribution $\mathbf{V}$ is

$$\mathbf{V}=\left[u\left(x,y\right),v\left(x,y\right),0\right].$$

(9)

The governing expressions for 2D stretching flow of viscoelastic nanofluids are [5,7,37]:

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

(10)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{\partial }^{2}u}{\partial y^2}-k_0\left(u\frac{{\partial }^{3}u}{\partial x\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}-\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right)-\frac{\sigma B_0^2}{\rho }u,$$

(11)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{\partial }^{2}T}{\partial y^2}+\frac{{\left(\rho c\right)}_p}{{\left(\rho c\right)}_f}\left(D_B^{*}\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}\right)+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right),$$

(12)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B^{*}\left(\frac{{\partial }^{2}C}{\partial y^2}\right)+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial y^2}\right),$$

(13)

$$u\frac{\partial a}{\partial x}+v\frac{\partial a}{\partial y}=D_A\left(\frac{{\partial }^{2}a}{\partial y^2}\right)-k_{c}a{b}^2,$$

(14)

$$u\frac{\partial b}{\partial x}+v\frac{\partial b}{\partial y}=D_B\left(\frac{{\partial }^{2}b}{\partial y^2}\right)+k_{c}a{b}^2,$$

(15)

$$u=u_w\left(x\right)=cx,v=0,T=T_w,C=C_w,D_A\frac{\partial a}{\partial y}=k_{s}a,D_B\frac{\partial b}{\partial y}=-k_{s}a\mathrm{at}y=0,$$

(16)

$$u\to 0,T\to T_{\infty },C\to C_{\infty },a\to a_0,b\to 0\mathrm{as}y\to \infty .$$

(17)

Here v and $u$ stand for velocities in vertical and horizontal directions respectively, ${\left(\rho c\right)}_f$ for heat capacity of liquid, $\nu (=\mu /\rho )$ for kinematic viscosity, ${\alpha }_1$ for normal stress moduli, $\mu$ for dynamic viscosity, T for temperature, $\sigma$ for electrical conductivity, $\rho$ for density, $k_0=-{\alpha }_1/\rho$ for elastic parameter, $D_T$ for thermophoretic factor, $\alpha =k/{\left(\rho c\right)}_f$ for thermal diffusivity, C for concentration, $D_B^{*}$ for Brownian factor, k for thermal conductivity, ${\left(\rho c\right)}_p$ for effective heat capacity of nanoparticles, $C_w$ and $T_w$ for wall concentration and temperature respectively and $C_{\infty }$ and $T_{\infty }$ for ambient fluid concentration and temperature respectively. Here $k_0<0$ stands for second grade fluid, $k_0>0$ for elastico-viscous fluid and $k_0=0$ for Newtonian fluid. Selecting [5,7,37]:

$$\begin{array}{c}u=cx{f}^{\prime }\left(\zeta \right),v=-{\left(c\nu \right)}^{1/2}f\left(\zeta \right),\zeta ={\left(\frac{c}{\nu }\right)}^{1/2}y,\\ \theta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\zeta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},a=a_{0}r\left(\zeta \right),b=a_{0}h\left(\zeta \right).\end{array}$$

(18)

Expression (10) is identically verified and Equations (11)–(17) give [5,7,37]:

$$f^{‴}+f{f}^{″}-f^{\prime 2}-k_1^{*}\left(2f^{\prime }{f}^{‴}-f^{\prime {\prime }^2}-f{f}^{iv}\right)-M^2{f}^{\prime }=0,$$

(19)

$${\theta }^{″}+Pr\left(f{\theta }^{\prime }+N_b{\theta }^{\prime }{\varphi }^{\prime }+N_t{\theta }^{{\prime }^2}\right)=0,$$

(20)

$${\varphi }^{″}+Scf{\varphi }^{\prime }+\frac{N_t}{N_b}{\theta }^{″}=0,$$

(21)

$$\frac{1}{S{c}_b}{r}^{″}+f{r}^{\prime }-Kr{h}^2=0,$$

(22)

$$\frac{\delta }{S{c}_b}{h}^{″}+f{h}^{\prime }+Kr{h}^2=0,$$

(23)

$$f=0,f^{\prime }=1,\theta =1,\varphi =1,r^{\prime }=K_{s}r,\delta h^{\prime }=-K_{s}r\mathrm{at}\zeta =0,$$

(24)

$$f^{\prime }\to 0,\theta \to 0,\varphi \to 0,r\to 1,h\to 0\mathrm{as}\zeta \to \infty .$$

(25)

Here $k_1^{*}$ stands for viscoelastic parameter, $\delta$ for ratio of mass diffusion coefficients, $N_t$ for thermophoresis parameter, K for homogeneous-reaction strength, M for magnetic parameter, $Sc$ for Schmidt number, $S{c}_b$ for Schmidt number (for heterogeneous–homogeneous reactions), $N_b$ for Brownian motion parameter, $K_s$ for heterogeneous-reaction strength and Pr for Prandtl number. We set these definitions as

$$\begin{array}{c}{k}_1^{*}=-k_0\left(\frac{c}{\nu }\right),M^2=\frac{\sigma B_0^2}{\rho c},Pr=\frac{\nu }{\alpha },\delta =\frac{D_B}{D_A},K=\frac{k_c{a}_0^2}{u_w},K_s=\frac{k_s}{D_A{a}_0}\sqrt{\frac{c}{\nu }},\\ Sc=\frac{\nu }{D_B^{*}},S{c}_b=\frac{\nu }{D_A},N_b=\frac{{\left(\rho c\right)}_p{D}_B^{*}\left(C_w-C_{\infty }\right)}{{\left(\rho c\right)}_f\nu },N_t=\frac{{\left(\rho c\right)}_p{D}_T\left(T_w-T_{\infty }\right)}{{\left(\rho c\right)}_f\nu T_{\infty }}.\end{array}$$

(26)

Considering that $D_A=D_B$ we have $\delta =1$ and thus

$$r\left(\zeta \right)+h\left(\zeta \right)=1.$$

(27)

Now Eqsuations (22) and (23) give

$$\frac{1}{S{c}_b}{r}^{″}+f{r}^{\prime }-K{(1-r)}^{2}r=0,$$

(28)

with boundary conditions

$$r^{\prime }\left(0\right)=K_{s}r\left(0\right),r(\infty )\to 1.$$

(29)

Coefficient of skin friction and local Sherwood and Nusselt numbers are

$${\mathrm{Re}}_x^{1/2}{C}_f=\left(1-3k_1^{*}\right)f^{{}^{″}}\left(0\right),{\mathrm{Re}}_x^{-1/2}S{h}_x=-{\varphi }^{{}^{\prime }}\left(0\right),{\mathrm{Re}}_x^{-1/2}N{u}_x=-{\theta }^{{}^{\prime }}\left(0\right),$$

(30)

in which ${\mathrm{Re}}_x=u_{w}x/\nu$ denotes the local Reynolds number.

3. Solution Methodology

By considering suitable boundary conditions on the system of equations, a numerical solution is developed using NDSolve in Mathematica. Shooting method is used via NDSolve. This method is very helpful in case of small step-size featuring negligible error. As a consequence, both x and y varied uniformly by a step-size of $0.01$ [40].

4. Graphical Results and Discussion

Effects of magnetic parameter $M,$ homogeneous-reaction strength $K,$ Schmidt number $Sc,$ Schmidt number (for heterogeneous–homogeneous reactions) $S{c}_b,$ thermophoresis parameter $N_t,$ heterogeneous-reaction strength $K_s,$ Prandtl number Pr and Brownian motion parameter $N_b$ on concentration $\varphi \left(\zeta \right),$ concentration rate $r\left(\zeta \right)$ and temperature $\theta \left(\zeta \right)$ for both second grade and elastico-viscous fluids are sketched in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

Figure 2 depicts impact of magnetic parameter M on temperature $\theta \left(\zeta \right)$. Here $M\ne 0$ is for hydromagnetic flow situation and $M=0$ corresponds to hydrodynamic flow case. Temperature $\theta \left(\zeta \right)$ is higher for hydromagnetic flow in comparison to hydrodynamic flow for both second grade and elastico-viscous fluids. Physically magnetic parameter depends upon Lorentz force. Lorentz force is an agent which resists the motion of fluid and therefore temperature $\theta \left(\zeta \right)$ enhances.

Figure 3 displays variations in temperature $\theta \left(\zeta \right)$ for increasing Prandtl number Pr. Temperature $\theta \left(\zeta \right)$ decays for larger Pr for both second grade and elastico-viscous fluids. Physically Prandtl number involves thermal diffusivity. Larger Prandtl number corresponds to weaker thermal diffusivity which produces a decay in temperature $\theta \left(\zeta \right).$

Figure 4 depicts impact of Brownian motion parameter $N_b$ on temperature $\theta \left(\zeta \right)$. Larger $N_b$ produces an increment in temperature $\theta \left(\zeta \right)$ for both second grade and elastico-viscous fluids. Larger Brownian motion parameter $N_b$ has stronger Brownian diffusivity and weaker viscous force which increased the temperature $\theta \left(\zeta \right).$

Figure 5 shows that larger thermophoresis parameter $N_t$ leads to higher temperature $\theta \left(\zeta \right)$ for both second grade and elastico-viscous fluids. Larger $N_t$ causes strong thermophoresis force which tends to shift nanoparticles from hot to cold zone and therefore temperature $\theta \left(\zeta \right)$ increases.

Impact of magnetic parameter M on concentration $\varphi \left(\zeta \right)$ is displayed in Figure 6 Concentration $\varphi \left(\zeta \right)$ is upgraded for increasing estimations of M for both second grade and elastico-viscous fluids. Furthermore, the concentration $\varphi \left(\zeta \right)$ shows similar trend for both second grade and elastico-viscous fluids.

Figure 7 depicts that concentration $\varphi \left(\zeta \right)$ is decreased for larger Schmidt number $Sc$ for both second grade and elastico-viscous fluids. Schmidt number $Sc$ has an inverse relation with Brownian diffusivity. Larger Schmidt number leads to weaker Brownian diffusivity which produces weaker concentration $\varphi \left(\zeta \right).$

Impact of Brownian motion $N_b$ on concentration $\varphi \left(\zeta \right)$ is shown in Figure 8 Bigger $N_b$ produces a reduction in concentration $\varphi \left(\zeta \right)$ for both second grade and elastico-viscous fluids. Physically Brownian force tries to push particles in opposite direction of concentration gradient and make nanofluid more homogeneous. Therefore, higher the Brownian force, lower the concentration gradient and more uniform concentration $\varphi \left(\zeta \right).$

Figure 9 displays that how thermophoresis $N_t$ affects concentration $\varphi \left(\zeta \right)$. Here concentration $\varphi \left(\zeta \right)$ is upgraded for higher estimations of $N_t$ for both second grade and elastico-viscous fluids. Furthermore, the concentration $\varphi \left(\zeta \right)$ shows similar trend for both second grade and elastico-viscous fluids.

Figure 10 displays that how Schmidt number $S{c}_b$ affects concentration rate $r\left(\zeta \right).$ Here concentration rate $r\left(\zeta \right)$ is upgraded for higher estimations of Schmidt number $S{c}_b$ for both second grade and elastico-viscous fluids. Furthermore, the concentration rate $r\left(\zeta \right)$ shows similar trend for both second grade and elastico-viscous fluids.

From Figure 11 it is noted that larger homogeneous-reaction K displays a decay in concentration rate $r\left(\zeta \right)$ for both second grade and elastico-viscous fluids. Larger homogeneous-reaction K corresponds to higher chemical reaction which consequently decreases the concentration rate $r\left(\zeta \right).$

Figure 12 depicts that larger heterogeneous-reaction $K_s$ produces higher concentration rate $r\left(\zeta \right)$ for both second grade and elastico-viscous fluids. Here heterogeneous-reaction parameter $K_s$ has an inverse relation with mass diffusivity which produces an enhancement in concentration rate $r\left(\zeta \right).$

Table 1. Skin-friction coefficient for various estimations of viscoelastic and magnetic parameters.

M	$-{\mathit{C}}_{\mathit{f}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$	$-{\mathit{C}}_{\mathit{f}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$
	${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=\mathbf{0}\mathbf{.}\mathbf{1}$	${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=-\mathbf{0}\mathbf{.}\mathbf{1}$
$0.0$	$0.7379$	$1.2395$
$0.2$	$0.7525$	$1.2640$
$0.5$	$0.8250$	$1.3858$

displays skin-friction $-C_f{\mathrm{Re}}_x^{1/2}$ subject to varying $k_1^{*}$ and $M$. Here skin-friction has higher estimations for larger M for both second grade and elastico-viscous fluids.

Table 2. Comparative data of skin-friction coefficient for various estimations of viscoelastic parameter when $M=0.$

${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}$	$-{\mathit{C}}_{\mathit{f}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$
	Numerical	HAM [5]
$0.0$	$1.0000$	$1.00000$
$0.1$	$0.7379$	$0.73786$
$0.2$	$0.4472$	$0.44721$
$0.3$	$0.1195$	$0.11952$

depicts comparison for various estimations of $k_1^{*}$ with homotopy analysis method (HAM). Table 2 presents a good agreement of numerical solution with existing homotopy analysis method (HAM) solution in a limiting sense.

Table 3. Local Nusselt number for various estimations of viscoelastic, Brownian motion and thermophoresis parameterswhen $Sc=Pr=1.0$ and $M=0.2.$

${\mathit{N}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{t}}$	${\mathbf{Nu}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$	${\mathbf{Nu}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$
		${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=\mathbf{0}\mathbf{.}\mathbf{1}$	${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=-\mathbf{0}\mathbf{.}\mathbf{1}$
$0.1$	$0.1$	$0.5232$	$0.5425$
$0.2$	-	$0.4970$	$0.5153$
$0.3$	-	$0.4716$	$0.4890$
$0.2$	$0.1$	$0.4970$	$0.5153$
-	$0.2$	$0.4827$	$0.5002$
-	$0.3$	$0.4689$	$0.4856$

depicts local Nusselt number $N{u}_x{\mathrm{Re}}_x^{-1/2}$ subject to varying $k_1^{*}$, $N_b$ and $N_t.$ Here larger $N_b$ and $N_t$ correspond to lower local Nusselt number for both second grade and elastico-viscous fluids.

Table 4. Local Sherwood number for various estimations of viscoelastic, Brownian motion and thermophoresis parameterswhen $Sc=Pr=1.0$ and $M=0.2.$

${\mathit{N}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{Sh}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$	${\mathit{Sh}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$
		${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=\mathbf{0}\mathbf{.}\mathbf{1}$	${\mathit{k}}_{\mathbf{1}}^{\mathbf{*}}=-\mathbf{0}\mathbf{.}\mathbf{1}$
$0.1$	$0.1$	$0.2100$	$0.2300$
$0.2$	-	$0.3994$	$0.4203$
$0.3$	-	$0.4622$	$0.4834$
$0.2$	$0.1$	$0.3994$	$0.4203$
-	$0.2$	$0.2435$	$0.2647$
-	$0.3$	$0.0982$	$0.1205$

shows local Sherwood number $S{h}_x{\mathrm{Re}}_x^{-1/2}$ subject to varying $k_1^{*},$ $N_b$ and $N_t$. Here larger $N_t$ produces lower local Sherwood number while opposite trend is noted via $N_b$ for both second grade and elastico-viscous fluids.

5. Conclusions

Magnetohydrodynamic flow of viscoelastic nanofluids bound by a linear stretchable surface with heterogeneous–homogeneous reactions are analyzed. Both concentration $\varphi \left(\zeta \right)$ and temperature $\theta \left(\zeta \right)$ are enhanced via higher M. Larger Brownian motion $N_b$ displays opposite trend for concentration $\varphi \left(\zeta \right)$ and temperature $\theta \left(\zeta \right)$. Larger thermophoresis number $N_t$ produces higher concentration $\varphi \left(\zeta \right)$ and temperature $\theta \left(\zeta \right)$. Temperature $\theta \left(\zeta \right)$ is reduced when Prandtl number enhances. Prandtl number is considered to control the rate of heat transfer in engineering and industrial processes. The suitable value of Prandtl number is very essential to control the rate of heat transfer in engineering and industrial processes. Larger homogeneous-reaction K depicts a reduction in concentration rate $r\left(\zeta \right).$ Larger heterogenous-reaction $K_s$ and Schmidt number $S{c}_b$ lead to higher concentration rate $r\left(\zeta \right).$ Skin friction is enhanced for larger magnetic parameter $M.$ Reverse trend of local Sherwood number is seen for $N_t$ and $N_b.$ Local Nusselt number is decreased for thermophoresis $N_t$ and Brownian motion $N_b$ parameters. Furthermore, the present analysis is reduced to Newtonian fluid flow case when $k_1^{*}=0$.