Role of Chemically Magnetized Nanofluid Flow for Energy Transition over a Porous Stretching Pipe with Heat Generation/Absorption and Its Stability

Abstract

The laminar movement in an expanding and contracting permeable pipe or surface has recently attracted the attention of many scholars owing to its application in engineering and biological processes. The objective of the current study is to examine the influence of chemical processes on magnetized nanofluid flow over extending or shrinking permeable pipes with a heat reservoir. The flow equations are renovated into first ODEs by introducing the new variable and then numerically solved by RK4 with a shooting procedure. The effect of emerging factors on the flow features is observed using graphs and elaborated in detail. From the analysis, the temperature is raised when the heat source is increased in both cases of wall expansion or contraction but declines in the case of heat sinks. In the case of a heat source, the temperature rises as the Hartmann and Prandtl numbers are enhanced, but in the case of a heat sink, the temperature falls. In the presence of heat sinks and injections, when the thermophoresis factor is increased, the concentration of nanoparticles is increased in both wall expansion and contractions. In both situations of wall extension or contraction, along with injection, the concentration of nanoparticles is a decreasing function of $Nb$, while the concentration of nanoparticles is an increasing function in the case of a heat source. Additionally, for the confirmation of the RK4 code, the total average square residue error and average square residue error are also presented. For the stability analysis, the current work is compared with published work, and excellent agreement is established. The novelty of the present study is to investigate the effect of chemical reaction on magnetized nanofluid flow over extending and shrinking porous pipes with heat generation and absorption.

1. Introduction

The establishment of nanofluid innovation is a crucial field of research in physics, industry, mathematics, and chemical science. For the majority of applications with a practical purpose, designers and scientists work to successfully communicate sufficient knowledge of the heat transfer mechanism in nanofluids. Nanofluids are essential in a variety of applications, such as heat exchangers, freezers, hybrid-driven motors, food processing, and chips. The term “nanofluid” was originally used by Choi et al. [1]. Nanoparticles are now the subject of major scientific investigation because of their many applications in biological, optical, and electrical areas. They may be discovered in nanometals [2,3,4,5,6,7] and graphite, as well as oxides, carbides, nitrides, and metals such as copper and aluminum. Buongiorno investigated the diffusion of heat in nanofluids (NFs) [8]. Moreover, Buongiorno [9] concluded that Brownian and thermophoresis diffusion would be important when turbulent impacts are absent. The unsteady BLF over a flowing sheet was examined by Rosca et al. [10] using Buongiorno’s model. Kuznetsov et al. [11] studied the significance of bioconvection on MHD tangent hyperbolic nanofluid flow of irregular thickness across a slender elastic surface. The micropolar dusty fluid with coriolis force effects on dynamics of MHD rotating fluid when lorentz force is significant by Lou et al. [12]. The magnetic BLF toward an extended sheet, including nanoparticles, was examined analytically by Mustafa et al. [13] using HAM. Alsaedi et al. [14] have achieved an analytical solution for the SPF when the heat source passes via a convective sheet. Chamkha et al. [15] studied the convective BLF of NFs approaching a vertical sheet. Malvandi et al. [16] discussed NFFs (nanofluid flows) employing a vertical pipe, utilizing Buongiomo’s model (BM). Across a vertical tube, Akbari et al. [17] examined a fully evolved NFF. Ellahi [18] found a mathematical solution for the magnetic and changing viscosity viscous fluid within a pipe. An MHD NFF, incorporating the slip impact at the border, was seen by Uddin et al. [19] across the extended sheet. To explore the BLF traveling via a vertical stretched channel, Xu et al. [20] and Malik et al. [21] studied the BM and Casson nanofluids, respectively. The nonlinear movements of axisymmetric ternary hybrid nanofluids in a thermally radiated expanding or contracting permeable Darcy Walls with different shapes and densities using simple linear regression by Raju et al. [22]. Further examined were the Brownian and thermophoresis variables. Ramesh et al. [23] developed a flow of hybrid CNTs past a rotating sphere subjected to thermal radiation and thermophoretic particle deposition. Presently, Srinivas et al. [24] have researched the MHD flow of NFs in a permeability-expanding pipe. Hedayati et al. [25] very recently investigated titania water-related nanofluids across a ferromagnetic vertical cylindrical conduit. Malvandi et al. [26] also looked at a alumina water-based nanofluid across a vertical channel using the Lorentz force effect. Chemical engineering, metallurgy, radioactive nuclear safety, photovoltaic collectors, and other nanotechnology and scientific fields have been shown to significantly impact heat transport [27,28,29]. The impact of chemical efficiency on blood flow using Walter’s B model across a tapered artery was taken into consideration by Nadeem et al. [30]. Hayat et al.’s [31] examined Maxwell fluids’ biochemical processes. Abdul et al. [32] examined the chemical reactions taking place over a stretchy sheet that was saturated with nanofluids, utilizing the BLF. Kameswaran et al. [33] investigated uniform and heterogeneous chemical reactions across a porous stretching surface. The MHD laminar BLF, with a slip impact over a stretched permeable surface with chemical reactions, was numerically explored by Uddin et al. [34]. For free convection flow (FCF) across a horizontal surface containing nanofluids, an analytical solution was discovered [35]. Recently, Srinivas et al. [36] examined a viscoelastic fluid across a chemically stretched pipe. Uddin et al. [37] investigated the MHD FCBLF with NF chemical reactivity via a vertical surface. The laminar movement in an escalating or shrinking permeable pipe or surface has recently attracted the curiosity of many scholars owing to its application in engineering and biological developments, for example, in the transportation of natural fluids through stretching vessels, the synchronous rhythm of absorbent diaphragms, the respirational system, and the deterioration of the red-hot sheet in rock-solid engines [38,39,40]. The thermal flux and heat transfer have been studied by many researchers (see [41,42,43,44,45]). The unsteady fluid flow over semi-infinite stretching pipes with a heat source was examined by Boutros et al. [46] via the Lie group approach. Zeeshan [47] investigated the energy activation analysis for Maxwell fluid comprising molybdenum disulfide and graphene nanoparticles in engine-based fluid, enclosing the effect of isothermal wall temperature. Rasheed et al. [48] scrutinized the movement of micropolar fluid over an extended surface with thermal radiation influence. Zeeshan et al. [49] obtained the numerical solution for entropy generation by scrutinizing the second-order nanofluid’s thin film flow with error and stability analyses. Recently, the influence of heat and transfer analyses of nanofluids over a horizontal surface with thermal and magnetic field effects was investigated by Zeeshan et al. [50]. Raza et al. [51] investigated the movement of MHD Casson liquid over a porous sheet with extended and stationary walls. Khan et al. [52] obtained the exact solution of a Casson model movement situated with dust particles through a stretching surface enclosing the Lorentz forces. Similarly, Chabani et al. [53] numerically investigated a magnetized hybrid nanofluid over a porous trapezoidal inclusion.

The above study reveals that no effort towards the influence of chemical processes on magnetohydrodynamic NFFs over a stretchable permeable pipe enclosing the effect of a heat reservoir has been scrutinized so far. Such deliberation has a significant value in science and engineering study, including chips, refrigerators, hybrid powered motors, food improvement, heat exchangers, and so on. Keeping the overhead observations in view, the purpose of the recent study is to observe the inspiration of chemical processes on magnetized nanofluid flows over extending or shrinking permeable pipes with a heat reservoir. The flow characteristics are renovated into first ODEs by introducing the new variable and then numerically elucidated using the Runge–Kutta fourth-order method with a shooting technique [54]. The effect of emerging parameters on the flow features is observed using graphs and elaborated in detail. Additionally, the confirmation of RK4 is compared with HAM. For the stability analysis, the current work is likened to the available literature, and exceptional correlation is established.

2. Formulation of the Problem

The unsteady NFF of an electrically conducting fluid in a semi-infinite length over an equally porous pipe is considered. The pipe has at a radius, $\widehat{ a}\left(t\right)$. The wall is the function of time $t$ that is expanding and shrinking with time. Figure 1 demonstrates the geometry of the present problem, in which its origin is taken at the center of the pipe, where $z$-axis is parallel to the wall and $r$ is normal to the wall.

The central equations are under these presumptions [16,17,18,19,20,21,22]:

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

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial z}+v\frac{\partial u}{\partial r}=-\frac{1}{{\rho }_f}\frac{\partial p}{\partial z}+v\left[\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{\partial }^{2}u}{\partial z^2}\right]-\frac{\sigma B_0^2}{{\rho }_f}u$$

(2)

$$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial z}+v\frac{\partial v}{\partial r}=-\frac{1}{{\rho }_f}\frac{\partial p}{\partial r}+v\left[\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{\partial }^{2}v}{\partial z^2}-\frac{v}{r^2}\right]$$

(3)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial z}+v\frac{\partial T}{\partial r}=\beta \left[\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}\right]+\tau \left[D_B\left(\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}+\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}\right)+\frac{D_T}{T_m}\left\{{\left(\frac{\partial T}{\partial r}\right)}^2+{\left(\frac{\partial T}{\partial z}\right)}^2\right\}\right]+\frac{Q_0}{{\left(\rho C_p\right)}_f}\left(T-T_0\right)$$

(4)

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial z}+v\frac{\partial C}{\partial r}=D_B\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{{\partial }^{2}C}{\partial z^2}\right)+\frac{D_T}{T_m}\left\{\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}\right\}-kC$$

(5)

where $z$ and $r$ represent directions, and $u$, $v$ represent the components of the velocity. The fluid density is ${\rho }_f$, thermal diffusivity is $\beta$, time is $t$, kinematic viscosity is $\nu$, dimensional pressure is $p$, magnetic field strength is $B_0$, and electrical conductivity is $\sigma$, $C_p$ represents the specific heat, the average (mean) temperature is $T_m$, Brownian and thermophoretic diffusion coefficients are $D_B$ and $D_T$, respectively, and $T$ and $C$ stand for temperature and nanoparticle concentration, respectively; $\tau =\frac{{\left(\rho C_p\right)}_p}{\left(\rho C_p\right)f}$, while $k$ represents the first-order chemical reaction rate ($k<0$ for a generative reaction (GR), $k>0$ for a destructive reaction (DR), and $k=0$for no reaction).

The pertinent boundary constraints are

$$u=0, v=-v_w=-A\widehat{a}, T=T_w, C=C_w r=a\left(t\right)$$

(6)

$$\frac{\partial u}{\partial r}=0, v=0, \frac{\partial T}{\partial r}=0, \frac{\partial C}{\partial r}=0 \mathrm{at} r=0$$

(7)

$$u=0, v=0 \mathrm{at} z=0.$$

(8)

The wall permeability is represented by the A (injection/suction coefficient) in Equation (6), where$T_w$, $C_w$ are the wall’s temperature and concentration, respectively.

Add a stream function that meets the continuity requirement in Equation (1)

$$\psi =\nu zf\left(\eta ,t\right)$$

(9)

where the dimensionless radial location is represented by $\eta =\frac{r}{a}$. You can write the radial and axial velocity components as

$$\begin{gathered} u=\frac{1}{r}\frac{\partial \psi }{\partial r}=\frac{\nu z{f}_{\eta }\left(\eta ,t\right)}{a^2\eta } \\ v=\frac{1}{r}\frac{\partial \psi }{\partial z}=-\frac{\nu f_{\eta }\left(\eta ,t\right)}{a\eta } \end{gathered}$$

(10)

One may obtain this by replacing Equation (10) with Equations (2) and (3) after removing pressure.

$$\begin{gathered} {\eta }^2{f}_{\eta \eta \eta \eta }+\left(\alpha {\eta }^3-2\eta \right)f_{\eta \eta \eta }+\left(\alpha {\eta }^2+3\right)f_{\eta \eta }-\left(\alpha \eta +\frac{3}{\eta }\right)f_{\eta }+f_{\eta }^2-\eta f_{\eta }{f}_{\eta \eta }+\eta f{f}_{\eta \eta \eta } \\ -3f{f}_{\eta \eta }+\frac{3}{\eta }f{f}_{\eta }-M^2\left({\eta }^2{f}_{\eta \eta }-\eta f_{\eta }\right)-\frac{a^2}{\nu }{(\frac{f_{\eta }}{\eta })}_{\eta t}{\eta }^3=0. \end{gathered}$$

(11)

where $M=\frac{\sqrt{\sigma }{B}_{0}a}{\sqrt{\mu }}$ is the Hartmann number (HN), and $\mu$ is the viscosity. The non-dimensional wall dilation rate is defined as $\alpha =\frac{a\widehat{a}}{\nu }$, being positive for extension and negative for shrinkage. The associated boundary constraints from Equations (6) through (8) convert into

$$\begin{gathered} f\left(0,t\right)=0, f\left(1,t\right)=R, f_{\eta }\left(1,t\right)=0, \\ \underset{\eta \to 0}{\lim }\frac{\partial }{\partial \eta }(\frac{1}{\eta }\frac{\partial f\left(\eta ,t\right)}{\partial \eta })=0 \end{gathered}$$

(12)

$R=\frac{a{\nu }_w}{\nu }=A\alpha$ defines the permeation Reynolds number (PRN) as follows: $R$ is positive for inoculation but negative for suction, as you can see. By separately applying the transformations outlined by Si et al. [9] and Uchida and Aoki [7], a comparable solution with regard to both time and space may be created. It follows that ${\left(\frac{f_{\eta }}{\eta }\right)}_{\eta t}=0$for constant $\alpha$ and $f=f\left(\eta \right).$ To fulfil this requirement, the expansion ratio’s $\alpha$ value must be determined by the beginning value.

$$\alpha =\frac{a\widehat{a}}{v}=\frac{a_0{\widehat{a}}_0}{v}=constant \mathrm{or} \frac{\widehat{a_0}}{\widehat{a}}=\frac{a}{a_0}$$

(13)

where $a_0$, ${\widehat{a}}_0$ define the initial radius and growth rate correspondingly. By integrating Equation (13) with regard to time, the temporal similarity transformation may be accomplished. The outcome is

$$\frac{a}{a_0}=\sqrt{1+2v\alpha t{a}_0^{-2}}.$$

(14)

It is demonstrated that the injection coefficient $A$ is constant since $v_w=A\widehat{a}$ may be used to calculate an equation for the injection velocity. It is evident from Equations (13) and (14),

$$\frac{{\widehat{a}}_0}{\dot{a}}=\frac{v_w\left(0\right)}{v_w\left(t\right)}=\sqrt{1+2v\alpha t{a}_0^{-2}}.$$

(15)

$$\begin{gathered} {\eta }^3{f}^{⁗}+\alpha \left({\eta }^4{f}^{‴}+{\eta }^3{f}^{″}-{\eta }^{2}f\right)-2{\eta }^2{f}^{‴}+3\eta f^{″}-3f^{\prime }+\eta f^{\prime }{}^2-3\eta f{f}^{″}+3f{f}^{\prime } \\ +3{\eta }^{2}f{f}^{‴}-{\eta }^2{f}^{\prime }{f}^{″}-M^2\left({\eta }^3{f}^{″}-{\eta }^2{f}^{\prime }\right)=0. \end{gathered}$$

(16)

These boundary constraints are

$$f\left(0, t\right)=0, f^{\prime }\left(1, t\right)=0, f\left(1\right)=R, \underset{\eta \to 0}{\lim }{\left(\frac{f^{\prime }}{\eta }\right)}^{\prime }=0.$$

(17)

One can normalize Equations (10), (16), and (17) by inputting Equation (18) and ignoring the “*”; we get

$${\psi }^{\mathrm{*}}=\frac{\psi }{a\dot{a}}, u^{\mathrm{*}}=\frac{u}{\widehat{a}}, v^{\mathrm{*}}=\frac{v}{\widehat{a}}, z^{\mathrm{*}}=\frac{z}{a}, f^{\mathrm{*}}=\frac{f}{R}.$$

(18)

$$\begin{gathered} {\eta }^3{f}^{⁗}+\alpha \left({\eta }^4{f}^{‴}+{\eta }^3{f}^{″}-{\eta }^{2}f\right)-2{\eta }^2{f}^{‴}+3\eta f^{″}-3f^{\prime }+\eta R{f}^{\prime }{}^2-3\eta Rf{f}^{″}+ \\ 3Rf{f}^{\prime }+3{\eta }^{2}Rf{f}^{‴}-{\eta }^2{f}^{\prime }{f}^{″}-M^2\left({\eta }^3{f}^{″}-{\eta }^2{f}^{\prime }\right)=0. \end{gathered}$$

(19)

$$f\left(0\right)=0, f^{\prime }\left(0\right)=0, f\left(1\right)=1, \underset{\eta \to 0}{\lim }{\left(\frac{f^{\prime }}{\eta }\right)}^{\prime }=0.$$

(20)

It should be noticed that Equation (19) describes the situation that Majdalani et al. [13] have outlined when $\alpha =0$ and $M=0$ exist. The relationship between the temperature and the concentration in the pipe is as follows:

$$T=T_0+\left(T_w-T_0\right)\theta \left(\eta \right), C=C_0+\left(C_w-C_0\right)\varphi \left(\eta \right),$$

(21)

where $T_0$ and $C_0$ are the reference temperatures and concentrations of nanoparticles at the center. From Equation (21), the dimensionless temperature and nanoparticle concentration are

$$\theta =\frac{T-T_0}{T_w-T_0}, \varphi =\frac{C-C_0}{C_w-C_0}.$$

(22)

Equation (21) is substituted into Equations (4) and (5) to get

$$\eta {\theta }^{″}+\alpha Pr{\eta }^2{\theta }^{\prime }+RPrf{\theta }^{\prime }+Nb\eta {\theta }^{\prime }{\varphi }^{\prime }+Nt\eta {\theta }^{\prime }{}^2+{\theta }^{\prime }+QPr\eta \theta =0$$

(23)

$$\eta {\varphi }^{″}+\alpha PLePr{\eta }^2{\varphi }^{\prime }+RLePrf{\varphi }^{\prime }+\eta \frac{Nt}{Nb}{\theta }^{″}+\frac{Nt}{Nb}{\theta }^{\prime }+{\varphi }^{\prime }+\gamma LePr\eta \varphi -LePr{K}_1\eta =0$$

(24)

with the boundary constraints

$${\theta }^{\prime }\left(0\right)=0, \theta \left(1\right)=1, {\varphi }^{\prime }\left(0\right)=0, \varphi \left(1\right)=1.$$

(25)

Here $Pr=\frac{v}{\beta },$ $Nb=\frac{\tau D_B\left(C_w-C_0\right)}{\beta }$, $Nt=\frac{\tau D_T\left(T_w-T_0\right)}{T_m\beta },$ $Le=\frac{\beta }{D_B},$ $Q=\frac{Q_0{a}^2}{{\left(\rho C_p\right)}_{f}v},$and $\gamma =\frac{k_1{a}^2}{v}$ represent the Prandtl number, Brownian motion, the thermophoresis factor, the Lewis number, the sink/heat source (i.e., $Q<0$ for a heat sink and $Q>0$ for a heat source), and the chemical reaction factor, respectively, and $K_1=\frac{k{C}_0{a}^2}{\nu \left(C_w-C_0\right)}.$

3. Numerical Procedure and Convergence

The scheme of Equations (23)–(25) is solved by RK4. For this solution, Equations (23) and (25) are first transformed to the conventional first ODEs by introducing new variables and then solved numerically by keeping step size $∆\mathsf{\eta }=0.01$ and obtaining the convergence criteria up to 10⁻⁵. For endorsement of the consequences, the HAM is also functional, and brilliant settlement is originated as shown in (Figure 2). The procedure of the numerical method is given in (Figure 3). Detailed information on this method is given in [31]. Additionally, the current effort is likened to the previous and brilliant settlement originated, as revealed in

Table 1. Variation of radial velocity in the suction case for the viscous fluid; $M=0, R=-50$.

$\mathit{\alpha }$	$\mathit{\eta }$	Srinivas et al. [49]	25th–Order Approximation
3	0.92763	−1.1398871	−1.147523
2	0.99429	−1.166290	−1.166291
0	0.87134	−1.179535	−1.179537
−3	0.87134	−1.182623	−1.182623
−2	0.87134	−1.187371	−1.187371

and

Table 2. Variation of radial velocity in the suction case for the viscous fluid; $M=0, R=-100$.

$\mathit{\alpha }$	$\mathit{\eta }$	Srinivas et al. [49]	25th–Order Approximation
3	0.97289	−1.173991	−1.176380
2	0.97134	−1.176393	−1.178475
0	0.87134	−1.177947	−1.179692
−3	0.87134	−1.178881	−1.178881
−2	0.87134	−1.179290	−1.179290

.

4. Stability Analysis

The numerical solution has been obtained for the nonlinear differential equation for the temperature and concentration profiles. For the stability analysis of the mathematical problem, the present work is compared with the published work. Table 1 and Table 2 show that for up to 25 orders of approximation, the comparison of the present work and the published work reported by Srinivas et al. [49] shows excellent agreement.

5. Error Explanation

For most linear first-order differential equations, the RK4 approach yields solutions that are typically trustworthy. Significant errors are quite improbable, but that is because the conclusions are based on computer sampling and error calculations. Examining results using a solution that is derived with working precision that is higher than the Machine Precise standard is often helpful. The RK4 method is used to compute the answer to the issue with the usual work precision, and the same method is used to calculate the error with working precision-22. As errors are often rather small, it is useful to evaluate them on a logarithmic scale. The following graphs show the incorrect solutions we generated for the various physical parameters utilized in the model.

We initially conducted an error evaluation to check the validity of the process before establishing any physical predictions. This is the purpose for the creation of Figure 4, Figure 5, Figure 6 and Figure 7. The minimum mistake 10⁻³⁰ of the RK4 program is corrected during the solution by using the ND-Solve Mathematica package to reduce the overall average squared residual error (ASRE). We initially modified the thermal radiation factor $R_d$ and fixed $Gr=Gm=1, Pr=5, Q=0.3, Ec=0.4, Sc=0.2,$ and $Sc=1.5$ to observe the error for numerous orders of approximation. The greatest ASRE at various interpolation orders is shown in Figure 4 and Figure 5.

In Figure 4, it is seen that for $\alpha =-1$, total ASRE and ASRE are decreasing as the order of approximation increases. Additionally, for $\alpha =0.1$, the error is streakily decreasing compared to the cases for $\alpha =0.8$, as exposed in Figure 5. By changing the magnetic parameter of $M=0$ and $M=0.1,$ setting $Gr=Grm=2, Pr=3, Q=0.5, Ec=0.1, Prm=0.5, and Sc=0.7,$ similarly distinct plots are given in Figure 6 and Figure 7, respectively. Averaged squared residual errors and total averaged squared residual errors both decrease for $\alpha =0.1, 0.8$and$M=0, 0.1$ as the order of approximation increases. It is also observed that for $\alpha =M=0.1$ the error streakily reduces, as related to the cases for $\alpha =0.8$ and $M=0.1$ and shown in Figure 6 and Figure 7.

6. Analysis and Discussion

This section examines the consequence of several physical factors on the flow characteristics, such as the Nusselt number, Sherwood number, non-dimensional temperature, and nanoparticle concentration. Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 provide a graphic representation of the outcomes against $0\le \eta \le 1$. The effect of numerous factors such as the thermophoresis factor $Nt$, the heat source/sink factor $Q$, the wall extension ratio, the Hartmann number $M$, the Prandtl number $Pr$, the Brownian motion factor $Nb$ and the permeation parameter $R$, the Lewis number $Le$, and the chemical reaction parameter are discussed in detail. For a better understanding of the physical interpretation of the physical parameters, we chose $Pr=6, Nb=Nt=0.3, Le=0.6, Q=0.3, \gamma =0.5,$ and $M=0.1$. It should be emphasized that the variables $Nb$ and $Nt$ describe the magnitudes of Brownian movement and thermophoresis impacts, respectively. The greater quantities of $Nb$ and $Nt$ have stronger consequences. Hence, $Nb$ and $Nt$ can have any value between $0\le Nb$ $<\infty$. Figure 8a, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14b illustrate the influence of various parameters, including the sink parameter or heat source, the Brownian motion factor, the thermophoresis factor, the ratio of the wall extension, the Hartmann number, the permeation Reynolds factor, and the Prandtl number, on the temperature field. The effects of the Brownian factor on the temperature field for both situations of injection, including wall extension and shrinking, are exposed in Figure 8a,b. It is noted that the temperature rises in injection with wall expansion and contraction by enhancing $Nb$ for $Q<0$. In the injection situation, the concentration of nanoparticles migrates from the wall to the liquid, which causes a considerable rise in the temperature field. For $Q>0$, the opposite behavior is observed. Figure 9a,b illustrates how the heat source/sink parameter affects the temperature in the circumstances of both injections, combined with wall shrinking and expansion. It is observed that the heat source raises the temperature in both situations of injection, along with wall contraction and moderation, but a heat sink lowers it. The influence of the thermophoresis factor on energy is depicted in Figure 10a,b for the injection situation, along with wall extension and contraction. For a certain rise in $Nt$, it is observed that the temperature is enhanced. Figure 11a,b show the relationship between temperature and the wall expansion ratio. In the event of wall extension, regardless of injection or suction, if it increases $\alpha$ in the presence of a heat generation (i.e., $Q<0$), then the temperature increases; if $\left|\alpha \right|$ grows in the case of wall shrinkage, then the temperature decreases. However, in the event of a heat source, the behavior is the opposite (for $Q>0$). The influence of the Harmann number (HN) on the temperature field is shown in Figure 12a,b. It has been noted that nanofluids exhibit similar properties to ordinary fluids in terms of temperature regarding the Harmann number. As $Ha$ grows, the temperature rises in the existence of a heat source for both the expansion and contraction of the wall. In a heat sink, the reverse observation is observed. Figure 13a,b explore the effect of the Prandtl number on the temperature. It is essential to remember that the Prandtl number describes liquid metals and oils. Larger values of $Pr$ are associated with high viscosity oils, whereas smaller values of $Pr$ define liquid metals with low viscosity and thermal conductivity. Here, $Pr=7114, 4$, and $21$ corresponds to water, water at 4 °C, and human blood, individually. It is clear that in both scenarios of wall expansion and contraction associated with inoculation, the temperature falls as $Pr$ grows (i.e., rising thermal diffusivity) in the existence of a heat sink, but it improves in the context of a heat source. The relationship between the permeation Reynolds number and the temperature is seen in Figure 14a,b. It has been found that the boundary layer thickness reduces with increasing injection, and as a consequence, the temperature declines in both scenarios of injection, together with wall extension and contraction. The reverse effect is observed in the case of suction, along with wall extension and contraction.

Figure 15a,b, Figure 16, Figure 17 and Figure 18 show the influence of the Lewis number, the wall expansion ratio, thermophoresis, Brownian motion, and the chemical factor on the concentration of nanoparticles. Figure 15a,b demonstrate that the concentration field declines with the growing amount of $Nb$ in the occurrence of a heat sink, but the opposite trend is observed in the existence of heat generation and destruction for wall extension and shrinkage. The consequence of the thermophoresis factor on the concentration of nanoparticles is shown in Figure 16a,b. It is evident that for both developments of wall extension and shrinkage with injection, nanoparticle concentration is increased with increasing $Nt$ in the existence of a heat sink. From a physical standpoint, a larger mass flux is produced by a rise in the thermophoresis parameter due to the temperature difference; as a consequence, the concentration enhances. In the manifestation of a heat source, the reverse behavior is examined in the same figure. Figure 17a,b exhibit the influence of a chemical factor, $\gamma$, on the concentration field, where the parameter is negative for a generative reaction and positive for a destructive reaction. It is noted that for an increment in $\gamma$, there is a drop in nanoparticle concentration in the case of the destructive chemical process parameter ($\gamma >0$). Additionally, in the event of a generative chemical process ($\gamma <0$), the behavior is the opposite. Figure 18a,b show how the Lewis quantity $Le$ effects $\varphi$. It is obvious that the function $\varphi$ is a shrinking function of $Le$. When the values of the Lewis number are increased, the concentration of mass transfer is increased, which lowers nanoparticle concentration. The effect of the wall extension ratio $\alpha$ on $\varphi$ is seen in Figure 19. When $\alpha$ increases, the nanoparticle concentration enhances for wall expansion in the occurrence of injection, although for the situation of wall contraction, it decreases as $\left|\alpha \right|$ increases.

Figure 20a–c shows the influence of the Brownian and thermophoresis factors on the Nusselt number $Nu$ against $Nt$. The $Nu$ at the pipe wall is found to be an increasing function of $Nt$. It is evident from Figure 20a that $Nu$ rises when $Nb$ rises near the wall for $Q<0$ but falls in situation $Q>0$ (Figure 20b). It is also analyzed from Figure 20c that $Nu$ rises for a given rise in $M$ at the surface of the wall. Figure 21a,b are plotted to check the inspiration of the thermophoresis and the Brownian factors versus the Sherwood number $Sh.$ It is obvious that the $Sh$ rises as$Nt$ at the surface of the wall increases. Figure 21a witnesses that that $Sh$ reduces as $Nb$ enhances at the wall. Figure 21b shows that $Sh$ declines for a specified growth in $\gamma$near the wall.

7. Key Notes

In this study, we investigate the influence of chemical reactions and heat reservoirs on MHD nanofluid flow in a porous expanding or contracting pipe. The characteristics equation of the flow cylinder coordinates is transformed to an ordinary differential equation and converted by using suitable transformation and then solved numerically by using the RK4 method. For the stability analysis, the present work is compared with the previous work. For the confirmation of the mathematical modeling, the error estimation and the residue error are also calculated, and it is found that the error is too small, which validates our solution. The novelty of the present study is to investigate the effect of chemical reactions on magnetized nanofluid flow over an extending and shrinking porous pipe with heat generation and absorption; in limiting cases, the present work is compared with the published work, and outstanding agreement is found.

The following summarizes the key conclusions:

• It is observed that for both situations of wall extension or contraction with injection, the temperature is the increasing function of the thermophoresis and Brownian motion factors.
• It is analyzed that the temperature is raised when the heat source is increased in both cases of wall expansion or contraction but declines in the case of a heat sink.
• In the case of a heat source, the temperature rises as the Hartmann and Prandtl numbers are enhanced, but in the case of a heat sink, the temperature falls.
• In the presence of heat sinks and injections when the thermophoresis factor is increased, the concentration of nanoparticles is increased in both wall expansion and contraction.
• In both situations of wall expansion or contraction, along with injection, the concentration of nanoparticles is a decreasing function of $Nb$, while the concentration of nanoparticles is an increasing function in the case of a heat source.
• It is also observed that $\varphi$ is increased for $\gamma <0$ and decreased for$\gamma >0$.