**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 *uw* (*x*) = *cx* denotes wall velocity along *x*-axis (see Figure 1). Homogeneous-reaction for cubic catalysis is [37]:

$$A + B \to \mathfrak{B}, \; rate = k\_{\varepsilon} ab^2. \tag{1}$$

At catalyst surface heterogeneous-reaction is [37]:

$$A \to B, \text{ } rate = k\_s a. \tag{2}$$

**Figure 1.** Flow configuration.

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

$$
\operatorname{div} \mathbf{V} = 0,\tag{3}
$$

$$
\rho \frac{d\mathbf{V}}{dt} = \operatorname{div} \mathbf{o} + \rho \mathbf{b}. \tag{4}
$$

Cauchy stress tensor of second-order fluid is

$$\mathbf{o} = -p\mathbf{I} + \mu \mathbf{A}\_1 + a\_1 \mathbf{A}\_2 + a\_2 \mathbf{A}\_{1\prime}^2 \tag{5}$$

in which **A**<sup>1</sup> and **A**<sup>2</sup> stand for 1st and 2nd Rivlin-Ericksen tensors respectively i.e.,

$$\mathbf{A}\_1 = \begin{pmatrix} \gcd \mathbf{V} \end{pmatrix}^\* + \begin{pmatrix} \gcd \mathbf{V} \end{pmatrix},\tag{6}$$

$$\mathbf{A}\_2 = \frac{d\mathbf{A}\_1}{dt} + \left(\grad \mathbf{V}\right)^\* \mathbf{A}\_1 + \mathbf{A}\_1 \left(\grad \mathbf{V}\right),\tag{7}$$

where *α*<sup>1</sup> and *α*<sup>2</sup> stand for material constants, **b** for body force, *<sup>d</sup> dt* for material derivative and *p* for pressure. Material moduli satisfy following relationships for second grade fluid:

$$
\mathfrak{a}\_1 \ge 0, \ \mu \ge 0, \ \mathfrak{a}\_1 + \mathfrak{a}\_2 = 0,\tag{8}
$$

in which ∗ stands for matrix transpose and velocity distribution **V** is

$$\mathbf{V} = \begin{bmatrix} \mu \ (\mathbf{x}, \mathbf{y}) \ , \upsilon \ (\mathbf{x}, \mathbf{y}) \ , \mathbf{0} \end{bmatrix} . \tag{9}$$

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

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

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} - k\_0 \left( u \frac{\partial^3 u}{\partial x \partial y^2} + v \frac{\partial^3 u}{\partial y^3} + \frac{\partial u}{\partial x} \frac{\partial^2 u}{\partial y^2} - \frac{\partial u}{\partial y} \frac{\partial^2 u}{\partial x \partial y} \right) - \frac{\sigma B\_0^2}{\rho} u,\tag{11}$$

$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = a\frac{\partial^2 T}{\partial y^2} + \frac{(\rho c)\_p}{(\rho c)\_f} \left( D\_B^\* \left( \frac{\partial T}{\partial y} \frac{\partial C}{\partial y} \right) + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial y} \right)^2 \right), \tag{12}$$

$$u\frac{\partial \mathcal{C}}{\partial x} + v\frac{\partial \mathcal{C}}{\partial y} = D\_B^\* \left(\frac{\partial^2 \mathcal{C}}{\partial y^2}\right) + \frac{D\_T}{T\_\infty} \left(\frac{\partial^2 T}{\partial y^2}\right),\tag{13}$$

$$u\frac{\partial a}{\partial x} + v\frac{\partial a}{\partial y} = D\_A \left(\frac{\partial^2 a}{\partial y^2}\right) - k\_c a b^2,\tag{14}$$

$$u\frac{\partial b}{\partial x} + v\frac{\partial b}{\partial y} = D\_B \left(\frac{\partial^2 b}{\partial y^2}\right) + k\_c a b^2,\tag{15}$$

$$\mu = u\_w \left( \mathbf{x} \right) = c \mathbf{x}, \ v = 0, \ T = T\_{\mathbf{w} \prime} \ \mathbf{C} = \mathbf{C}\_{\mathbf{w} \prime} \ D\_A \frac{\partial a}{\partial y} = k\_s a, \ D\_B \frac{\partial b}{\partial y} = -k\_s a \text{ at } y = 0,\tag{16}$$

$$u \to 0, \quad T \to T\_{\infty}, \quad \mathbb{C} \to \mathbb{C}\_{\infty}, \quad a \to a\_{0\prime} \quad b \to 0 \quad \text{as} \quad y \to \infty. \tag{17}$$

Here *v* and *u* stand for velocities in vertical and horizontal directions respectively, (*ρc*)*<sup>f</sup>* for heat capacity of liquid, *ν*(= *μ*/*ρ*) for kinematic viscosity, *α*<sup>1</sup> for normal stress moduli, *μ* for dynamic viscosity, *T* for temperature, *σ* for electrical conductivity, *ρ* for density, *k*<sup>0</sup> = −*α*1/*ρ* for elastic parameter, *DT* for thermophoretic factor, *α* = *k*/(*ρc*)*<sup>f</sup>* for thermal diffusivity, *C* for concentration, *D*<sup>∗</sup> *<sup>B</sup>* for Brownian factor, *k* for thermal conductivity, (*ρc*)*<sup>p</sup>* for effective heat capacity of nanoparticles, *Cw* and *Tw* for wall concentration and temperature respectively and *C*∞ and *T*∞ for ambient fluid concentration and temperature respectively. Here *k*<sup>0</sup> < 0 stands for second grade fluid, *k*<sup>0</sup> > 0 for elastico-viscous fluid and *k*<sup>0</sup> = 0 for Newtonian fluid. Selecting [5,7,37]:

$$\begin{array}{l} \mathsf{u} = \mathsf{c} \mathsf{x} f'(\zeta), \; \mathsf{v} = -\; (\mathsf{c} \mathsf{v})^{1/2} \; f(\zeta), \; \zeta = \left(\frac{\mathsf{c}}{\mathsf{v}}\right)^{1/2} \mathsf{y}, \\\ \theta(\zeta) = \frac{T - T\_{\infty}}{\mathsf{T}\_{w} - T\_{\infty}}, \; \phi(\zeta) = \frac{\mathsf{C} - \mathsf{C}\_{\infty}}{\mathsf{C}\_{w} - \mathsf{C}\_{\infty}}, \; a = a\_{0} r(\zeta), \; b = a\_{0} h(\zeta). \end{array} \tag{18}$$

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

$$f^{\prime\prime\prime} + ff^{\prime\prime} - f^{\prime 2} - k\_1^\* \left( 2f^{\prime}f^{\prime\prime\prime} - f^{\prime\prime} - ff^{iv} \right) - M^2 f^{\prime} = 0,\tag{19}$$

$$\left(\theta^{\prime\prime} + \text{Pr}\left(f\theta^{\prime} + \text{N}\_{b}\theta^{\prime}\phi^{\prime} + \text{N}\_{l}\theta^{\prime^2}\right) = 0,\tag{20}$$

$$
\xi^{\prime\prime} + \mathrm{Scf}\,\phi^{\prime} + \frac{N\_{\mathrm{f}}}{N\_{\mathrm{b}}}\theta^{\prime\prime} = 0,\tag{21}
$$

$$\frac{1}{Sc\_b}r^{\prime\prime} + fr^{\prime} - \mathcal{K}rh^2 = 0,\tag{22}$$

$$\frac{\delta}{Sc\_b}h'' + fh' + \mathcal{K}rh^2 = 0,\tag{23}$$

$$f = 0, \; f' = 1, \; \theta = 1, \; \phi = 1, \; r' = \mathcal{K}\_s r, \; \delta h' = -\mathcal{K}\_s r \text{ at } \mathcal{J} = 0,\tag{24}$$

$$f' \to 0, \; \theta \to 0, \; \phi \to 0, \; r \to 1, \; h \to 0 \; \text{as} \; \zeta \to \infty. \tag{25}$$

Here *k*∗ <sup>1</sup> stands for viscoelastic parameter, *δ* for ratio of mass diffusion coefficients, *Nt* for thermophoresis parameter, *K* for homogeneous-reaction strength, *M* for magnetic parameter, *Sc* for Schmidt number, *Scb* for Schmidt number (for heterogeneous–homogeneous reactions), *Nb* for Brownian motion parameter, *Ks* for heterogeneous-reaction strength and Pr for Prandtl number. We set these definitions as

$$\begin{array}{l} \text{s.} \, k\_1^\* = -k\_0 \left( \frac{\varepsilon}{\mathcal{V}} \right), \, M^2 = \frac{\sigma D\_0^2}{\rho \mathfrak{c}}, \, \text{Pr} = \frac{\nu}{\mathfrak{a}}, \, \delta = \frac{D\_\mathcal{R}}{D\_A}, \, \text{K} = \frac{k\_\mathcal{L} a\_0^2}{\underline{\mathcal{V}}\_w}, \, \text{N}\_\mathcal{S} = \frac{k\_\mathcal{s}}{D\_A a\_0} \sqrt{\frac{\varepsilon}{\mathcal{V}}}\\ \text{Sc} = \frac{\mathcal{V}}{\mathcal{D}\_\mathcal{B}}, \, \text{Sc}\_\mathcal{b} = \frac{\nu}{\mathcal{D}\_\mathcal{A}}, \, \text{N}\_\mathcal{b} = \frac{(\rho c)\_p D\_\mathcal{B}^\* (\mathbb{C}\_w - \mathbb{C}\_\infty)}{(\rho c)\_f \underline{\mathcal{V}}}, \, \text{N}\_\mathcal{l} = \frac{(\rho c)\_p D\_\mathcal{T} (T\_w - T\_\infty)}{(\rho c)\_f \underline{\mathcal{V}} \underline{\mathcal{T}}\_\infty} . \end{array} \tag{26}$$

Considering that *DA* = *DB* we have *δ* = 1 and thus

$$r(\zeta) + h(\zeta) = 1.\tag{27}$$

Now Eqsuations (22) and (23) give

$$\frac{1}{Sc\_b}r^{\prime\prime} + fr^{\prime} - K(1 - r)^2r = 0,\tag{28}$$

with boundary conditions

$$r'\left(0\right) = \mathcal{K}\_{\mathfrak{s}}r\left(0\right),\ r\left(\infty\right) \to 1. \tag{29}$$

Coefficient of skin friction and local Sherwood and Nusselt numbers are

$$\operatorname{Re}\_x^{1/2} \mathbb{C}\_f = \left(1 - 3k\_1^\*\right) f'\left(0\right),\\\operatorname{Re}\_x^{-1/2} \operatorname{Sh}\_x = -\boldsymbol{\phi}'\left(0\right),\\\operatorname{Re}\_x^{-1/2} \operatorname{Nu}\_x = -\boldsymbol{\theta}'\left(0\right),\tag{30}$$

in which Re*<sup>x</sup>* = *uwx*/*ν* denotes the local Reynolds number.
