**2. Problem Description**

Let us examine a mixed convective 3D nanoliquid flow by a pivoting disk with slip features. Arrhenius activation energy, magnetic field and binary chemical reaction are also accounted for. A disk at *z* = 0 rotates with constant angular velocity Ω (see Figure 1). Brownian dispersion and thermophoretic impacts are additionally present. The velocities are (*u*, *v*, *w*) in the headings of expanding (*r*, *ϕ*, *z*) respectively. The associated boundary-layer equations are [11,37]:

$$
\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,
\tag{1}
$$

$$u\frac{\partial u}{\partial r} - \frac{v^2}{r} + w\frac{\partial u}{\partial z} = \nu \left( \frac{\partial^2 u}{\partial z^2} + \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{r^2} \right) - \frac{\sigma B\_0^2}{\rho\_f} u + \mathbf{g}^\* (\boldsymbol{\beta}\_T (T - T\_\infty) + \boldsymbol{\beta}\_\mathbb{C} (\mathbb{C} - \mathbb{C}\_\infty)), \tag{2}$$

$$u\frac{\partial v}{\partial r} + \frac{uv}{r} + w\frac{\partial v}{\partial z} = \nu \left(\frac{\partial^2 v}{\partial z^2} + \frac{\partial^2 v}{\partial r^2} + \frac{1}{r}\frac{\partial v}{\partial r} - \frac{v}{r^2}\right) - \frac{\sigma B\_0^2}{\rho\_f} v,\tag{3}$$

$$u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z} = \nu \left(\frac{\partial^2 w}{\partial r^2} + \frac{\partial^2 w}{\partial z^2} + \frac{1}{r}\frac{\partial w}{\partial r}\right),\tag{4}$$

$$\begin{split} u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} &= \quad a\_m \left( \frac{\partial^2 T}{\partial r^2} + \frac{\partial^2 T}{\partial z^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) \\ &+ \frac{(\rho c)\_p}{(\rho c)\_f} \left( D\_B \left( \frac{\partial T}{\partial r}\frac{\partial C}{\partial r} + \frac{\partial T}{\partial z}\frac{\partial C}{\partial z} \right) + \frac{D\_T}{T\_{\text{co}}} \left( \left( \frac{\partial T}{\partial z} \right)^2 + \left( \frac{\partial T}{\partial r} \right)^2 \right) \right), \end{split} \tag{5}$$

$$\begin{split} u\frac{\partial\mathbb{C}}{\partial r} + w\frac{\partial\mathbb{C}}{\partial z} &= \,^D D\_B \left( \frac{\partial^2 \mathbb{C}}{\partial r^2} + \frac{\partial^2 \mathbb{C}}{\partial z^2} + \frac{1}{r}\frac{\partial \mathbb{C}}{\partial r} \right) \\ &+ \frac{D\_T}{T\_{\infty}} \left( \frac{\partial^2 T}{\partial r^2} + \frac{\partial^2 T}{\partial z^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) - k\_r^2 \left( \mathbb{C} - \mathbb{C}\_{\infty} \right) \left( \frac{T}{T\_{\infty}} \right)^n \exp\left( -\frac{E\_d}{\kappa T} \right), \end{split} \tag{6}$$

$$u = L\_1 \frac{\partial u}{\partial z}, \; v = r\Omega + L\_1 \frac{\partial v}{\partial z}, \; w = 0, \; T = T\_w + L\_2 \frac{\partial T}{\partial z}, \; C = C\_w + L\_3 \frac{\partial C}{\partial z} \text{ at } z = 0,\tag{7}$$

$$u \to 0, \quad v \to 0, \quad T \to T\_{\infty \prime} \quad \mathbb{C} \to \mathbb{C}\_{\infty} \quad \text{as } z \to \infty. \tag{8}$$

**Figure 1.** Schematic diagram of the problem.

Here *u*, *v* and *w* stand for velocity components in directions of *r*, *ϕ* and *z*; *ρ<sup>f</sup>* , *μ* and *ν* = *μ*/*ρ<sup>f</sup>* are for fluid density, dynamic and kinematic viscosities, respectively. *L*<sup>1</sup> stands for velocity slip factor; *C*∞ for ambient concentration; *g*∗ for acceleration due to gravity; *T*∞ for ambient temperature; *β<sup>T</sup>* for thermal expansion factor; (*ρc*)*<sup>p</sup>* for effective heat capacity of nanoparticles; *σ* for electrical conductivity; *Ea* for activation energy; *L*<sup>3</sup> for concentration slip factor; (*ρc*)*<sup>f</sup>* for heat capacity of liquid; *β<sup>C</sup>* for concentration expansion factor, *C* for concentration; *L*<sup>2</sup> for thermal slip factor; *n* for fitted rate constant; *DT* for thermophoretic factor, *α<sup>m</sup>* = *k*/(*ρc*)*<sup>f</sup>* and *k* for thermal diffusivity and thermal conductivity, respectively; *kr* for reaction rate; *T* for fluid temperature; *DB* for Brownian factor; and *κ* for Boltzmann constant. Selecting [37]:

$$\begin{aligned} u &= r\Omega f'(\zeta), \; w = -(2\Omega\nu)^{1/2} f(\zeta), \; \theta(\zeta) = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}, \\ \phi(\zeta) &= \frac{\mathbb{C} - \mathbb{C}\_{\infty}}{\mathbb{C}\_w - \mathbb{C}\_{\infty}}, \; \zeta = \left(\frac{2\Omega}{\nu}\right)^{1/2} z, \; \upsilon = r\Omega g(\zeta). \end{aligned} \tag{9}$$

Equation (1) is now verified while Equations (2)–(8) yield [11,37]:

$$2f^{\prime\prime\prime} + 2ff^{\prime\prime} - f^{\prime} + g^2 - (Ha)^2 f^{\prime} + \lambda\_T(\theta + \lambda\_C\phi) = 0,\tag{10}$$

$$2\mathbf{g}^{\prime\prime} + 2f\mathbf{g}^{\prime} - 2f^{\prime}\mathbf{g} - (Ha)^{2}\mathbf{g} = 0,\tag{11}$$

$$\frac{1}{\text{Pr}}\boldsymbol{\theta}^{\prime\prime} + f\boldsymbol{\theta}^{\prime} + \text{N}\_{b}\boldsymbol{\theta}^{\prime}\boldsymbol{\phi}^{\prime} + \text{N}\_{l}\boldsymbol{\theta}^{\prime^2} = \text{0},\tag{12}$$

$$\frac{1}{Sc}\phi'' + f\phi' + \frac{1}{Sc}\frac{N\_t}{N\_b}\theta'' - \sigma \left(1 + \delta\theta\right)''\phi \exp\left(-\frac{E}{1+\delta\theta}\right) = 0,\tag{13}$$

$$f(0) = 0, \ f'(0) = a f'(0), \ g(0) = 1 + a g'(0), \ \theta(0) = 1 + \beta \theta'(0), \ \phi(0) = 1 + \gamma \phi'(0), \tag{14}$$

$$f'(\infty) \to 0, \ g(\infty) \to 0, \; \theta(\infty) \to 0, \; \phi(\infty) \to 0. \tag{15}$$

Here *λ<sup>T</sup>* stands for thermal buoyancy number, *Nt* for thermophoresis parameter, *α* for velocity slip parameter, Pr for Prandtl number, *λ<sup>C</sup>* for concentration buoyancy number, *Sc* for Schmidt parameter, *Ha* for Hartman number, *β* for thermal slip parameter, *σ* for chemical reaction number, *Nb* for Brownian parameter, *δ* for temperature difference parameter, *γ* for concentration slip parameter and *E* for non-dimensional activation energy. These parameters are defined by

$$\begin{split} (Ha)^2 &= \frac{\rho \mathcal{E}\_0^2}{\Omega \rho\_f}, \Pr = \frac{\nu}{a\_m}, \ n = L\_1 \sqrt{\frac{2\Omega}{\nu}}, \ \gamma = L\_3 \sqrt{\frac{2\Omega}{\nu}}, \\ N\_l = \frac{(\rho c)\_p D\_T (T\_w - T\_\infty)}{(\rho c)\_f \sqrt{T\_\infty}}, \ \mathcal{S} \underline{c} = \frac{\nu}{D\_\mathcal{B}}, \ N\_b = \frac{(\rho c)\_p D\_\mathcal{B} (T\_w - C\_\infty)}{(\rho c)\_f \nu}, \ \lambda\_T = \frac{\mathcal{S}' \mathcal{S}\_\mathcal{T} \beta\_T (T\_w - T\_\infty)}{\Omega}, \\ \lambda\_\mathcal{C} = \frac{\beta\_\mathcal{C} (\mathcal{C}\_w - \mathcal{C}\_\infty)}{\beta\_T (T\_w - T\_\infty)}, \ \beta = L\_2 \sqrt{\frac{2\Omega}{\nu}}, \ \sigma = \frac{k\_\mathcal{T}^2}{\Pi'}, \ \delta = \frac{T\_w - T\_\infty}{T\_\Omega}, \ \mathcal{E} = \frac{\mathcal{E}\_\mathbf{u}}{\overline{\pi} T\_\infty}. \end{split} \tag{16}$$

The coefficients of skin friction and Sherwood and Nusselt numbers are

$$\operatorname{Re}\_r^{1/2} \mathbb{C}\_f = f^{\prime\prime}(0), \ \operatorname{Re}\_r^{1/2} \mathbb{C}\_\emptyset = \mathfrak{g}^{\prime}(0), \ \operatorname{Re}\_r^{-1/2} \operatorname{Sh} = -\mathfrak{g}^{\prime}(0), \ \operatorname{Re}\_r^{-1/2} \operatorname{Nu} = -\theta^{\prime}(0), \tag{17}$$

where Re*<sup>r</sup>* = 2(Ω*r*)*r*/*ν* depicts local rotational Reynolds number.

#### **3. Solution Methodology**

By employing suitable boundary conditions on the system of equations, a numerical solution was constructed considering NDSolve in Mathematica. The shooting method was employed via NDSolve. This method is very helpful in the situation of a smaller step-size featuring negligible error. As a consequence, both the *z* and *r* varied uniformly by a step-size of 0.01 [20].

#### **4. Graphical Results and Discussion**

This segment displays variations of various physical flow parameters, such as the thermophoresis parameter *Nt*, Hartman number *Ha*, thermal slip parameter *β*, chemical reaction parameter *σ*, Brownian motion parameter *Nb*, concentration slip parameter *γ* and activation energy *E*, on concentration *φ*(*ζ*) and temperature *θ* (*ζ*) distributions. Figure 2a displays the effect of Hartman number *Ha* on temperature *θ* (*ζ*). Temperature *θ* (*ζ*) is enhanced for higher estimations of *Ha*. The effect of thermal slip *β* on temperature *θ* (*ζ*) is shown in Figure 2b. Greater *β* shows diminishing trend of *θ* (*ζ*) and associated warmth layer. The impact of *Nt* on temperature *θ* (*ζ*) is explored in Figure 2c. An increment in *Nt* leads to stronger temperature field *θ* (*ζ*). Figure 2d depicts change in temperature *θ* (*ζ*) for varying Brownian motion number *Nb*. Physically, the Brownian motion of nanoparticles is enhanced by increasing Brownian motion number *Nb*. Therefore dynamic vitality is altered into thermal vitality, which depicts an increment in temperature *θ* (*ζ*) and the respective warmth layer. Figure 3a shows that how the Hartman number *Ha* influences concentration *φ*(*ζ*). For a greater Hartman number *Ha*, both concentration *φ*(*ζ*) and the concentration layer are upgraded. Figure 3b displays that concentration *φ*(*ζ*) is weaker for a greater concentration slip. Figure 3c demonstrates how thermophoresis *Nt* influences concentration *φ*(*ζ*). By improving the thermophoresis parameter *Nt*, the concentration *φ*(*ζ*) and associated layer are upgraded. Figure 3d depicts effect of Brownian motion *Nb* on concentration *φ*(*ζ*). It is noted that higher concentration *φ*(*ζ*) is developed by utilizing greater Brownian parameter *Nb*. Figure 3e explains effect of non-dimensional activation energy *<sup>E</sup>* on concentration *<sup>φ</sup>*(*ζ*). An increment in *<sup>E</sup>* rots change Arrhenius work - *T T*∞ *n* exp - <sup>−</sup> *Ea κT* , which inevitably builds up a generative synthetic reaction due to which concentration *φ*(*ζ*) increases. Figure 3f introduces the fact that an increment in chemical response number *σ* causes a rot in concentration *φ*(*ζ*). Figure 4a,b displays the effects of *Nt* and *Nb* on Re−1/2 *<sup>r</sup> Nu*. It is noted that Re−1/2 *<sup>r</sup> Nu* decreases for greater *Nt* and *Nb*. Contributions of *Nt* and *Nb* on Re−1/2 *<sup>r</sup> Sh* are explored in Figure 5a,b. Here Re−1/2 *<sup>r</sup> Sh* is increasing the factor of *Nb* while it is decreasing the factor of *Nt*.

**Figure 2.** (**a**) Variations of temperature distribution *θ*(*ζ*) for Hartman number *Ha*; (**b**) variations of temperature distribution *θ*(*ζ*) for thermal slip parameter *β*; (**c**) variations of temperature distribution *θ*(*ζ*) for thermophoresis parameter *Nt*; (**d**) variations of temperature distribution *θ*(*ζ*) for Brownian motion parameter *Nb*.

**Figure 3.** *Cont.*

**Figure 3.** (**a**) Variations of concentration distribution *φ*(*ζ*) for Hartman number *Ha*; (**b**) variations of concentration distribution *φ*(*ζ*) for concentration slip parameter *γ*; (**c**) variations of concentration distribution *φ*(*ζ*) for thermophoresis parameter *Nt*; (**d**) variations of concentration distribution *φ*(*ζ*) for Brownian motion parameter *Nb*; (**e**) variations of concentration distribution *φ*(*ζ*) for activation energy *E*; (**f**) variations of concentration distribution *φ*(*ζ*) for chemical reaction parameter *σ*.

**Figure 4.** (**a**) Variations of Nusselt number Re−1/2 *<sup>r</sup> Nu* for thermophoresis parameter *Nt*; (**b**) variations of Nusselt number Re−1/2 *<sup>r</sup> Nu* for Brownian motion parameter *Nb*.

**Figure 5.** (**a**) Variations of Sherwood number Re−1/2 *<sup>r</sup> Sh* for thermophoresis parameter *Nt*; (**b**) variations of Sherwood number Re−1/2 *<sup>r</sup> Sh* for Brownian motion parameter *Nb*.

#### **5. Conclusions**

Mixed convective 3D nanoliquid flow by a rotating disk subject to activation energy, magnetohydrodynamics and a binary chemical reaction was studied. Here, the flow field was considered to contain the chemically reacting species. Moreover, the mass transport mechanism was developed via modified Arrhenius function for the activation energy. Activation energy is the minimum quantity of energy needed by reactants to examine a chemical reaction. The source of the activation energy needed to initiate a chemical reaction is typically heat energy from the surroundings. Furthermore, the scientific system obtained was solved numerically via shooting method. A stronger temperature distribution was seen for *Nb* and *Nt*. Both the concentration and temperature display increasing behavior for greater *Ha*. Higher *γ* exhibits a decreasing trend for concentration field. Concentration *φ*(*ζ*) depicts decreasing behavior for larger *σ*. Higher activation energy *E* shows stronger concentration *φ*(*ζ*). Concentration *φ*(*ζ*) displays reverse behavior for *Nb* and *Nt*.

**Funding:** This research was funded by the Deanship of Scientific Research, King Khalid University, Abha, Saudi Arabia under grant number (R.G.P.2./26/40).

**Acknowledgments:** The author extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program under grant number (R.G.P.2./26/40).

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Nomenclature**


