The influence of thermophoresis and Brownian motion on maxwell nanofluids utilizing Cattaneo-Christov double diffusion theory

Abstract

The primary objective of this work is to examine the concentration and temperature boundary layers in a Maxwell nanofluid containing nanoparticles, influenced by thermophoresis & Brownian motion. The main findings are to enhance understanding of nanoscale heat with mass transport mechanisms, with potential applications in advanced thermal management systems and nanofluid-based technologies. The contribution of Cattaneo-Christov double diffusion theory captures time-delayed thermal effects. A system of partial differential equations (PDEs) is transformed into ordinary differential equations (ODEs) using similarity transformation (ST). The analysis considers the collective impacts of thermophoresis, Brownian motion, and viscoelasticity. The study elucidates the complex relationship between double diffusion phenomena and nanofluid behavior through analytical and numerical techniques. Dimensionless mathematical problems are numerically solved using MATLAB’s built-in function bvp4c, with the behavior of flow-controlling parameters presented through graphical and tabular data.

Introduction

Nonmaterials suspended in a base liquid are called nanofluids, which aim to increase the combined heat and mass transformation process. Its potential applications are in the fields of biology and medicine. Furthermore, the mobility of the necessary nonmaterial’s in the structures is significantly reliant on cancer therapy theories, drug delivery systems, nanotechnologies, fermentation, and medicine fluids traveling via micro fluidic devices in freezers and thermal systems rely only on heat transfer particles given by nonmaterial’s, well-known phenomena. To achieve peak performance, nanofluid dynamics is a vital concept to grasp in any industry that deals with non-particle suspensions in any form. Since, Choi and Eastman ¹ invented the term “nanofluid” in 1995, several research have been conducted to investigate the thermos-physical characteristics, synthesis, manufacturing, thermal fluid features, and applications of nanofluid. Nonmaterial science provided an appropriate working material to improve thermal performance. First, three different kinds of nanoparticles were placed in a continuous phase liquid, and their viscosity and thermal conductivity were studied. These particles have a size range of 1–100 nm. Nanofluid contains solid nanoparticles of size 1–100 nm in regular liquid. The thermal conductivity properties of nanomolecules, including silicon, copper, the metal aluminum, and titanium, is extensively utilized. Large heat exchangers in automobiles and industries need more production; consequently, nanofluid is kept together with the necessary nanomolecules. Sheikholeslami ² investigated the numerical solution of a curved cavity utilizing nanoparticles. Lahmar et al. ³ examined the flow & heat-transmission properties of the compressed nanofluid between two parallel plates. In the existence force, Waqas et al. ⁴ inspected the magneto cross flow of Oldroyd-B nanoliquid. Rashidi et al. ⁵ investigated the magnetic field-relevant Copper–water nanoliquids stagnation point boundary layer movement. Slip flow of Maxwell nanofluid with Cattaneo-Christov (CC) heat flux past a vertical surface surrounded by a porous medium was investigated by Al Rashdi et al. ⁶ Nabwey et al. ⁷ provides the analysis of micropolar Williamson and Maxwell nanofluids above a perpendicular cylinder with heat and mass transport. Maple 23 software was hired to solve the coupled resulting nonlinear differential equations resulting from appropriate similarity transformations. Saleem and Faheem ⁸ discussed the synchronized impacts of melting heat transfer Williamson magnetized nanofluid containing microorganisms over a leading edge. A numerical bvp4c approach was used by them.

Many other researchers consider the different models and physical boundary conditions to discussed the flow situation. Mentioned can be made to.^9–20 Ayodeji et al. ²¹ used velocity slip and viscous dissipation to sightsee the flow of nanofluid against a stretched sheet. Platt ²² investigated nanofluids have piqued the interest of researchers due to their potential for enhancing heat transfer and other features in a variety of engineering applications. However, challenges remain in terms of large-scale production, cost-effectiveness, and ensuring long-term stability. Ongoing research aims to address these challenges and explore new applications for nanofluid. Kuznetsov ²³ highlighted the key research for the stability of nonmaterial, including motile gyrotactic bacteria. Nayak et al. ²⁴ studied the effect of temperature, solutal, microorganism profiles and momentum on the flow of a biologically inert nanoliquid over a stretched surface. Nima et al. ²⁵ investigated the melting properties of fluids that are not Newtonian in bio-convective swimming organisms passing through non-Darcy porous surfaces with changing fluid flow. Ben Henda et al., ²⁶ Ijaz Khan et al., ²⁷ and Khan et al. ²⁸ relate to more current work on nanoparticles bioconvection patterns and altered flow geometry and technique. On multiple occasions, the steadiness of nanoparticles suspensions has been effectively increased in the presence of gyrostatic bacteria. Imran et al. ²⁹ suggested the properties of melting in non-Newtonian (NF) fluid passing through absorbent materials that are not Darcy with fluid flow that varies in swimming microorganisms that are bioconvective. Jayadevamurthy et al. ³⁰ examined the biological convection stream of a hybrid nanofluid through a moving rotating disk with numerically and MHD has various practical applications. Without the need of mechanical moving parts, MHD generators may directly transfer the kinetic energy of a high-velocity conducting fluid into electrical energy for power generation. MHD propulsion systems have been proposed for spacecraft and submarines. MHD pumps and valves can be used in industries where handling liquid metals is necessary. MHD is also relevant to astrophysics, space science, and the study of natural phenomena involving conducting fluids. Magneto hydrodynamics is a complex and interdisciplinary field that continues to be an active area of research. Comprehending the conductivity of fluids in the presence of magnetic fields holds significant significance for several scientific and technical disciplines. Freidoonimehr et al. ³¹ examined MHD transient laminar flow of moving nanoparticles across a porous media sheet under free convection. Gangadhar et al. ³² examined MHD outer velocity flow stability across a stretched cylinder in a number of scenarios. Several scholars have looked into stretched flow problems.

For instance, Mukhopadhyay ³³ examined the MHD boundary layer (BL) slip flow along a stretched cylinder. Due to its many applications, particularly in engineering, petroleum, mechanical manufacturing, geophysical scientific fields and agriculture. Magneto hydrodynamics has recently become more and more well-known. Typically, one that is orthogonal to a liquid flow, in order to generate the Lorentz force, this strength acts in opposition to the fluid’s flow, influencing its velocity. The generation of information on fluids using nanoparticles and the analysis of magneto-hydrodynamic methods are currently extremely valuable and applicable in a wide range of research sectors, including manufacturing and science. Among the applications are wound therapy, an optical modulator, and a magneto-hydrodynamic power-based transformer. Furthermore, the Lorentz power is discussed in this area’s hypothetical evaluation. Magneto-hydrodynamic forces influence this type of force, which is an extremely valuable concept for regulating cooling processes. Shoaib et al. ³⁴ examined the spinning motion using mathematical modeling of a hybrid nanofluid that is magnetohydrodynamic and radiates heat nonlinearly over a stretched surface. Tian et al. ³⁵ examined the two fins’ two- and three-dimensional shapes and how they affected the heat-sink efficiency of a magneto hydrodynamic hybrid nanofluid that was running either non-slip or slip-slip. Haq et al. ³⁶ studied the process of Eyring Powell nanoliquids’ mixed-convective MHD transport, which results in gyro tactic microorganisms in a stretched cylinder. The effects of thermodynamics and magnetism on nonliquid flow in an annular, porous circular zone created by two particles in motion were examined by Siddiqui and Chamkha. ³⁷ Salem and Fathy ³⁸ investigated how various factors affected the flow of mass & heat transfer via a permeable material that radiates heat toward a stretched sheet, and the flow toward the stagnation point. Najib et al. ³⁹ examined mass transport across a stretching/shrinking cylinder and stagnation point flow. Numerous researchers provide a details studies.^40–44

Novelty

This work presents a fresh interpretation of Maxwell nanofluid boundary layer behavior by integrating Brownian motion & thermophoresis into the Cattaneo-Christov (CC) double diffusion model. Unlike traditional Fourier and Fick laws, the Cattaneo-Christov model includes finite speeds for heat & mass propagation, offering a more accurate depiction of thermal and concentration fields in non-Newtonian fluids. This study is the first to use this comprehensive diffusion model to Maxwell nanofluids and illustrates how boundary layer formation is influenced by non-Fickian mass diffusion and non-Fourier heat conduction. The findings provide fresh insights into the interactions between thermal and concentration boundary layers in viscoelastic nanofluids, improving our understanding and suggesting new ways to optimize thermal management systems in engineering applications.

Mathematical modeling

Here propose unsteady 2-D incompressible flowing of Maxwell fluid through a stretching cylinder with radius $R_1$ . Let us cylindrical polar coordinated be adopted in such a manner that (axis of cylinder) z-axis and (normal to the z-axis) r-axis respectively. Stretching velocity on time dependent of the cylinder is considered $u_w(t,z)=\frac{\mathrm{az}}{1-\gamma t}$ where $\alpha ,\gamma >0$ . The phenomena of heat & mass in a flow are investigated in the Cattaneo-Christov heat and mass flux context as given in flow diagram in Figure 1.

Figure 1.

Flow diagram.

Governing equation

The governing flow equation for the proposed model are expressed as Khan et al. ⁴⁴

$$\frac{\partial \left(\mathrm{ru}\right)}{\partial z}+\frac{\partial \left(\mathrm{rw}\right)}{\partial r}=0,$$

(1)

$$\begin{array}{c}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial z}+w\frac{\partial u}{\partial r}\hfill \\ =\nu \{\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{\partial }^{2}u}{\partial r^2}\}-{\lambda }_1\left\{\begin{array}{c}\frac{{\partial }^{2}u}{\partial t^2}+2u\frac{{\partial }^{2}u}{\partial t\partial z}+2w\frac{{\partial }^{2}u}{\partial r\partial t}\hfill \\ +2\mathrm{uw}\frac{{\partial }^{2}u}{\partial z\partial r}+w^2\frac{{\partial }^{2}u}{\partial r^2}+u^2\frac{{\partial }^{2}u}{\partial z^2}\hfill \end{array}\right\},\hfill \end{array}$$

(2)

$$\begin{array}{c}\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial z}+w\frac{\partial T}{\partial r}+{\lambda }_T\hfill \\ \left\{\begin{array}{c}\frac{{\partial }^{2}T}{\partial t^2}+\frac{\partial T}{\partial z}\frac{\partial u}{\partial t}+\frac{\partial w}{\partial t}\frac{\partial T}{\partial r}+2w\frac{{\partial }^{2}T}{\partial t\partial r}+u^2\frac{{\partial }^{2}T}{\partial z^2}\hfill \\ +2\mathrm{uw}\frac{{\partial }^{2}T}{\partial r\partial z}+2u\frac{{\partial }^{2}T}{\partial r\partial z}+u\frac{\partial u}{\partial z}\frac{\partial T}{\partial z}+w^2\frac{{\partial }^{2}T}{\partial r^2}\hfill \\ +w\frac{\partial w}{\partial r}\frac{\partial T}{\partial r}+w\frac{\partial u}{\partial r}\frac{\partial T}{\partial z}+u\frac{\partial w}{\partial z}\frac{\partial T}{\partial r}\hfill \end{array}\right\}\hfill \\ =\{\frac{1}{\left(\rho c_p\right)}\frac{1}{r}\frac{\partial }{\partial r}\left\{K\left(T\right)r\frac{\partial T}{\partial r}\right\}\hfill \end{array}$$

(3)

$$\begin{array}{c}\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial z}+w\frac{\partial C}{\partial r}+{\lambda }_c\hfill \\ \left\{\begin{array}{c}\frac{{\partial }^{2}C}{\partial t^2}+\frac{\partial C}{\partial z}\frac{\partial u}{\partial t}+\frac{\partial C}{\partial r}\frac{\partial w}{\partial t}+2u\frac{{\partial }^{2}C}{\partial t\partial z}\hfill \\ +2\mathrm{uw}\frac{{\partial }^{2}C}{\partial r\partial z}+2w\frac{{\partial }^{2}C}{\partial t\partial r}+w^2\frac{{\partial }^{2}C}{\partial r^2}+u^2\frac{{\partial }^{2}C}{\partial z^2}\hfill \\ +u\frac{\partial C}{\partial z}\frac{\partial u}{\partial z}+w\frac{\partial C}{\partial z}\frac{\partial u}{\partial r}+u\frac{\partial C}{\partial r}\frac{\partial w}{\partial z}+w\frac{\partial C}{\partial r}\frac{\partial w}{\partial r}\hfill \end{array}\right\}\hfill \\ =D_B\frac{1}{r}\frac{\partial }{\partial r}\left\{r\frac{\partial C}{\partial r}\right\}\hfill \end{array}$$

(4)

Boundary condition

The problem of Maxwell fluid flow is demonstrated under the following boundary constraints.

$$\begin{array}{l}[u(t,z,r]=u_w\left(t,z\right)=\frac{az}{\left(1-\gamma t\right)},w\left(t,z,r\right)=0,\\ -k\frac{\partial T}{\partial r}=h_f\left(T_w-T\right),D_B{C}_r+\frac{D_T}{T_{\infty }}\frac{\partial T}{\partial r}=0,atr=R_1\\ u\to 0,T\to T_{\infty },C\to C_{\infty },asr\to \infty ]\end{array}$$

(5)

So, $T_w$ and $C_w$ are surface temperature & concentration and $T_{\infty }$ and $C_{\infty }$ are ambient temperature and concentration, $\nu$ indicates the kinematic viscosity, $\rho$ be the density of fluid, and $C_p$ signifies the specific heat capacity.

In the governing equation, $u$ and $w$ indicates the velocity components in r z-direction respectively, ${\alpha }_1$ stand for thermal diffusivity of nanofluid, ${\lambda }_i$ and $i$ = 1, 2 signifies the coefficient of thermal relaxation to time and retardation to time etc, $\nu$ is the kinematics viscosity, $\tau$ demonstrates the ratio of nanoparticles heat capacity to the host fluid capacity, $T$ and $C$ are depicts the temperature and concentration of nanoparticles respectively, and $D_B$ is coefficients of Brownian diffusion.

Similarity transformation

The (PDEs) partial differential equations driving the flow characteristics are subjected to a series of notable similarity transformations in this section, resulting in the transformation of the original system into a set of nonlinear dimensionless ordinary differential equations (ODEs). These changes are critical in simplifying the mathematical description of the fluid dynamics being studied. ⁴⁴

$$\begin{array}{c}u=\frac{\mathit{az}}{1-\mathit{\gamma }\mathit{t}}\mathit{f}\prime (\mathit{\zeta }),w=-\frac{{\mathit{R}}_1}{\mathit{r}}\sqrt{\frac{\mathit{a}\upsilon }{(1-\mathit{\gamma }\mathit{t})}}\mathit{f}(\mathit{\zeta }),\hfill \\ \mathit{\zeta }=\sqrt{\frac{\mathit{a}}{\mathit{\upsilon }(1-\mathit{\gamma }\mathit{t})}}\left(\frac{{\mathit{r}}^2-{\mathit{R}}_1^2}{2{\mathit{R}}_1}\right),\mathit{\theta }\left(\mathit{\zeta }\right)=\frac{\mathit{T}-{\mathit{T}}_{\infty }}{{\mathit{T}}_{\mathit{w}}-{\mathit{T}}_{\infty }},\hfill \\ \multicolumn{1}{c}{\varphi \left(\mathit{\zeta }\right)=\frac{\mathit{C}-{\mathit{C}}_{\infty }}{{\mathit{C}}_{\mathit{w}}-{\mathit{C}}_{\infty }}.}\end{array}$$

(6)

Dimensionless equation

The detail procedure for the transforming (1–4) into the dimensionless form has been communicated in the upcoming discussion.

$$\begin{array}{c}(2\lambda \zeta +1)f″\prime +2\lambda ff″-\frac{S}{2}\zeta f″-\frac{7{\beta }_1\zeta S^{2}f″}{4}-\frac{{\beta }_1}{4}{\zeta }^2{S}^{2}f″\prime \hfill \\ -sf\prime -2{\beta }_1{S}^{2}f\prime -f\prime 2+ff″+2{\beta }_{1}ff\prime f″-\frac{\lambda {\beta }_1}{2\lambda \zeta +1}f″f^2\hfill \\ -2{\beta }_{1}Sf\prime -{\beta }_{1}S\zeta ff″\prime +S{\beta }_1\zeta ff″\prime +3S{\beta }_{1}ff″-{\beta }_1{f}^{2}f″\prime =0.\hfill \end{array}$$

(7)

In the above equations $\lambda =\frac{1}{R_1}.\sqrt{v(1-\gamma t)/a}$ investigated the curvature parameter ${\beta }_1=\frac{{\lambda }_{1}a}{(1-\gamma t)}$ is the Maxwell fluid parameter, and $S=\frac{\gamma }{a}$ e the unsteadiness parameter, and the below procedure for the transformation of Temperature equation into the dimensionless form

$$\begin{array}{c}(2\lambda \zeta +1)\theta ″+\mathrm{pr}(f\theta \prime -\frac{S}{2}\zeta \theta \prime )+(2\lambda \zeta +1)(\theta \theta ″+{\theta \prime }^2)\epsilon \hfill \\ +2\lambda \theta \prime +2\lambda \epsilon \theta \theta \prime -\mathrm{pr}{\beta }_t\hfill \\ (\frac{3}{4}{S}^2\zeta \theta \prime -\frac{3}{2}\mathrm{Sf}\theta \prime -\frac{1}{2}S\zeta f\prime \theta \prime +\frac{1}{4}{S}^2{\zeta }^2\theta ″-S\zeta f\theta ″+{\theta }^{″f^2}+\theta \prime ff\prime =0)\hfill \end{array}$$

(8)

Here, $pr=\frac{v}{{\alpha }_1}$ Prandtl number, ${\beta }_t=\frac{\alpha {\lambda }_t}{(1-\gamma t)}$ the thermal relaxation time parameter.

$$\begin{array}{c}(2\lambda \zeta +1)\varphi ″+\mathrm{Lepr}(f\varphi \prime -\frac{s}{2}\zeta \varphi \prime )+(2\lambda \zeta +1)(\varphi \prime \varphi ″-\varphi {\prime }^2)\epsilon \hfill \\ +2\lambda \varphi \prime +2\lambda \epsilon \varphi \prime \varphi ″-\mathrm{prLe}{\beta }_c\hfill \\ [\frac{3}{4}{S}^2\zeta \varphi \prime -\frac{3}{2}\mathrm{Sf}f\prime -\frac{1}{2}S\zeta f\prime \varphi \prime +\frac{1}{4}{\zeta }^2{S}^2\varphi ″-\mathrm{Sf}\zeta \varphi ″+{\varphi }^{″f^2}+\varphi \prime ff\prime ]=0.\hfill \end{array}$$

(9)

In the above equations ${\beta }_c=\frac{{\lambda }_{c}a}{(1-\gamma t)}$ the mass relaxation time parameter, $\mathrm{Le}=\frac{{\alpha }_1}{D_B}$ the Lewis number.

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}\mathrm{Nb}\varphi \prime \left(0\right)+\mathrm{Nt}\theta \prime \left(0\right)=0,\theta \prime \left(0\right)=-\mathrm{Bi}[1-\theta \left(0\right)],\hfill \\ \multicolumn{1}{c}{f\prime \left(\infty \right)\to 0,\theta \left(\infty \right)\to 0,\varphi \left(\infty \right)\to 0,}\end{array}\right)}\end{array}$$

(10)

In the above equations $\lambda =\frac{1}{R}.\sqrt{\frac{v(1-\gamma t)}{a}}$ investigated the curvature parameter ${\beta }_1=\frac{{\lambda }_{1}a}{(1-\gamma t)}$ is the Maxwell fluid parameter, and $S=\left(\frac{\gamma }{a}\right)$ the unsteadiness parameter, $\mathrm{pr}(=\frac{\nu }{{\alpha }_1})$ the Prandtl number, $\lambda =\frac{1}{R}\sqrt{\frac{v(1-\gamma t)}{a}}$ the curvature parameter, $S(=\frac{\gamma }{a})$ is the unsteadiness parameter, ${\beta }_t=\frac{\alpha {\lambda }_t}{(1-\gamma t)}$ the thermal relaxation time parameter, ${\beta }_c(=\frac{{\lambda }_{c}a}{(1-\gamma t)})$ the mass relaxation time parameter, $\mathrm{Le}=\frac{{\alpha }_1}{D_B}$ the Lewis number, $\mathrm{Nb}=\frac{\tau D_B(C_w-C_{\infty })}{v}$ is the Brownian diffusion coefficient, $\mathrm{Nt}=\frac{\tau D_T(T_w-T_{\infty })}{v{T}_{\infty }}$ represents the Thermophoresis parameter, $\mathrm{Bi}-\frac{h_f}{k}\sqrt{\frac{v(1-\gamma t)}{a}}$ the Biot number.

Numerical scheme

The shooting technique using the bvp4c tool is applied to achieve the numerical results of coupled (ODEs) ordinary differential equations with boundary conditions (BC). The analysis is performed using the software MATLAB and solve numerically and graphically. The higher order of (ODEs) must be reduced in first order for this case. This is accomplished by introducing new variables.

Let

$$\begin{array}{c}f=q_1,f\prime =q_2,f″=q_3,f″\prime ={q\prime }_3,\hfill \\ \theta =q_4,\theta \prime =q_5,\theta ″={q\prime }_5,\varphi =q_6,\varphi \prime =q_7,\hfill \\ \varphi ″={q\prime }_7\hfill \end{array}$$

(11)

$$\begin{array}{c}q{\prime }_3=\hfill \\ \frac{\left[\begin{array}{c}-2\lambda q_3+\frac{S}{2}\zeta q_3+s{q}_2+{q_2}^2-q_1{q}_3+\frac{7}{4}{\beta }_1{S}^2\zeta q_3+2{\beta }_1{S}^2{q_2}^2\hfill \\ \multicolumn{1}{c}{+2{\beta }_{1}S{q_2}^2+{\beta }_{1}S\zeta q_2{q}_3-3{\beta }_1{q}_1{q}_3-2{\beta }_1{q}_1{q}_2{q}_3+\frac{\lambda {\beta }_1}{(1+2\lambda \zeta )}{q}_1^2{q}_3}\end{array}\right]}{(1+2\lambda \zeta )-\frac{{\beta }_1}{4}{\zeta }^2{S}^2+S{\beta }_1\zeta q_1-{\beta }_1{q^2}_1}\hfill \end{array}$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{q{\prime }_5=\frac{\left[\begin{array}{c}-\mathrm{pr}(q_1{q}_5-\frac{s}{2}\zeta q_5)-(1+2\lambda \zeta )(-{q_5}^2)\epsilon -2\lambda q_5\hfill \\ \multicolumn{1}{c}{+\mathrm{pr}{\beta }_t\{\frac{3}{4}{s}^2\zeta q_5-\frac{3}{2}s{q}_1{q}_2-\frac{1}{2}s\zeta q_1{q}_5+q_5{q}_1{q}_2\}}\end{array}\right]}{((1+2\lambda \zeta )+q_4)+2\lambda \epsilon q_4-\mathrm{pr}{\beta }_t(+\frac{1}{4}{s}^2{\zeta }^2-s\zeta q_1+{q_1}^2)}}\end{array}$$

(13)

$$\begin{array}{c}q{\prime }_7=\hfill \\ \frac{-2\lambda q_7-\mathrm{pr}(q_1{q}_7-\frac{s}{2}\zeta q_7)+\mathrm{pr}{\beta }_c[\frac{3}{4}{s}^2\zeta q_7-\frac{3}{2}s{q}_1{q}_2-\frac{1}{2}s\zeta q_2{q}_7+q_7{q}_1{q}_2]}{((1+2\lambda \zeta )+\epsilon q_7)+(1+2\lambda \zeta )(-{q_7}^2)\epsilon +2\lambda \epsilon q_7+\mathrm{pr}{\beta }_c(\frac{1}{4}{s}^2{\zeta }^2-s\zeta q_1+{q_1}^2)}\hfill \end{array}$$

(14)

Result and discussion

The importance of flow regulating factors on subjective flow fields is examined in length in this specific area.

Velocity profile

Figure 2 depicted to examine the effects of relaxation parameter ${\beta }_1$ against velocity of rate type fluid $\chi \left(\zeta \right)$ . Larger fluid relaxation parameter ${\beta }_1$ decreases the velocity $\chi \left(\zeta \right)$ .

Figure 2.

Influences of ${\beta }_1$ on $f\prime$ .

Temperature field

In this slice the behavior of temperature distribution $\mathrm{Nr}$ versus rheological prominent parameters are illustrated in Figures 3–8.

Figure 3.

Influences of $\mathrm{Pr}$ on $\theta$ .

Figure 4.

Influences of $\mathrm{Nt}$ against $\theta$ .

Figure 5.

Influences of $\mathrm{Bi}$ against $\theta$ .

Figure 6.

Influences of ${\beta }_T$ against $\theta$ .

Figure 7.

Influences of ${\beta }_1$ against $\theta$ .

Figure 8.

Influences of $\mathrm{Pr}$ against $\varphi$ .

The nature of Prandtl number $\left(\mathrm{Pr}\right)$ in contrast with thermal field of species $\theta \left(\zeta \right)$ is characterized in Figure 3. The thermal field $\theta \left(\zeta \right)$ shows by increasing Prandtl number $\mathrm{Pr}$ . The outcomes of thermal distribution $\theta \left(\zeta \right)$ against Thermophoresis parameter $\mathrm{Nt}$ in interpreted in Figure 4. Here, temperature field $\theta \left(\zeta \right)$ upsurges with larger thermophoresis parameter $\mathrm{Nt}$ . Thermophoresis forces cause the nanoparticles to physically migrate from warm to cold regions in a fluid. Plots (Figure 5) displayed the impact of Biot number $\mathrm{Bi}$ versus temperature distribution $\theta \left(\zeta \right)$ . Larger fluid relaxation parameter ${\beta }_1$ decreases the velocity $\theta \left(\zeta \right)$ . Temperature field $\theta \left(\zeta \right)$ boosted up for larger Biot number $\mathrm{Bi}$ . Physically Biot number enhances the heat transfer rate as well as heat efficiency of fluid. Figure 6 captured to examine the influences of relaxation parameter ${\beta }_1$ against temperature of rate type fluid Figure 7 captured to examine the influences of thermal relaxation parameter ${\beta }_1$ versus temperature of rate type fluid increased.

Concentration field

It is noteworthy that $\left(\mathrm{Pr}\right)$ Prandtl number reduced the concentration of nanoparticles profile, $\varphi \left(\zeta \right)$ . The outcome of $\mathrm{Pr}$ Prandtl number versus solutal domain of different species $\varphi \left(\zeta \right)$ is displayed in Figures 8 and 9 explain the properties of nanoparticles concentration $\varphi$ against $\mathrm{Nb}$ Brownian motion parameter. The volumetric concentration of nanoparticles $\varphi$ dwindles in magnitude with improvement in Brownian diffusion coefficient $\mathrm{Nb}$ the effects of thermophoresis $\left(\mathrm{Nt}\right)$ parameter on solutal field of nanoparticles $\varphi \left(\zeta \right)$ is discussed in Figure 10. It is reported that the concentration $\varphi \left(\zeta \right)$ is enriched with improve Thermophoresis parameter $\mathrm{Nt}$ . Larger amount of activation energy parameter physically improves the chemical reaction process. The significance of Lewis number $\mathrm{Le}$ via nanoparticles concentration $\varphi \left(\zeta \right)$ is reported through Figure 11. It is observed that the solutal field $\varphi \left(\zeta \right)$ diminishes by escalates $\mathrm{Le}$ Lewis number. Physically, Lewis number is directly tied to the mass diffusivity. Hence by enlarging Lewis number, the concentration field declines. Characteristics of Biot number $\left(\mathrm{Bi}\right)$ on nanoparticles concentration $\varphi \left(\zeta \right)$ is highlighted in Figure 12. In this figure it is clearly noted volumetric concentration of nanoparticles $\varphi \left(\zeta \right)$ rises by enlarging Biot number $\left(\mathrm{Bi}\right)$ .

Figure 9.

Influences of $\mathrm{Nb}$ against $\varphi$ .

Figure 10.

Influences of $\mathrm{Nt}$ against $\varphi$ .

Figure 11.

Influences of $\mathrm{Le}$ against $\varphi$ .

Figure 12.

Influences of $\mathrm{Bi}$ via $\varphi$ .

Engineering quantities behavior

This section examines how several parameters, such as the ${\mathrm{Re}}^{1/2}\mathrm{Sh}$ local Sherwood number, ${\mathrm{Re}}^{1/2}\mathrm{Nu}$ local Nusselt number, and ${\mathrm{Re}}^{1/2}{C}_f$ skin friction coefficient, influence engineering values. Figures 13 and 14 show the stimulus of the $\left(\mathrm{pr}\right)$ Prandtl number on the Lewis $\left(\mathrm{Le}\right)$ number through the local Sherwood number and local Nusselt numbers. The graphical findings show that the local Sherwood and Nusselt numbers fall as the Prandtl and Lewis numbers increase. Figures 15 and 16 demonstrate the methods of local Nusselt number and skin fraction coefficient for bioconvection Rayleigh umber using the buoyancy ratio parameter. The local Nusselt number decreases by the bioconvection Rayleigh quantity, as seen by the buoyancy ratio parameter. Additionally, as the bioconvection Rayleigh number rises, the buoyancy ratio parameter increases and the skin friction coefficient rises as well.

Figure 13.

${\mathrm{Re}}^{1/2}\mathrm{Sh}$ influences versus $\mathrm{pr}$ .

Figure 14.

${\mathrm{Re}}^{1/2}\mathrm{Nu}$ versus of $\mathrm{pr}$ .

Figure 15.

${\mathrm{Re}}^{1/2}\mathrm{Nu}$ versus of $\mathrm{Nr}$ .

Figure 16.

${\mathrm{Re}}^{1/2}{C}_f$ versus of $\mathrm{Nr}$ .

Remarks and conclusion

A problem of the unstable flow of Maxwell fluid across the stretched cylinder in 2D has finally been discussed. The Cattaneo-Christov hypothesis is utilized in the context of mass and heat transfer systems. The thermal conductivity of the fluid is considered to vary (temperature dependent). This study investigates the bioconvective flow of thyrotrophic nanofluid containing microorganisms in the presence of a bent magnetic field. The following are the main findings of the offered analysis:

• It is determined that increases the value of relaxation parameter than the velocity profile reductions.

• Increasing the value of Bioconvection Rayleigh number then the velocity is decreases.

• It is significant to note that the velocity profile will growth as the mixed convection parameter values increase. The buoyancy ratio parameter value was raised, causing a dip in the velocity profile.

• Raising the mixed convection parameter increases the velocity of the profile, whereas raising the buoyancy ratio parameter decreases it. The values of dimensionless parameters are increasing then the velocity will be increases.

• Discusses the effects of microorganisms’ difference parameter and microorganism stratification Biot number on microorganism profile.