Numerical exploration of radiative heat transfers in a magneto bioconvection Maxwell fluid passing through a spinning disk

Abstract

When applied to a rotating disk, Maxwell nanofluids have a wide range of applications in various fields. They provide increased heat transfer efficiency in cooling systems, which is essential for preserving the ideal operating temperatures of spinning gears like disk brakes and turbines. Furthermore, Maxwell nanofluids in microfluidic devices provide improved fluid manipulation and control, improving performance in applications such as microscale pumps and lab-on-a-chip systems. This article has numerically examined the magneto-bio-convection flow of Maxwell, which includes nanofluid, past a disk surface based on these applications. We present the model equations in PDE format, then shift them to ODEs using the appropriate variables. We utilize the bvp4c approach to obtain numerical solutions to the modeled equations. We also take into account the effects of thermal radiation, heat sources, Brownian motion, thermophoresis, chemical reactivity, and activation energy. We find that a higher stretching/shrinking variable has enhanced the radial velocity profile, while simultaneously diminishing the axial and angular velocity field. While the velocity profiles in the axial direction and redial direction have reduced with larger magnitude of Maxwell fluid variable. The thermal distributions have risen due to higher magnetic, Brownian motion, thermophoresis, heat sources, and thermal radiation components. Thermophoresis, magnetic and thermal radiation, heat sources, and Brownian motion all contribute to an increase in heat transfer rates.

Introduction

A Maxwell fluid is a type of viscoelastic fluid, which means it displays both viscous as well as elastic features. As Maxwell fluid flow, characterized by its behavior governed by the Maxwell model, manifests in a dynamic relationship between elasticity and viscosity, profoundly influencing velocity profiles and heat transfer characteristics. ¹ Abdelsalam et al. ² studied comparatively the rheological features of convective Maxwell flow fluid on a porous sheet. Yasin et al. ³ inspected Maxwell fluid flow on a sheet with slip constraints subject to Hall effects and Darcy’s law. Contrasting to Newtonian fluids, Maxwell fluid exhibits time-dependent responses to applied stresses, transitioning from elastic to viscous behavior over a characteristic relaxation time was explored by Muhammad et al. ⁴ This transition from elastic to viscous behavior results in unique velocity distributions compared to purely viscous fluids. In regions of high stress or strain rates where deformation occurs rapidly, the elastic component dominates, causing the fluid to resist deformation and store energy. Consequently, the velocity distribution deviates from the parabolic profile typically observed in Newtonian fluids, with higher velocities observed at the center of the flow channel and reduced velocities near solid boundaries. This alteration in velocity distribution arises from the elastic properties of the Maxwell fluid, which resist deformation near boundaries, leading to velocity gradients that differ from those predicted by classical fluid dynamics. ⁵ Additionally, the time-dependent nature of Maxwell fluid flow introduces complexities in velocity evolution, with the fluid transitioning between elastic and viscous responses. ⁶ This dynamic behavior can influence flow stability, turbulence characteristics, and the development of velocity profiles over time. Faizan et al. ⁷ conducted research on the Maxwell nanofluid as it traveled across a stretching/shrinking disc in the direction of a stagnation point. Reddy et al. ⁸ investigated the flow characteristics of three distinct nanofluids on a stretching surface in the presence of a non-uniform heat source, as well as the influence of thermal radiation.

Magnetohydrodynamics (MHD) is a multidisciplinary field that examines electrically conducted fluids, with magnetic fields effects. In MHD, the fluid’s motion is affected not only by conventional fluid dynamics but also by electromagnetic forces arising from interactions between the fluid’s motion and the magnetic field. ⁹ These electromagnetic forces significantly impact velocity and temperature distributions within the fluid. For instance, when a conducted fluid flows in a magnetic field, it experiences Lorentz forces, which can alter the fluid’s velocity profile by inducing drag forces and modifying flow patterns. ¹⁰ Furthermore, MHD effects can influence temperature distributions through joule heating, where electrical currents induced in the fluid by the magnetic field dissipate energy as heat. ¹¹ This phenomenon can lead to non-uniform temperature distributions within the fluid, with regions experiencing stronger magnetic fields exhibiting higher temperatures due to increased Joule heating as observed by Bilal et al. ¹² The impressions of MHD on velocity and thermal distributions are essential in several fields, like astrophysics, ¹³ plasma physics, ¹⁴ and engineering, where accurate characterization of fluid behavior under the influence of magnetic fields is critical for augmenting performance and attaining preferred consequences. ¹⁵ Khan et al. ¹⁶ conducted a comprehensive study on the production of entropy and thermal flow analysis for MHD fluid flow on a gyrating cylinder, considering the impacts of Joule heating. Their research investigated the multifaceted association among fluid dynamics, electromagnetic forces, and thermal effects in the presence of a magnetic field. Through their analysis, they demonstrated that the velocity profiles exhibited a notable decline, attributed to the influence of electromagnetic forces induced by magnetic field. Li et al. ¹⁷ conducted an investigation into the numerical solution for heat transfer near a stagnation point of Maxwell nanofluid flow over a porous rotating disk. Reddy et al. ¹⁸ investigated the MHD stagnation point flow of a Williamson hybrid nanofluid over a porous extended sheet.

Heat sources for fluid flow on a surface can arise from various mechanisms, including external heating sources like electrical heaters or solar radiation, as well as internal sources like chemical reactions or frictional heating. These heat sources play a significant role in influencing the thermal transference manner within the fluid. ¹⁹ In regions where the heat source is present, the fluid temperature increases due to absorption of thermal energy, leading to the formation of thermal layers at boundaries characterized by higher temperatures near the surface. ²⁰ This temperature gradient drives convective heat transfer, causing the heated fluid to transfer thermal energy to cooler regions through advection and diffusion. Additionally, the existence of a heat source induces changes in flow patterns and velocity distributions, further impacting the heat transfer process.^21,22 The influences of heat sources on fluid and thermal flows are crucial for various engineering applications.^23,24 Pasha et al. ²⁵ studied the effects of nonlinear radiative magnetized fluid flow with impacts of heat sink, source, and chemical reactivity and observed that thermal distribution have augmented with growth in the strength of source factor. Sajid et al. ²⁶ debated on trihybrid nanofluid flow on a porous sheet using radiation and thermal source factor. Kumar and Sharma ²⁷ studied optimization of entropy for fluid flow on a gyrating disk with impacts of magnetic field and non-uniform thermal source and observed that thermal distribution has augmented with development in thermal source and magnetic factors.

Fluid flow with Brownian motion and thermophoresis encompasses a complex relationship that significantly influence particle dynamics and transport in various systems. Brownian motion, driven by haphazard collisions between suspended particles and surrounding fluid molecules, leads to erratic particle movement that affects dispersion and mixing. At higher temperatures, the kinetic energy of particles increases, intensifying Brownian motion and enhancing particle diffusion rates. ²⁸ Consequently, temperature gradient induces changes in particle distribution and concentration profiles within the fluid. ²⁹ Thermophoresis, meanwhile, involves the directed motion of particles in response to temperature gradients. This phenomenon is influenced by factors like particle mass, temperature gradient magnitude, and fluid properties. ³⁰ In regions of temperature variation, thermophoretic forces act on particles, causing them to migrate toward zones of greater or lesser temperatures depending on their features. ³¹ Temperature gradients also influence fluid properties like thickness and density, which in turn affect the thermophoretic behavior of particles. Furthermore, concentration gradients play a vital role in both Brownian motion and thermophoresis by influencing the spatial distribution and transport of particles within the fluid. ³² Variations in particles’ concentration impact collision frequencies and thus the extent of Brownian motion, while concentration gradients also affect the thermophoretic mobility of particles. Moreover, interactions between Brownian motion, thermophoresis, and fluid flow leads to complex phenomena like particles’ accumulation, deposition, and dispersion, with implications for various industrial and natural processes ranging from nanoparticle transport in biological systems to aerosol dynamics in atmospheric science. ³³ The coupled effects of Brownian motion, thermophoresis, temperature, and concentration gradients is crucial for precisely modeling and calculating particle behavior in complex fluid environments, with implications for fields including environmental engineering, nanotechnology, and biophysics. ³⁴ Shahzad et al. ³⁵ studied impacts of thermophoresis as well as Brownian motion on Darcy-Forchheimer flow of fluid through two disks with Cattaneo-Christov model. Many such study can be seen in References ^36–40.

Fluid flow with chemical reaction is a dynamic process where the movement of the fluid medium interacts intimately with chemical kinetics, resulting in distinct concentration profiles within the flow system of fluid. ⁴¹ As the fluid flows, it transports reactants and products throughout the domain, influencing the distribution of fluid’s concentrations as noticed by Nasir et al. ⁴² Convective transport, driven by fluid velocity and turbulence, accelerates the mixing of reactants chemically, facilitating collisions and promoting reaction rates in regions of high flow as observed by Li et al. ⁴³ Chemical reactions occurring within the fluid alter the concentrations of fluid, leading to spatial variations in species concentrations over time. ⁴⁴ Khan et al. ⁴⁵ studied fluid flow on a movable needle with impacts of chemical reactivity Hall current and magnetic field. Many applications of fluid flow with impressions of chemical reactivity involve, cooling towers, catalysis and fog dispersion, etc. as discussed by He et al. ⁴⁶ Factors like reaction rate constants, reactant concentrations, and temperature profoundly impact the extent and rate of chemical reactions, further shaping the concentration profiles. The interaction between fluid flow and chemical reaction dynamics results in augmentation of concentration profiles as observed by Razaq et al. ⁴⁷ Mathematical modeling combined with physical approaches in case of fluid flow with impacts of chemical reactions, are essential tools to design of efficient and sustainable processes in fields like chemical engineering, environmental science, and materials science. Mishra et al. ⁴⁸ simulated numerically nanofluid flow with significance of chemical reactions.

Nanofluids flows over a revolving disk have variety of uses. Nanofluids have the potential to upsurge the heat transfer rates in the field of thermal management, which makes them useful for cooling systems in spinning machinery such as industrial disk brakes or computer hard drives. Furthermore, nanofluids can enhance the lubricating qualities of lubrication systems, lowering wear and friction on rotating parts like gears and bearings. Furthermore, nanofluids in aerospace applications can help reduce drag and enhance the aerodynamic performance of propellers or spinning disks in aircraft engines. All things considered, a variety of industries have used nanofluids over spinning disks to improve heat transmission, lubrication, and aerodynamics. The current analysis investigates the laminar, time-independent and incompressible flow of Maxwell nanofluid over a gyratory disk surface. With the help of a rotating and stretching disk, we will study the thermal and solutal energy transport in the Von Kármán swirling flow of Maxwell nanofluids in this effort with various flow effects.

Problem formulation

Assume that the laminar and time-independent flow of Maxwell nanofluid over a spinning disk surface. A cylindrical coordinate system is assumed to analyze the flow analysis by assuming the u, v, w along r-, φ-, z-directions. Here, r, φ, and z are the radial, azimuthal, and axial directions. The stretching velocity of the disk is denoted by $u_w=\mathrm{cr}$ where $c$ is the stretching constant. The disk swirls with an angular velocity $v=\mathrm{\Omega }r$ where $\mathrm{\Omega }$ , is the swirling rate (Figure 1). A magnetic field of strength $B_0$ is applied normal (z-direction) to the fluid flow. Furthermore, the effects of thermal radiation, heat source, thermophoresis, Brownian motion, chemical reaction, and Activation energy are taken into consideration. The main equation take the form as appended below^49–51:

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

(1)

$$\begin{array}{c}u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}-\frac{1}{r}{v}^2={\nu }_f\frac{{\partial }^{2}u}{\partial z^2}-\lambda \hfill \\ \left(\begin{array}{c}{u}^2\frac{{\partial }^{2}u}{\partial r^2}+2\mathrm{uw}\frac{{\partial }^{2}u}{\partial r\partial z}+w^2\frac{{\partial }^{2}u}{\partial z^2}\hfill \\ -2\frac{1}{r}\mathrm{uv}\frac{\partial v}{\partial r}-2\frac{1}{r}\mathrm{vw}\frac{\partial v}{\partial z}+\frac{1}{r}{v}^2\frac{\partial u}{\partial r}+\frac{1}{r^2}u{v}^2\hfill \end{array}\right)\hfill \\ -\frac{{\sigma }_f{B}_0^2}{{\rho }_f}(u+\lambda w\frac{\partial u}{\partial z}),\hfill \end{array}$$

(2)

$$\begin{array}{c}u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{1}{r}\mathrm{uv}={\nu }_f\frac{{\partial }^{2}v}{\partial z^2}-\lambda \hfill \\ \left(\begin{array}{c}{u}^2\frac{{\partial }^{2}v}{\partial r^2}+2\mathrm{uw}\frac{{\partial }^{2}v}{\partial r\partial z}+w^2\frac{{\partial }^{2}v}{\partial z^2}+2\frac{1}{r}\mathrm{uv}\frac{\partial u}{\partial r}\hfill \\ +2\frac{1}{r}\mathrm{vw}\frac{\partial u}{\partial z}-\frac{2}{r^2}{u}^{2}v-\frac{1}{r^2}{v}^3+\frac{1}{r^2}{v}^2\frac{\partial v}{\partial r}\hfill \end{array}\right)\hfill \\ -\frac{{\sigma }_f{B}_0^2}{{\rho }_f}(v+\lambda w\frac{\partial v}{\partial z}),\hfill \end{array}$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\frac{k_f}{{\left(\rho C_p\right)}_f}\frac{{\partial }^{2}T}{\partial z^2}+\tau (D_B\frac{\partial T}{\partial z}\frac{\partial C}{\partial z}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2)}\\ -\frac{1}{{\left(\rho C_p\right)}_f}\frac{\partial q_r}{\partial z}+\frac{Q_0}{{\left(\rho C_p\right)}_f}(T-T_{\infty }),\hfill \end{array}$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_B\frac{{\partial }^{2}C}{\partial z^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial z^2}}\\ -k_r^2(C-C_{\infty }){\left(\frac{T}{T_{\infty }}\right)}^n\exp \left(\frac{-E_a}{K^{*}T}\right),\hfill \end{array}$$

(5)

with boundary conditions:

$$\begin{array}{c}{u}_w=\mathrm{cr},w=0,v=\mathrm{\Omega }r,T=T_w,C=C_w\mathrm{at}z=0,\hfill \\ u\to 0,v\to 0,C\to C_{\infty },T\to T_{\infty }\mathrm{as}z\to \infty .\hfill \end{array}$$

(6)

Figure 1.

Graphical view of the flow phenomenon.

Above $q_r$ is described as:

$$q_r=-\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\frac{\partial T}{\partial z}.$$

(7)

The similarity transformations are defined as ⁵¹ :

$$\begin{array}{c}u=F\left(\xi \right)r\mathrm{\Omega },v=G\left(\xi \right)r\mathrm{\Omega },w=H\left(\xi \right)\sqrt{\mathrm{\Omega }{\nu }_f},\hfill \\ \theta \left(\xi \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\xi \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\xi =\sqrt{\frac{\mathrm{\Omega }}{{\nu }_f}}z.\hfill \end{array}$$

(8)

By implementing equation (8) we have from above

$$H\prime \left(\xi \right)+2F\left(\xi \right)=0,$$

(9)

$$\begin{array}{c}F″\left(\xi \right)+G^2\left(\xi \right)-F^2\left(\xi \right)-H\left(\xi \right)F\prime \left(\xi \right)-M(F\left(\xi \right)+\alpha F\prime \left(\xi \right)H\left(\xi \right))\hfill \\ -\alpha (F″\left(\xi \right)H^2\left(\xi \right)+2F\left(\xi \right)F\prime \left(\xi \right)H\left(\xi \right)-2G\left(\xi \right)G\prime \left(\xi \right)H\left(\xi \right))=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}G″\left(\xi \right)-2F\left(\xi \right)G\left(\xi \right)-G\prime \left(\xi \right)H\left(\xi \right)-M(G\left(\xi \right)+\alpha G\prime \left(\xi \right)H\left(\xi \right))\hfill \\ -\alpha (H^2\left(\xi \right)G″\left(\xi \right)+2F\left(\xi \right)H\left(\xi \right)G\prime \left(\xi \right)+2G\left(\xi \right)F\prime \left(\xi \right)H\left(\xi \right))=0,\hfill \end{array}$$

(11)

$$\begin{array}{c}(1+\frac{4}{3}\mathrm{Rd})\theta ″\left(\xi \right)-\mathrm{Pr}H\left(\xi \right)\theta \prime \left(\xi \right)+\mathrm{Pr}\mathrm{Nb}\theta \prime \left(\xi \right)\varphi \prime \left(\xi \right)\hfill \\ +\mathrm{Pr}\mathrm{Nt}\theta {\prime }^2\left(\xi \right)+Q\mathrm{Pr}\theta \left(\xi \right)=0,\hfill \end{array}$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{\varphi ″\left(\xi \right)-\mathrm{ScH}\left(\xi \right)\varphi \prime \left(\xi \right)+\frac{\mathrm{Nt}}{\mathrm{Nb}}\theta ″\left(\xi \right)-\mathrm{Sc}{K}_r{(1+\delta \theta \left(\xi \right))}^n}\\ \exp (-\frac{E}{(1+\delta \theta \left(\xi \right))})\varphi \left(\xi \right)=0,\hfill \end{array}$$

(13)

with constraints at boundary:

$$\begin{array}{c}F(\xi =0)=\gamma ,G(\xi =0)=1,H(\xi =0)=0,\hfill \\ \theta (\xi =0)=1,\varphi (\xi =0)=1,\hfill \\ F(\xi \to \infty )\to 0,G(\xi \to \infty )\to 0,\theta (\xi \to \infty )\to 0,\hfill \\ \varphi (\xi \to \infty )\to 0.\hfill \end{array}$$

(14)

In above equations, $M(=\frac{{\sigma }_f{B}_0^2}{{\rho }_f\mathrm{\Omega }})$ is magnetic factor, $Q(=\frac{Q_0}{{\left(\rho C_p\right)}_f\mathrm{\Omega }})$ is heat source factor, $\mathrm{Rd}(=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k_f{k}^{*}})$ thermal radiation factor, $\alpha (=\lambda \mathrm{\Omega })$ is Maxwell parameter, $\mathrm{Nt}(=\frac{\tau D_T(T_w-T_{\infty })}{{\nu }_f{T}_{\infty }})$ is thermophoresis factor, $\mathrm{Nb}(=\frac{\tau D_B(C_w-C_{\infty })}{{\nu }_f})$ is Brownian motion factor, $\mathrm{Sc}(=\frac{{\nu }_f}{D_B})$ is Schmidt number, $K_r(=\frac{k_r^2}{c})$ is chemical reaction factor, $\mathrm{Pr}(=\frac{{\nu }_f{\left(\rho C_p\right)}_f}{k_f})$ is Prandtl number, $E(=\frac{E_a}{K^{*}{T}_{\infty }})$ is Activation energy factor, and $\delta (=\frac{T_w-T_{\infty }}{T_{\infty }})$ is temperature difference factor.

The Nusselt and Sherwood number are defined as:

$$\sqrt{{\mathrm{Re}}_r}{C}_f=\frac{\sqrt{{\tau }_{\mathrm{rz}}^2+{\tau }_{\theta z}^2}}{{\rho }_f},N{u}_r=-(1+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k_f{k}^{*}})\frac{r{\frac{\partial T}{\partial z}|}_{z=0}}{(T_w-T_{\infty })}.$$

(15)

$$S{h}_r=-\frac{r{\frac{\partial C}{\partial z}|}_{z=0}}{(C_w-C_{\infty })}.$$

(16)

Using equation (8), (15) and (16) are reduced as:

$$\sqrt{{\mathrm{Re}}_r}\mathrm{Cf}=\sqrt{f{\prime }^2(0)+g{\prime }^2(0)},\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}=-(1+\frac{4}{3}\mathrm{Rd})\theta \prime (\xi =0),$$

(17)

$$\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}=-\varphi \prime (\xi =0).$$

(18)

Here ${\mathrm{Re}}_r(=\frac{r^2\mathrm{\Omega }}{{\nu }_f})$ is the local Reynolds number.

Numerical solution

To solve the modeled equations by means of numerical method, we have chosen bvp4c MATLAB built-in function. This method is applicable to solve boundary value problems arising in physical and engineering fields. To ensure the accuracy of the bvp4c, error tolerance of 10⁻⁴ is chosen. To apply this method, let us assume that:

$$\begin{array}{cc}\multicolumn{1}{c}{H\left(\xi \right)}& ={\mathrm{\Upsilon }}_1,H\prime \left(\xi \right)={\mathrm{\Upsilon }}_1\prime ,\hfill \\ \multicolumn{1}{c}{F\left(\xi \right)}& ={\mathrm{\Upsilon }}_2,F\prime \left(\xi \right)={\mathrm{\Upsilon }}_3,F″\left(\xi \right)={\mathrm{\Upsilon }}_3\prime ,\hfill \\ \multicolumn{1}{c}{G\left(\xi \right)}& ={\mathrm{\Upsilon }}_4,G\prime \left(\xi \right)={\mathrm{\Upsilon }}_5,G″\left(\xi \right)={\mathrm{\Upsilon }}_5\prime ,\hfill \\ \multicolumn{1}{c}{\theta \left(\xi \right)}& ={\mathrm{\Upsilon }}_6,\theta \prime \left(\xi \right)={\mathrm{\Upsilon }}_7,\theta ″\left(\xi \right)={\mathrm{\Upsilon }}_7\prime ,\hfill \\ \multicolumn{1}{c}{\varphi \left(\xi \right)}& ={\mathrm{\Upsilon }}_8,\varphi \prime \left(\xi \right)={\mathrm{\Upsilon }}_9,\varphi ″\left(\xi \right)={\mathrm{\Upsilon }}_9\prime .\hfill \end{array}$$

(19)

Then

$${\mathrm{\Upsilon }}_1\prime =-2{\mathrm{\Upsilon }}_1,$$

(20)

$${\mathrm{\Upsilon }}_3\prime =-\frac{\left\{\begin{array}{c}-{\left({\mathrm{\Upsilon }}_2\right)}^2+{\left({\mathrm{\Upsilon }}_4\right)}^2-M({\mathrm{\Upsilon }}_4+\alpha {\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_3)-\hfill \\ \multicolumn{1}{c}{{\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_3-2\alpha {\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_2{\mathrm{\Upsilon }}_3+2\alpha {\mathrm{\Upsilon }}_4{\mathrm{\Upsilon }}_5{\mathrm{\Upsilon }}_2}\end{array}\right\}}{(1-\alpha {\left({\mathrm{\Upsilon }}_1\right)}^2)},$$

(21)

$${\mathrm{\Upsilon }}_5\prime =-\frac{\left\{\begin{array}{c}-2{\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_2-M({\mathrm{\Upsilon }}_4+\alpha {\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_5)-{\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_5\hfill \\ \multicolumn{1}{c}{-2\alpha {\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_2{\mathrm{\Upsilon }}_5-2\alpha {\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_3{\mathrm{\Upsilon }}_4}\end{array}\right\}}{(1-\alpha {\left({\mathrm{\Upsilon }}_1\right)}^2)},$$

(22)

$${\mathrm{\Upsilon }}_7\prime =-\frac{\{-\mathrm{Pr}{\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_7+\mathrm{Pr}\mathrm{Nb}{\mathrm{\Upsilon }}_9{\mathrm{\Upsilon }}_7+\mathrm{Pr}\mathrm{Nt}{\left({\mathrm{\Upsilon }}_7\right)}^2+Q\mathrm{Pr}{\mathrm{\Upsilon }}_6\}}{(1+4/3\mathrm{Rd})},$$

(23)

$$\begin{array}{c}{\mathrm{\Upsilon }}_9\prime =\hfill \\ -\{-\mathrm{Sc}{\mathrm{\Upsilon }}_1{\mathrm{\Upsilon }}_9+\frac{\mathrm{Nt}}{\mathrm{Nb}}{\mathrm{\Upsilon }}_7\prime -\mathrm{Sc}{K}_r{(1+\delta {\mathrm{\Upsilon }}_6)}^n\exp (-\frac{E}{(1+\delta {\mathrm{\Upsilon }}_6)}){\mathrm{\Upsilon }}_8\},\hfill \end{array}$$

(24)

with boundary conditions:

$$\begin{array}{c}{\mathrm{\Upsilon }}_1-0,{\mathrm{\Upsilon }}_2-\gamma ,{\mathrm{\Upsilon }}_4-1,{\mathrm{\Upsilon }}_6-1,{\mathrm{\Upsilon }}_8-1,\hfill \\ {\mathrm{\Upsilon }}_2-0,{\mathrm{\Upsilon }}_4-0,{\mathrm{\Upsilon }}_6-0,{\mathrm{\Upsilon }}_8-0.\hfill \end{array}$$

(25)

The applied numerical technique has the following advantages:

• • A variety of BVPs, including stiff and highly nonlinear problems, may be solved using the robust solver bvp4c. It makes use of collocation and adaptive mesh refinement techniques to guarantee precise answers for a variety of issue kinds.

• • In comparison to previous approaches, the bvp4c solver requires comparatively few function evaluations to solve BVPs, indicating its computational efficiency. Because of its efficiency, it may be used to solve problems involving a lot of equations or intricate geometries.

• • By employing fourth-order collocation techniques, bvp4c offers very accurate solutions to BVPs. It uses error control techniques and automatically modifies the mesh density to make sure the calculated solution satisfies the required accuracy standards.

• • bvp4c is a versatile and adaptable tool that can handle mixed boundary conditions, discontinuities, and singularities in systems of nonlinear differential equations.

Validation

To insure the accuracy, the present results are compared with Sparrow and Gregg ⁵² for varying $\mathrm{Pr}$ when $M=0$ , $\alpha =0$ , $Q=0$ , $\mathrm{Rd}=0$ , $\mathrm{Nt}=0$ , $\mathrm{Nb}=0$ , $\mathrm{Sc}=0$ , $K_r=0$ , and $E=0$ . The present results are displayed in Table 1. We can see from this table that the present and past results are in great agreement which ensure the accuracy of the applied numerical method.

Table 1.

Validation of present results with published results by Sparrow and Gregg ⁵² for varying $\mathrm{Pr}$ when $M=0$ , $\alpha =0$ , $Q=0$ , $\mathrm{Rd}=0$ , $\mathrm{Nt}=0$ , $\mathrm{Nb}=0$ , $\mathrm{Sc}=0$ , $K_r=0$ , and $E=0$ .

$\mathrm{Pr}$	Sparrow and Gregg 52	Present
1.0	0.3925	0.3926097547
10.0	1.1341	1.1341001232
100.0	2.6871	2.6871110345

Discussion of results

This section shows the obtained results from the present analysis and their physical explanation. The obtained results are presented in Figures 2 to 17 and Tables 2 and 3. The default values of factor used in the study are selected as $M=0.5$ , $\gamma =0.1$ , $\alpha =0.1$ , $Q=0.5$ , $\mathrm{Rd}=0.2$ , $\mathrm{Nt}=0.1$ , $\mathrm{Nb}=0.1$ , $\mathrm{Pr}=0.72$ , $\mathrm{Sc}=1.0$ , $K_r=0.1$ , $E=1.0$ , and $\delta =1.0$ . Figures 2 and 3 show the effect of stretching velocity factor ( $\gamma$ ) on radial velocity ( $F\left(\xi \right)$ ) and Azimuthal velocity ( $G\left(\xi \right)$ ) profiles, respectively. From these Figures, we can see that the greater $\gamma$ enhances ( $F\left(\xi \right)$ ) and decreases the velocity profiles in angular direction. The reason is that the stretching rate heightens in radial direction via higher ratio factor and the swirling rate reduces in angular direction due to higher ratio factor. Figures 4 and 5 demonstrate the influence of Maxwell parameter ( $\alpha$ ) on radial velocity ( $F\left(\xi \right)$ ) and Azimuthal velocity ( $G\left(\xi \right)$ ) profiles, respectively. From these Figures, we can see that the greater $\alpha$ reduces the velocity panels in radial and angular directions. These results have scientific justification since the fluid grows increasingly solid-like with larger values of $\alpha$ due to the significant relaxation time value in the fluid. As a result, there is less movement that is fluid in every direction. Figures 6 to 8 show the effect ( $M$ ) on ( $F\left(\xi \right)$ ), ( $G\left(\xi \right)$ ), and $\theta \left(\xi \right)$ profiles. We can see from these Figures that the greater $M$ lessens the velocity panels in radial and angular directions. Further, from Figure 8, we can see that the greater values of $M$ increases the thermal distribution. The Lorentz force, which works perpendicular to both the velocity and magnetic field vectors, is responsible for the drop in velocity profiles in both radial and tangential directions caused by the magnetic factor. The fluid flow is resisted by this force, which causes the velocity components in those directions to drop. Similarly, the higher Lorentz force enhances the heat transmission rate at the sheet surface which results higher temperature profile. The coefficient of heat production, denoted by $Q$ has a significant impact on the dimensionless temperature profile of the fluid, denoted by $\theta \left(\xi \right)$ . According to Figure 9, it is obvious that as $Q$ increases, so does the temperature $\theta \left(\xi \right)$ . The thermal layer width at boarder line will grow with upsurge in the fluid temperature $\theta \left(\xi \right)$ . The external heat source has a major influence on thermal gradient of fluid, which ultimately causes an upsurge in $\theta \left(\xi \right)$ as well as the thermal state of the fluid. The width of thermal layer at borderline grows to a greater extent when a substantial quantity of heat energy is created by the fluid particles themselves. Figure 10 illustrates the temperature fluctuations caused by radiation $\mathrm{Rd}$ . It is observed that the $\theta \left(\xi \right)$ of the Maxwell nanofluid climb as the amount of $\mathrm{Rd}$ heightens. As $\mathrm{Rd}$ shows that $\mathrm{Rd}\propto \frac{1}{k^{*}}$ so increases the amount of $\mathrm{Rd}$ means the reduction in $k^{*}$ causes more heat is transported to working fluid. In addition to this, temperature and thermal boundary layer thickness both upsurge as a result of larger values of the thermal radiation parameter. When the thermophoresis constraint $\mathrm{Nt}$ growths, the $\theta \left(\xi \right)$ also augment as shown in Figure 11. The reason behind this fact is that the fluid’s thermal resistance tends to increase due to the thermophoretic distribution, which causes the non-dimensional fluid temperature to rise. Specifically, the temperature difference is the root cause of this phenomenon. As the thermophoresis constraint $\mathrm{Nt}$ upsurges, particles in the fluid migrate from the colder area to the warmer one which causes the temperature profile to climb. Here, the thermophoresis constraint $\mathrm{Nt}$ is directly related to the temperature distribution. The nanoparticle concentration profile $\varphi \left(\xi \right)$ is a function of $\mathrm{Nt}$ is shown in Figure 12. When $\mathrm{Nt}$ is raised, then the concentration field is upswing. Physically $\mathrm{Nt}$ is a key factor in the process of mass transmission, therefore $\mathrm{Nt}$ is of vital importance. Nanoparticles are more likely to be displaced from hotter zone to colder one when $\mathrm{Nt}$ is improved, leading to a rise in boundary layer width. Figure 13 portrays the effects of $\mathrm{Nb}$ on $\theta \left(\xi \right)$ . From the Figure it is seen that the $\theta \left(\xi \right)$ profiles get more pronounced as the values of $\mathrm{Nb}$ amplified. As $\mathrm{Nb}$ enhances, the solid nanoparticles’ irregular mobility also intensifies. The dynamic process strengthens the thermal layer by converting the interior energy of nanoparticles into heat and kinetic energy. Thus, as seen in Figure 16, the thermal properties rise as the $\mathrm{Nb}$ factor upturns. Figure 14 shows the effect of $\mathrm{Nb}$ on $\varphi \left(\xi \right)$ . As the value of $\mathrm{Nb}$ amplified, the profile $\varphi \left(\xi \right)$ declines. A positive value of $\mathrm{Nb}$ boost the nanoparticles’ random mobility, which ultimately enhances fluid mixing as a result the solutal concentration, becomes weakens and reduces mass diffusion. Consequently heightened $\mathrm{Nb}$ leads to increased random motion among nanoparticles. This enhanced mixing potentially yields a more uniform dispersion of nanoparticles and reduced concentration gradients, thereby thinning the concentration boundary layer and reducing its concentration. The behavior of $(K_r>0)$ on concentration profiles $\varphi \left(\xi \right)$ is illustrated in Figure 15. As the chemical reaction $K_r$ grows, the reactants are utilized leading to a reduction in their concentration within the boundary. As shown in Figure 16, the activation energy $E$ effect on $\varphi \left(\xi \right)$ may be imagined. The graph clearly shows that when $E$ is made larger, the concentration profile rises. The Arrhenius function degrades as $E$ increases exponentially, leading to an uptick in the concentration field $\varphi \left(\xi \right)$ and the encouragement of the generative chemical process. Physically when a chemical reaction takes place at low temperatures and high activation energies, the reaction rate decreases, making the process go more slowly. The effect of $\mathrm{Sc}$ on the concentration profile $\varphi \left(\xi \right)$ is revealed in Figure 17. This displays that upsurge in $\mathrm{Sc}$ declines $\varphi \left(\xi \right)$ . The reason behind this is that the Schmidt number $(\mathrm{Sc}=\frac{{\nu }_f}{D_B})$ is a measure of the ratio between momentum diffusivity and mass diffusivity. There is a negative relationship between mass diffusivity and momentum diffusivity, which results in reduced scalar diffusivity, as the Schmidt number $\mathrm{Sc}$ enhanced.

Figure 2.

Impact of $\gamma$ on $F\left(\xi \right)$ .

Figure 3.

Impact of $\gamma$ on $G\left(\xi \right)$ .

Figure 4.

Impact of $\alpha$ on $F\left(\xi \right)$ .

Figure 5.

Impact of $\alpha$ on $G\left(\xi \right)$ .

Figure 6.

Impact of $M$ on $F\left(\xi \right)$ .

Figure 7.

Impact of $M$ on $G\left(\xi \right)$ .

Figure 8.

Impact of $M$ on $\theta \left(\xi \right)$ .

Figure 9.

Impact of $Q$ on $\theta \left(\xi \right)$ .

Figure 10.

Impact of $\mathrm{Rd}$ on $\theta \left(\xi \right)$ .

Figure 11.

Impact of $\mathrm{Nt}$ on $\theta \left(\xi \right)$ .

Figure 12.

Impact of $\mathrm{Nt}$ on $\varphi \left(\xi \right)$ .

Figure 13.

Impact of $\mathrm{Nb}$ on $\theta \left(\xi \right)$ .

Figure 14.

Impact of $\mathrm{Nb}$ on $\varphi \left(\xi \right)$ .

Figure 15.

Impact of $K_r$ on $\varphi \left(\xi \right)$ .

Figure 16.

Impact of $E$ on $\varphi \left(\xi \right)$ .

Figure 17.

Impact of $\mathrm{Sc}$ on $\varphi \left(\xi \right)$ .

Table 2.

Effects of $\alpha ,\gamma$ , and $M$ on $\sqrt{{\mathrm{Re}}_r}\mathrm{Cf}$ .

$\alpha$	$\gamma$	$M$	$\sqrt{{\mathrm{Re}}_r}\mathrm{Cf}$
0.4			1.1031
0.9			1.2343
1.3			1.4019
1.9			1.5068
	0.2		1.5982
	0.7		1.6423
	1.1		1.7421
	1.5		1.8975
		0.8	1.9889
		1.4	2.1431
		1.7	2.1982
		2.1	2.3364

Table 3.

Effects of $\mathrm{Rd}$ , $Q$ , $M$ , $\mathrm{Nb}$ , and $\mathrm{Nt}$ on $\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}$ .

$\mathrm{Rd}$	$Q$	$M$	$\mathrm{Nb}$	$\mathrm{Nt}$	$\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}$
0.1					1.5322
0.2					1.5436
0.3					1.5525
0.4					1.5697
	0.1				1.2864
	0.2				1.3075
	0.3				1.3254
	0.4				1.3443
		0.1			1.6974
		0.2			1.7032
		0.3			1.7164
		0.4			1.7256
			0.1		1.6356
			0.2		1.6132
			0.3		1.5953
			0.4		1.5785
				0.1	1.1853
				0.2	1.2542
				0.3	1.3246
				0.4	1.3800

Figures 18 to 20 show the surface plot of $\sqrt{{\mathrm{Re}}_r}C{f}_r$ , $\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}$ , and $\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}$ . It is revealed that the enhancement against the rising values of $\alpha$ and $\gamma$ has been elaborated in Figure 18. Figure 19 shows that when $Q$ and $\mathrm{Rd}$ increases results enhanced with $\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}$ . $\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}$ grows rapidly with the larger values of $K_r$ and $\delta$ as seen in Figure 20.

Figure 18.

Impact of $\alpha ,\gamma$ on ${\mathrm{Re}}_r^{1/2}C{f}_r$ .

Figure 19.

Impact of $Q,\mathrm{Rd}$ on ${\mathrm{Re}}_r^{-1/2}N{u}_r$ .

Figure 20.

Impact of $\delta ,K_r$ on ${\mathrm{Re}}_r^{-1/2}S{h}_r$ .

Table 2 reveals the role of $\gamma$ , $\alpha$ , and M on $\sqrt{{\mathrm{Re}}_r}C{f}_r$ . It is noted that the greater values of $\gamma$ , $\alpha$ , and M enhances the surface drag. Table 3 shows the effects of $\mathrm{Rd}$ , $Q$ , $M$ , $\mathrm{Nb}$ , and $\mathrm{Nt}$ on $\frac{N{u}_r}{\sqrt{{\mathrm{Re}}_r}}$ . From this Table, we have observed that greater values of $\mathrm{Rd}$ , $Q$ , $M$ , and $\mathrm{Nt}$ enhances the rate of heat transfer at the sheet surface while the greater values of $\mathrm{Nb}$ reduces it. Table 4 shows the effects of $\mathrm{Nb}$ and $\mathrm{Nt}$ on $\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}$ . From this Table, we observed that the higher values of $\mathrm{Nb}$ increases the mass transfer rate and the higher values of $\mathrm{Nt}$ reduce the mass transfer rate.

Table 4.

Effects of $\mathrm{Nb}$ and $\mathrm{Nt}$ on $\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}$ .

$\mathrm{Nb}$	$\mathrm{Nt}$	$\delta$	$K_r$	$E$	$\frac{S{h}_r}{\sqrt{{\mathrm{Re}}_r}}$
0.1					1.6424
0.2					1.6633
0.3					1.6864
0.4					1.7035
	0.1				1.8675
	0.2				1.8589
	0.3				1.8064
	0.4				1.7532
		0.1			2.6543
		0.4			2.7456
		0.7			2.8012
		1.0			2.8489
			1.0		2.9235
			1.2		2.9476
			1.4		2.9645
			1.6		2.9865
				1.0	2.1043
				1.3	2.1224
				1.6	2.1654
				1.9	2.1834

Conclusions

In this effort, we have examined the laminar, time-independent, and incompressible flow of Maxwell nanofluid over a rotating disk surface. The problem is modeled in a cylindrical coordinate system. The model equations are formed as PDEs which then transformed into ODEs by using suitable similarity variables. The modeled equations are solved numerically by using bvp4c MATLAB built-in function. Furthermore, the effects of thermal radiation, heat source, thermophoresis, Brownian motion, chemical reaction, and Activation energy are taken into consideration. The final remarks are listed as:

• • It is found that the greater ratio factor has enhanced the velocity profile in radial direction while decreased the velocity profiles in angular direction.

• • The greater Maxwell factor has reduced the velocity profile in radial and angular directions.

• • The greater magnetic factor has reduced the velocity profile in radial and angular directions.

• • The greater magnetic, Brownian motion, thermophoresis, heat source, and thermal radiation factors have increased the thermal distributions.

• • The greater Schmidt number, chemical reaction, and Brownian motion factors have reduced the concentration profiles while the greater thermophoresis factor and activation energy factor have increased the concentration profiles.

• • The heat transfer rate is increased by the thermal radiation, heat source, magnetic, Brownian motion, and thermophoresis factors.

• • The higher Brownian motion factor has increased the mass transfer rate and the higher thermophoresis factor has reduced the mass transfer rate.