Hydrothermal dynamics of darcy Forchheimer ternary hybrid nanofluid flow past bended surface with active—passive controls

Abstract

This dynamic study investigates hydrothermal and rheological properties of two-dimensional ternary nanofluid flow by using active passive control mechanisms on nanoparticles navigating a porous curved surface. The working fluid contains nanometer sized spherical shaped particles of three different metals $(\mathrm{Co},\mathrm{Ag},\mathrm{Zn})$ , which are homogenously and stably dispersed in liquid dihydrogen monoxide as host fluid. A strong magnetic force of strength $B_{°}$ acts along radial direction of curved surface. Thermal radiation term in heat equation suggests a strong heat source in the vicinity. This unique amalgamation, which has not been addressed so far by any researcher incorporates Brownian motion and thermophoresis to construct the hydrothermal framework. The passive control of nanoparticles on surface influenced by Brownian movement and thermophoresis, which is considered more realistic in comparison to constant concentration hypothesis (active control), have been taken into account. The nanoliquid flow is governed by system of Partial Differential Equations (PDEs) that cater thermophysical properties of nanoparticles. After simplifying the system to Ordinary Differential Equations (ODEs) using suitable transform, it is solved numerically using bvp4c in MATLAB 2023a. The obtained graphical and numerical results are discussed in detail to explain physics of observed thermo-rheological behaviour and mass diffusivity. The study reveals that curvature and porosity factors have opposite effects on velocity profile. All involved parameters augment temperature profile for both active and passive controls except curvature factors. In all the cases, active control assures higher temperature over passive control. Dual consequences have been noted for active and passive controls for concentration panels in most cases. Nusselt number is found increasing for increasing nanoparticles concentration with ternary nanofluid having maximum heat transfer rate. This study significantly contributes to the knowledge of ternary nanofluidic thermo-rheological analysis on curved stretching surface having numerous applications in mechanical, thermal and industrial domains. It also offers insights for developing advanced coatings and materials in industrial and engineering context.

Introduction

Fluids and energy are the most essentials elements for human lives. These serve as vital foundations to human civilizations, constituting fundamental necessities for homes, industries, technological development and overall support to diverse process of modern-day livings. Growing energy needs necessitate prudent approaches for optimization of energy resources. Researchers has been diligently exploring innovative methods to optimize energy utilization by employing both experimental and theoretical approaches. One of such effective technique is to improve thermal efficiency of heat transferring fluids. Water being abundant in nature and cost effective is the most widely used heat careering fluid in industries; however, its low thermal conductivity makes it unsuitable for satisfying ultramodern high heating and cooling processes requirements. Other commonly used fluids such as oils and ethylene glycols etc. are also not good thermal conductors. As the metals are good conductors, one way of improving thermal conductivity of such fluids is adding small particles of metals in these fluids. Hamilton and Crosser ¹ in 1960s conducted some experiments on this idea; however, such solutions were having stability issues because metal powders are heavier than base fluid molecules. The nanotechnological advances by end 20th century made the nanosized particles available and in 1995 Choi and Eastman ² carried out series of experiments by adding nanoparticles of various metals in base fluids. They established that nanoparticles were stable and thermal conductivity was increased manifold even by very small nanoparticles volume fractions. Since their pioneer work, the field of nanofluids is evolving at very rapid pace and researchers are increasingly engaged in theoretical and experimental work related to this dynamic new type of fluids. Extensive investigations have been carried out in recent years to extract marvelous thermal features of nanofluids in diverse range of geometries and under various constraints. Researchers developed hybrid nanoliquid to intersperse properties of two different materials in base fluids. More recently, the development of ternary nanofluid, which blends three various types of nanoparticles in base fluid and possesses excellent thermal and rheological features has caught greater attention of research community. Abbasi et al. ³ analyzed ternary nanoliquid flow past curved shape geometry by focusing impacts of various shapes of nanoparticles and two different base fluids. They reported that shape and type of base fluid has considerable impact on density and boundary layer thickness of nanofluid on curved structure. In another relevant study, Heyhat et al. ⁴ deliberated influences of type and volume fraction of ternary nanofluid on density and thickness of boundary layer and revealed that both were increasing function of volume fraction of nanoparticle concentration. Abbasi et al. ⁵ numerically explored peristalsis flow of hybrid nanoliquid with curved structure of channel considering variable thermal conductivity and deduced that effect of variable thermal conductivity was significant on thermo-rheological properties. Wenhao et al. ⁶ carried out a comparative study of ternary nanoliquid with free, forced and mixed convection considering thermal radiations and slip conditions. Their results indicate that skin friction was minimum when slip conditions with free conviction was employed. Babu et al. ⁷ examined flow of ternary nanofluid across a permeable channel considering joule heating and observed that temperature was a decreasing function of Reynolds numbers. Some more recent studies on ternary nanofluids with various regulatory factors on stretching surfaces are given at Shahsavar et al.^8–13

The boundary layer concept was presented by Prandtl, ¹⁴ in 1904. Boundary layer flow over a continuous moving surface was first studied by Sakiadis. ¹⁵ The exact solution of two dimensional fluid flow past stretching surface was first computed by Crane ¹⁶ in 1970. Gupta and Gupta ¹⁷ extended the work of Crane by considering the suction and blowing factors. There have been countless studies on fluid flow at flat stretching surface, however, the curved stretching surface is very rarely investigated. Such surface has numerous applications in industries and engineering procedures where extraction takes place by movement of curved planes like hot rolling, annealing of copper wires, rubber and plastic sheets manufacturing etc. The everyday significance of flow past a curved surface can be exemplified by soap films that are widely employed in studying classical two-dimensional hydrodynamic phenomenon. Due to their extremely thin nature, these films adhere to the Navier-Stokes equations at lower Mach numbers. Moreover, curved surfaces find applications in curved jaws of flexible assembly machines. One prominent biological aspect is the lipid bilayer membrane, which forms boundary of many cells. Lipid bilayer exhibits hydrodynamic properties such as viscosity and diffusion, which have been validated through various experiments. Sajid et al. ¹⁸ numerically investigated flow of viscous fluid on curved stretching surface and noted that drag force was lesser and boundary layer was greater at curved surface in comparison to flat surface. In another study, Sajid et al. ¹⁹ examined flow of micropolar fluid past curved surface and noted that velocity was superior on curved surface comparison to flat surface. Rosca and Pop ²⁰ scrutinized the fluid flow generated by shrinking and stretching curved sheet. Naveed et al. ²¹ studied flow of fluid on curved surface considering magnetohydrodynamic. Waqas et al. ²² analyzed MHD hybrid nanofluid flow past curved sheet. Their results revealed that increase in curvature factor and volume fraction of nanoparticles amplifies velocity profile. Okechei et al. ²³ deliberated flow of viscous fluid over a rapidly stretching curved surface and observed that curvature factor was directly proportional to the shear stress. Some other researchers have also studied flow past curved surface.^24–28

Thermal radiation, which involves transfer of heat energy in the form of electromagnetic waves, is a fundamental mechanism of energy transfer. From physical point of view, these are photons emitted by atoms or molecules of a substance having temperature above absolute zero. This form of heat transfer is unique in a way that it occurs without the need of a physical medium. Thermal radiations play a crucial role in numerous applications including heating processes like microwave oven, industrial processes, solar energy systems, nuclear power plants and many more. Venkateswarlu et al. ²⁹ explored the heat transfer of hybrid nanofluid flow past a porous stretching surface considering thermal radiations and observed that thermal factor increases the temperature by 14.89 %. Harinath Reddy et al. ³⁰ investigated the influence of thermal radiations on stagnation point chemically reactive MHD hybrid nanofluid flow and deduced that temperature panel rises with higher radiation parameter. Thermal radiations are a prominent heat transfer parameter in numerous other studies involving heat transfer.^31–37

Earlier, it was believed that concentration of nanoparticles at the surface remains constant. However, Nield and Kuznetsov ³⁸ extended the idea by introducing a new boundary condition which incorporates two nanometric phenomenon, the Brownian motion and thermophoresis. The previously assumed hypothesis of constant concentration is now termed as active control while that influenced by the Brownian and thermophoresis ³⁹ effects is referred as passive control. Acharya et al. ⁴⁰ evaluated thermal analysis of nanofluid flow using active passive controls over a linear curved stretching surface and observed lower heat transport rate for passive control nanofluid flow. Revanna Lalitha et al. ⁴¹ explored effects of active passive controls on Jeffery nanofluid fluid flow over a stretching sheet and deduced that temperature boosts sharply for active controls as compared to passive controls. In another relevant study, Hayat et al. ⁴² employed active passive controls to examine thermal and rheological behaviour of MHD Jeffery fluids flow generated by nonlinear stretching sheet and observed increasing trend in thermal as well as nanoparticles concentration panels for both active and passive controls with rising thermophoresis while reverse behaviour was reported for increase in Brownian movement. Kalaivanan et al. ⁴³ scrutinized second grade nanofluid flow with activation energy using nanoparticles active and passive controls and observed that concentration and Nusselt number improve by greater activation energy. Moreover, impact of thermophoresis and Brownian motion on concentration was lower in active control. Khan et al. ⁴⁴ investigated the hydrothermal behaviour of nanoliquid past curved surface using active and passive controls and declared that Brownian and thermophoresis factors augment the thermal boundary layer.

Cobalt $\left(\mathrm{Co}\right)$ , Silver $\left(\mathrm{Ag}\right)$ , and Zinc $\left(\mathrm{Zn}\right)$ are versatile metals playing a critical role in both industrial and consumer applications. Cobalt is used as catalyst in plastic production, lithium-ion batteries, glass and ceramics as well as in biomedical in radiation therapy for cancer treatment. ⁴⁵ Silver, owing to its unique properties of high thermal and electrical conductivity, is used in electronic components, solar panels, medical devices and photographic films. ⁴⁶ Zinc is widely used in coating to prevent corrosion, in zinc-carbon batteries, in manufacturing of rubber and sunscreens etc. ⁴⁷ Ternary hybrid nanofluids containing $\mathrm{Co},\mathrm{Zn},\mathrm{and}\mathrm{Ag}$ in di-hydrogen monoxide as base fluid offer a powerful combination possessing improved thermos-rheological properties and magnetic functionality having wide ranging applications in numerous industries, including cooling systems, industrial processes and bio-medical applications.^48–50 This study considers the water based ternary nanofluids containing nanoparticles of Siver, Zinc, and Cobalt. This unique combination provides better stability and high thermal conductivity. Its ease of preparation, cost effectiveness and low toxicity make it a most suitable ternary hybrid nanofluid in heat transfer applications.

A close scrutiny of literature reviewed reveals us that hydrothermal investigation of loading ternary nanoparticles navigating a porous curved surface with active passive controls using non-Fourier approach has not yet been studied by any researcher. Motivated by the practical aspects of such investigation as conferred in former paras, it is apparent that this communication offers valuable insights into various heat and mass transport challenges. Extensive studies are available in literature on nanoliquid flow past flat stretching sheet; however, exploration of how specific factors within the flow influence the hydrothermal characteristics is relatively rare. Furthermore, flow over flat surface is a particular case of such flow over curved surface. Therefore, this embodies broader configuration. To address this research gap, this premier investigation is aimed at evaluating the following research objectives.

• (a) To determine the impact of curvature factor and porosity of medium on the flow and thermal pattern of ternary hybrid nanofluids past a curved stretching surface.

• (b) To explore influences of adding metal nanoparticles in base fluid on skin friction at the curved surface.

• (c) To analyze effects of different control mechanisms (active vs passive) on rheological and thermal profile in boundary layer.

• (d) To investigate impression of radiations intensity on nanoparticles concentration in active and passive control scenarios and under different conditions?

• (e) To evaluate bearing of Brownian motion and thermophoresis on nanoparticles concentration and mas transport rates in active versus passive strategies.

Model formulations

we have selected curvilinear coordinates system $(r,s)$ to represent physical model of problem keeping in view the curved shape of our geometry shown in Figure 1. The coordinate axis is chosen such that $s-$ component of velocity aligns with the curved surface which corresponds to $x-$ component of Cartesian coordinates system and fluid is flowing along this direction whereas $r-$ component of curvilinear coordinates system is in the radial direction perpendicular to curved surface and corresponds to $y-$ component of velocity. The boundary layer forms in radial direction. The curved surface is stretching with uniform velocity $U_{\mathrm{cs}}=\mathrm{as}$ which drives the fluid flow. This communication takes into consideration the loading of tri-hybrid nanoliquid which is blend of three different metals $(\mathrm{Co},\mathrm{Ag},\mathrm{Zn})$ in Di-hydrogen Mono-oxide as base fluid over the curved porous geometry for scrutinizing its thermo-rheological characteristics through active and passive controls of nanoparticles at solid-liquid boundary. Temperature of solid curved surface is $T_{\mathrm{cs}}$ whereas ambient temperature of nanoliquid is $T_{\infty }$ such that $T_{\mathrm{cs}}>T_{\infty }$ . Similarly, $C_{\mathrm{cs}}\mathrm{and}{C}_{\infty }$ are the corresponding concentration values. A strong magnetic field of strength $B_{°}$ acts in radial direction of curved surface. The energy equation incorporates the famous Buongiorno model taking into consideration the significant slip factors responsible for energy transfer in ternary nanofluid. Thermal radiations have also been taken under advisement for comprehensive thermal analysis of ternary nanofluid flow. Under the presumed constraints, governing equations of this model are written as.^18,20,51,52

$$\frac{\partial }{\partial r}\left(\tilde{r}v\right)+\mathrm{R}\frac{\partial u}{\partial s}=0$$

(1)

$$\left(\frac{u^2}{\tilde{r}}\right)=-\frac{1}{{\rho }_{\mathrm{thnf}}}\frac{\partial p}{\partial r},$$

(2)

$$\begin{array}{c}v\frac{\partial u}{\partial r}+\left(\frac{1}{\tilde{r}}\right)\mathrm{uv}+\left(\frac{R}{\tilde{r}}\right)u\frac{\partial u}{\partial s}=-\frac{1}{{\rho }_{\mathrm{thnf}}}\left(\frac{R}{\tilde{r}}\right)\frac{\partial p}{\partial s}+\hfill \\ {\nu }_{\mathrm{thnf}}(\frac{{\partial }^{2}u}{\partial r^2}+\frac{\partial u}{\partial r}\left(\frac{1}{\tilde{r}}\right)-u{\left(\frac{1}{\tilde{r}}\right)}^2)-\frac{{\sigma }_{\mathrm{thnf}}}{{\rho }_{\mathrm{thnf}}}{B_{°}}^{2}u-\frac{{\nu }_{\mathrm{thnf}}}{\stackrel{\prime }{\kappa }}u\hfill \end{array}$$

(3)

$$\begin{array}{c}\mathrm{v}\frac{\partial T}{\partial r}+\left(\frac{R}{\tilde{r}}\right)\mathrm{u}\frac{\partial T}{\partial s}=\frac{k_{\mathrm{thnf}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}[\left(\frac{1}{\tilde{r}}\right)\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}]+\frac{{\sigma }_{\mathrm{thnf}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}{B_{°}}^2{\mathrm{u}}^2\hfill \\ +\frac{{\left(\rho \mathrm{Cp}\right)}_p}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}(D_B\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2)-\frac{1}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}\left(\tilde{r}\right)}\frac{\partial }{\partial r}\left(\tilde{r}\right)q_r\hfill \end{array}$$

(4)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{v}\frac{\partial C}{\partial r}+\left(\frac{R}{\tilde{r}}\right)\mathrm{u}\frac{\partial C}{\partial s}=D_B[\frac{\partial C}{\partial r}\left(\frac{1}{\tilde{r}}\right)+\frac{{\partial }^{2}C}{\partial r^2}]}\\ +\frac{D_T}{T_{\infty }}(\frac{\partial T}{\partial r}\left(\frac{1}{\tilde{r}}\right)+\frac{{\partial }^{2}T}{\partial r^2})\hfill \end{array}$$

(5)

Figure 1.

Geometrical representation of the flow.

In above $\tilde{r}=r+R$ . The associated boundary conditions are given as.^40,53,54

$$\begin{array}{c}\left\{\begin{array}{c}u=U_{\mathrm{cs}}=\mathrm{as},v=0,T=T_{\mathrm{cs}},\hfill \\ \multicolumn{1}{c}{D_B\frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty }}\frac{\partial T}{\partial r}=0,\left(\mathrm{passive}\mathrm{control}\mathrm{of}\varphi \right)}\\ \multicolumn{1}{c}{C=C_w\left(\mathrm{active}\mathrm{control}\mathrm{of}\varphi \right)}\end{array}\right\}\mathrm{when}r=0,\hfill \\ u\to 0,\frac{\partial u}{\partial r}\to 0,T\to T_{\infty },C\to C_{\infty },\mathrm{when}r\to \infty \hfill \end{array}$$

(6)

Where, $u$ and $v$ are velocity components along $s$ , $r$ directions respectively, p stands for pressure, $\nu$ is the kinematics viscosity, $\mu$ is density, $\sigma$ is electrical conductivity, $K^{\prime }$ is porosity of medium, $k$ is thermal conductivity, $\rho \mathrm{Cp}$ designates heat capacitance, $D_T\mathrm{and}{D}_B$ characterize thermophoresis and Brownian motion, subscripts $\mathrm{tnf},\mathrm{and}f$ specify the ternary nanofluid and base fluid resp, $q_r$ means thermal radiation parameter defined by Rosseland approximation in the form. ⁵⁵

$$q_r=\frac{4{\sigma }^{*}\partial T^4}{3k^{*}\partial r}$$

(7)

Here ${\sigma }^{*}$ designates Stephan Boltzmann constant and $k^{*}$ embodies mean absorption coefficient. By using Taylor’ series to expand $T^4$ about $T_{\infty }$ and considering the $\delta T$ $(=T-T_{\infty })$ sufficiently small, we ignore square and higher order terms of $\delta T$ to get.

$$T^4\approx T_{\infty }^3(4T-3T_{\infty })$$

(8)

Using equations (8) in (7), we can have

$$q_r=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3K^{*}}\frac{\partial T}{\partial r}$$

(9)

Normalization analysis

In order to convert the governing equations into non-dimensional format, we employ the appropriate unitless functions as^53,56,57

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}u=U_{w}f\prime \left(\eta \right),v=\frac{-R}{\tilde{r}}\sqrt{a{\nu }_f}f\left(\eta \right),p=\rho {U_w}^{2}P\left(\eta \right),\eta =\sqrt{\frac{a}{{\nu }_f}}r,\hfill \\ \multicolumn{1}{c}{\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\{\begin{array}{c}\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_{\infty }},\left(\mathrm{for}\mathrm{passive}\mathrm{control}\right)\hfill \\ \multicolumn{1}{c}{\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\left(\mathrm{for}\mathrm{active}\mathrm{control}\right)}\end{array}}\end{array}\}}\end{array}$$

(10)

The renovated equations (1)–(5) by employing the stated similarity variables becomes as

$$P\prime -\frac{{f\prime }^2}{\tilde{\eta }}=0$$

(11)

$$\begin{array}{c}\frac{\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}}{\frac{{\rho }_{\mathrm{thnf}}}{{\rho }_f}}(f″\prime +\frac{f″}{\tilde{\eta }}-\frac{f\prime }{{\tilde{\eta }}^2})-\frac{\frac{{\sigma }_{\mathrm{thnf}}}{{\sigma }_f}}{\frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}}\mathrm{Mf}\prime +\frac{\mathrm{K}\mathrm{ff}″}{\tilde{\eta }}-\frac{\mathrm{K}{f\prime }^2}{\tilde{\eta }}\hfill \\ +\frac{\mathrm{K}f\prime f}{{\tilde{\eta }}^2}-\frac{{\rho }_f}{{\rho }_{\mathrm{thnf}}}\frac{2\mathrm{K}P}{\tilde{\eta }}-\frac{\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}}{\frac{{\rho }_{\mathrm{thnf}}}{{\rho }_f}}\lambda f\prime =0\hfill \end{array}$$

(12)

The pressure function can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{P\left(\eta \right)=\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}(\left(\frac{\tilde{\eta }}{2\mathrm{K}}\right)f″\prime +\frac{f″}{2\mathrm{K}}-\frac{f\prime }{2\mathrm{K}\tilde{\eta }})-\frac{{\sigma }_{\mathrm{thnf}}}{{\sigma }_f}\frac{\tilde{\eta }}{2\mathrm{K}}\mathrm{Mf}\prime }\\ +\frac{{\rho }_{\mathrm{thnf}}}{{\rho }_f}(\frac{1}{2}f″f-\frac{f^{\prime 2}}{2}+\frac{1}{2\tilde{\eta }}f\prime f)-\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}\frac{\tilde{\eta }}{2\mathrm{K}}\lambda f\prime \hfill \end{array}$$

(13)

In order to eliminate the pressure term from above equation (13) we differentiate it with respect to $\eta$ and substitute in equation (11), we have

$$\begin{array}{c}\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}(f^{\left(\mathrm{IV}\right)}+\frac{2f″\prime }{\tilde{\eta }}-\frac{f″}{{\tilde{\eta }}^2}+\frac{f\prime }{{\tilde{\eta }}^3})-\frac{{\sigma }_{\mathrm{Thnf}}}{{\sigma }_f}M(f″+\frac{f\prime }{\tilde{\eta }})\hfill \\ +\frac{{\rho }_{\mathrm{thnf}}}{{\rho }_f}(\frac{\mathrm{K}}{\tilde{\eta }}(\mathrm{ff}″\prime -f″f\prime )+\frac{\mathrm{K}}{{\tilde{\eta }}^2}(\mathrm{ff}″-f^{\prime 2})-\frac{\mathrm{K}}{{\tilde{\eta }}^3}\mathrm{ff}\prime )\hfill \\ -\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}\lambda (f″+\frac{f\prime }{\tilde{\eta }})=0\hfill \end{array}$$

(14)

$$\begin{array}{c}\left(\frac{\frac{K_{\mathrm{thnf}}}{K_f}}{\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}{{\left(\rho \mathrm{Cp}\right)}_f}}\right)\frac{1}{\mathrm{Pr}}(\theta ″+\frac{\theta \prime }{\tilde{\eta }})+\frac{\mathrm{K}f\theta \prime }{\tilde{\eta }}+\left(\frac{{\left(\rho \mathrm{Cp}\right)}_f}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}\right)\hfill \\ (\mathrm{Nb}\theta \prime \varphi \prime +\mathrm{Nt}{\theta \prime }^2)+\frac{\frac{{\sigma }_{\mathrm{thnf}}}{{\sigma }_f}}{\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}{{\left(\rho \mathrm{Cp}\right)}_f}}\mathrm{MEc}{f\prime }^2+\frac{\mathrm{Rd}}{\frac{\mathrm{Pr}{\left(\rho \mathrm{Cp}\right)}_{\mathrm{thnf}}}{{\left(\rho \mathrm{Cp}\right)}_f}}(\theta ″+\frac{\theta \prime }{\tilde{\eta }})=0\hfill \end{array}$$

(15)

$$\frac{1}{\mathrm{Le}}(\varphi ″+\frac{\varphi \prime }{\tilde{\eta }})+\frac{\mathrm{Nt}}{\mathrm{Nb}}(\theta ″+\frac{\theta \prime }{\tilde{\eta }})+\frac{\mathrm{K}f\varphi \prime }{\tilde{\eta }}=0$$

(16)

In equations (14)–(16) which represent the flow profile of this problem, $\tilde{\eta }=\eta +\mathrm{K}$ . The non-dimensional form of boundary conditions become:

$$\begin{array}{c}\left\{\begin{array}{c}f=0,f\prime =1,\theta =1,\hfill \\ \multicolumn{1}{c}{[\begin{array}{c}\mathrm{Nb}\varphi \prime +\mathrm{Nt}\theta \prime =0,\mathrm{for}\mathrm{passive}\mathrm{control}\hfill \\ \multicolumn{1}{c}{\varphi =1\mathrm{for}\mathrm{active}\mathrm{control}}\end{array}}\end{array}\right\}\mathrm{at}\eta =0\hfill \\ f\prime =f″=\theta =\varphi \to 0,\mathrm{as}\eta \to \infty \hfill \end{array}$$

(17)

In above equations the dimensionless factors are defined as:

Prandtl number	$\mathrm{Pr}=\frac{{\nu }_f{(\rho \mathrm{Cp})}_f}{k_f}$	Thermal radiation factor	$\mathrm{Rd}=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3K_f{K}^{*}}$
Brownian motion parameter active flow	$\mathrm{Nb}=\frac{\tau D_B{C}_{\infty }}{{\nu }_f}$	Brownian motion parameter active flow	$\mathrm{Nb}=\frac{\tau D_B(C_w-C_{\infty })}{{\nu }_f}$
Magnetic factor	$M=\frac{{\sigma }_f{B}_{°}^2}{{\rho }_{f}a}$	Eckert Number	$E_c=\frac{{U^2}_w}{{\left(\mathrm{Cp}\right)}_f(T_w-T_{\infty })}$
Porosity parameter	$\lambda =\frac{{\mu }_f}{{\rho }_{f}a{K}^{\prime }}$	Lewis number	$\mathrm{Le}=\frac{\alpha }{D_B}$
Curvature factor	$\mathrm{K}=R\sqrt{\frac{a}{{\nu }_f}}r$	Thermophoresis parameter	$\mathrm{Nt}=\frac{\tau D_T(T_w-T_{\infty })}{{\nu }_f{T}_{\infty }}$

Physical engineering parameters

The noteworthy parameters of engineering significance in this investigation include reduced skin friction $C{f}_s$ , which provides information about drag force effects at curved surface texture, Nusselt number $N{u}_s$ which is related to heat transport heat and Sherwood number $S{h}_s$ , representing the rate of mass transfer. These are generally characterized as:

$$C{f}_s=\frac{{\tau }_{\mathrm{rs}}}{\rho {U_w}^2},\mathrm{Where}{\tau }_{\mathrm{rs}}={{\mu }_{\mathrm{thnf}}(\frac{\partial u}{\partial r}-\frac{u}{r+R})|}_{r=0}$$

(18)

$$N{u}_s=\frac{s{q}_s}{k_f(T_w-T_{\infty })}\mathrm{Where}{q}_s={-k_{\mathrm{thnf}}\left(\frac{\partial T}{\partial r}\right)|}_{r=0}$$

(19)

$$S{h}_s=\frac{s{j}_w}{D_B(C_w-C_{\infty })}\mathrm{Where}{j}_w={-D_B\left(\frac{\partial C}{\partial r}\right)|}_{r=0}$$

(20)

In above, ${\tau }_{\mathrm{rs}}$ , $q_s$ , and $j_w$ are tangential shear stress, thermal flux, and mass flux at surface. Using similarity transformations, we have.

$$C{f}_r=C{f}_s{\left(\mathrm{Re}\right)}^{1/2}=\frac{{\mu }_{\mathrm{thnf}}}{{\mu }_f}(f″\left(0\right)-\frac{1}{K}f\prime \left(0\right))$$

(21)

$$N{u}_r=N{u}_s{{\left(\mathrm{Re}\right)}^{-}}^{1/2}=-\frac{K_{\mathrm{thnf}}}{K_f}\theta \prime \left(0\right)$$

(22)

$$S{h}_r=S{h}_s{{\left(\mathrm{Re}\right)}^{-}}^{1/2}=\{\begin{array}{c}\frac{-\varphi \prime \left(0\right)}{\varphi \left(0\right)}\left(\mathrm{passive}\mathrm{control}\mathrm{of}\varphi \right)\hfill \\ -\varphi \prime \left(0\right)\left(\mathrm{active}\mathrm{control}\mathrm{of}\varphi \right)\hfill \end{array}$$

(23)

$\mathrm{Re}=\frac{a{s}^2}{{\nu }_f}$ is local Reynolds number.

Thermophysical characteristics

To gain deeper insights into flow dynamics of ternary nanofluids across various physical conditions and flow scenarios, it is imperative to consider their thermophysical properties. Table 1 presents numerical values of key thermophysical properties of nanoparticles and base fluid.

Table 1.

Thermophysical properties values of used nanoparticles and base fluid.^58–60

Thermophysical properties	$H_{2}O$	$\mathrm{Co}$	$\mathrm{Ag}$	$\mathrm{Zn}$
Specific heat $\left(\mathrm{Cp}\right)$ $\left[J{k}^{-1}{g}^{-1}{K}^{-1}\right]$	4179	420	233	387
Density $\left(\rho \right)$ $\left[\mathrm{Kg}{m}^{-3}\right]$	997.1	8900	10,500	7135
Thermal conductivity $\left(k\right)$ $\left[W{m}^{-1}{K}^{-1}\right]$	0.613	100	429	116
Electrical conductivity ( $\sigma$ ) $\left[S{m}^{-1}\right]$	$5.50\times {10}^{-6}$	$1.602\times {10}^7$	$6.30\times {10}^7$	$1.69\times {10}^7$

Various relationships, theoretical models as well as experimental investigations are employed to compute thermophysical characteristics of nanoliquid. The relations for vital thermophysical characteristics are presented in Table 2.

Table 2.

Thermophysical properties for nanoparticles and base fluid.^61,62

Viscosity	$\frac{{\mu }_{\mathrm{tnf}}}{{\mu }_{\mathrm{bf}}}=\frac{1}{{(1-{\phi }_{\mathrm{Co}})}^{2.5}{(1-{\phi }_{\mathrm{Ag}})}^{2.5}{(1-{\phi }_{\mathrm{Zn}})}^{2.5}}$
Density	$\frac{{\rho }_{\mathrm{tnf}}}{{\rho }_{\mathrm{bf}}}=(1-{\phi }_{\mathrm{Zn}})[(1-{\phi }_{\mathrm{Ag}})\{(1-{\phi }_{\mathrm{Co}})+{\phi }_{\mathrm{Co}}\frac{{\rho }_{\mathrm{Co}}}{{\rho }_{\mathrm{bf}}}\}+{\phi }_{\mathrm{Ag}}\frac{{\rho }_{\mathrm{Ag}}}{{\rho }_{\mathrm{bf}}}]+{\phi }_{\mathrm{Zn}}\frac{{\rho }_{\mathrm{Zn}}}{{\rho }_{\mathrm{bf}}}$
Specific heat capacity	$\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{tnf}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{bf}}}=(1-{\phi }_{\mathrm{Zn}})[(1-{\phi }_{\mathrm{Ag}})\{(1-{\phi }_{\mathrm{Co}})+{\phi }_{\mathrm{Co}}\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{Co}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{bf}}}\}+{\phi }_{\mathrm{Ag}}\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{Ag}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{bf}}}]+{\phi }_{\mathrm{Zn}}\frac{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{Zn}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{bf}}}$
Thermal conductivity	$\frac{k_{\mathrm{tnf}}}{k_{\mathrm{bf}}}=\left[\frac{k_{\mathrm{Zn}}+2k_{\mathrm{hnf}}-2(k_{\mathrm{hnf}}-k_{\mathrm{Zn}}){\phi }_{\mathrm{Zn}}}{k_{\mathrm{Zn}}+2k_{\mathrm{hnf}}+(k_{\mathrm{hnf}}-k_{\mathrm{Zn}}){\phi }_{\mathrm{Zn}}}\right]\left(\frac{k_{\mathrm{hnf}}}{k_{\mathrm{bf}}}\right)$ where $\frac{k_{\mathrm{hnf}}}{k_{\mathrm{nf}}}=\frac{k_{\mathrm{Ag}}+2k_{\mathrm{nf}}-2(k_{\mathrm{nf}}-k_{\mathrm{Ag}}){\phi }_{\mathrm{Ag}}}{k_{\mathrm{Ag}}+2k_{\mathrm{nf}}+(k_{\mathrm{nf}}-k_{\mathrm{Ag}}){\phi }_{\mathrm{Ag}}}$ and $\frac{k_{\mathrm{nf}}}{k_{\mathrm{bf}}}=\frac{k_{\mathrm{Co}}+2k_{\mathrm{bf}}-2(k_{\mathrm{bf}}-k_{\mathrm{Co}}){\phi }_{\mathrm{Co}}}{k_{\mathrm{Co}}+2k_{\mathrm{bf}}+(k_{\mathrm{bf}}-k_{\mathrm{Co}}){\phi }_{\mathrm{Co}}}$
Electrical conductivity	$\frac{{\sigma }_{\mathrm{tnf}}}{{\sigma }_{\mathrm{bf}}}=\left[\frac{(1+2{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{Zn}}+(1-2{\phi }_{\mathrm{Zn}}){\sigma }_{\mathrm{hnf}}}{(1-{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{Zn}}+(1+{\phi }_{\mathrm{Zn}}){\sigma }_{\mathrm{hnf}}}\right]\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_{\mathrm{bf}}}$ where $\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_{\mathrm{nf}}}=\frac{(1+2{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{Ag}}+(1-2{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{nf}}}{(1-{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{Ag}}+(1+{\phi }_{\mathrm{Ag}}){\sigma }_{\mathrm{bf}}}$ and $\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_{\mathrm{bf}}}=\frac{(1+2{\phi }_{\mathrm{Co}}){\sigma }_{\mathrm{Co}}+(1-2{\phi }_{\mathrm{Co}}){\sigma }_{\mathrm{bf}}}{(1-{\phi }_{\mathrm{Co}}){\sigma }_{\mathrm{Co}}+(1+{\phi }_{\mathrm{Co}}){\sigma }_{\mathrm{bf}}}$

Numerical solution

The acquired system of ODEs being highly non-linear and coupled is not possible to solve analytically. In order to get a numerical solution, MATLAB bvp4c package has been employed on the system of equations (14)–(16) along with boundary conditions given by equation (17). The bvp4c is an efficient numerical technique which has been specifically designed to solve boundary value problems related to ODEs. This method employs a combination of shooting method and finite difference method to compute an approximate solution to the boundary value problem. This method requires expressing the higher order ODEs system and the associated boundary values to a first order ODEs system. Moreover, an initial guess of the solution is also required. The solver integrates the first order ODEs system taking the initial guess toward end of the given domain. At the end of domain, it computes the residual error. For the convergence of solution, this residual error must be less than a specified level. On reaching a specified level of convergence, results can be acquired for analysis in the form of numerical data or graphs as per the requirement. This method is easy to operate, computationally cost effective as it takes less time to execute as compared to other methos and is frequently used to solve highly non-linear ODEs systems.^63,64

Method validation

To confirm the accuracy of coded problem, Table 3 presents a comparison of obtained results for skin friction across different curvature parameter values with the data previously published in open literature by Rosca et al., ²⁰ Abbas et al. ⁶⁵ and Afridi et al. ⁶⁶ A fine agreement amongst results validates our code.

Table 3.

Comparison of reduced present model for $C{f}_r$ with previous published work for varying $K$ , keeping ${\phi }_1,{\phi }_2,{\phi }_3,\gamma \mathrm{and}M=0$ .

$K$	Abbas et al. 65	Afridi et al. 66	Present results
5	1.5763	1.15763	1.1576310
10	1.07349	1.07348	1.0734886
50	1.01405	1.01404	1.0140491
100	1.00704	1.00703	1.0070383
1000	1.00079	1.00079	1.0007993

Results and discussion

This segment constitutes most important portion of this pragmatic study, playing a crucial role in parametric examination of the overall flow regime. It contributes in better understanding the results of the pertinent physical quantities which have been obtained by employing the bvp4c numerical technique. Here the explanation covers how the parameters at play impact the velocity, temperature and mass transport. It is highlighted that prime objective of this study is was to investigation of thermo-rheological behaviour of water based ternary hybrid nanofluid containing nanoparticles of $\mathrm{Co},\mathrm{Zn},\mathrm{and}\mathrm{Ag}$ . Metal nanoparticles have been considered due to their higher thermal properties. This unique combination of metallic nanoparticles with water as base fluid is chosen owing to better stability, high thermal conductivity, ease of preparation, cost effectiveness and low toxicity. Brownian motion and thermophoresis being the most relevant nanometric phenomena are taken into account. The passive control of nanoparticles on the surface influenced by Brownian movement and thermophoresis, which is considered more realistic in comparison to constant concentration hypothesis (active control), have been taken into account. The profiles of velocity, temperature and nanoparticles concentration are plotted for varying sundry parameters as shown in Figures 3–10 keeping value of Prandtl number fixed $(\mathrm{Pr}=6.2)$ and varying values of other factors that have been annotated at each graph.

The values of skin friction, heat transfer rate and mass transport factors have been presented in the tabular form given in Tables 4–6. The findings and their implications are thoroughly discussed and explained, supported by logical justifications.

Table 4.

Numerical values of $C_{f_r}$ for varying $M,\lambda$ , and $K$ .

$K$	$\lambda$	$M$	$C_{f_r}$
			Nanofluid $(\mathrm{Co}/H_{2}O)$	Hybrid $(\mathrm{Co}+\mathrm{Ag}/H_{2}O)$	Ternary $(\mathrm{Co}+\mathrm{Ag}+\mathrm{Zn}/H_{2}O)$
1	0.1	0.1	1.6965	1.7252	1.7859
3			1.4648	1.4880	1.5419
5			1.3658	1.3868	1.3126
	0.1		2.3926	2.4458	2.5168
	0.2		2.8464	2.9126	2.9938
	0.3		3.2309	3.3077	3.3980
		1	2.3926	2.4458	2.5168
		2	2.8464	2.9126	2.9938
		3	3.2309	3.3077	3.3980

Table 5.

The numerical values of $\mathrm{Nu}$ for varying $M,\mathrm{Nb},\mathrm{Nt},\lambda$ and $K$ .

$M$	$K$	$\mathrm{Nb}$	$\mathrm{Nt}$	$\lambda$	$N{u}_r$
					Nanofluid $(\mathrm{Co}/H_{2}O)$	Hybrid $(\mathrm{Co}+\mathrm{Ag}/H_{2}O)$	Ternary $(\mathrm{Co}+\mathrm{Ag}+\mathrm{Zn}/H_{2}O)$
1	5	0.1	0.1	0.1	0.3718	0.3885	0.4032
2					0.3150	0.3299	0.3440
3					0.2751	0.2887	0.3021
	1				0.6448	0.6769	0.7054
	3				0.5121	0.5345	0.5519
	5				0.4545	0.4736	0.4879
		0.1			0.4577	0.4769	0.4911
		0.2			0.4212	0.4398	0.4540
		0.3			0.3873	0.4054	0.4194
			0.1		0.4577	0.4769	0.4911
			0.2		0.4375	0.4563	0.4706
			0.3		0.4184	0.4369	0.4510
				1	0.4577	0.4769	0.4911
				2	0.4496	0.4682	0.4823
				3	0.4420	0.4602	0.4742

Table 6.

The numerical values of $S{h}_r$ for varying $\mathrm{Le},\mathrm{Nb},\mathrm{Nt}\mathrm{and}\mathrm{BiT}$ for active and passive controls od nanoparticles.

$\mathrm{Le}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$S{h}_r$
			Nanofluid $(\mathrm{Co}/H_{2}O)$		Hybrid $(\mathrm{Co}+\mathrm{Ag}/H_{2}O)$		Ternary $(\mathrm{Co}+\mathrm{Ag}+\mathrm{Zn}/H_{2}O)$
			Active control	Passive control	Active control	Passive control	Active control	Passive control
5	0.1	0.1	1.4337	2.2909	1.4337	2.3013	1.4296	2.2983
6			1.6150	2.4685	1.6152	2.4785	1.6111	2.4761
7			1.7720	2.6292	1.7811	2.6390	1.7770	2.6368
	0.1		1.4337	2.2909	1.4337	2.3013	1.4296	2.2983
	0.2		1.5079	2.2909	1.5092	2.3013	1.5052	2.2983
	0.3		1.5319	2.2909	1.5336	2.3013	1.5297	2.2983
		0.1	1.4337	2.2909	1.4337	2.3013	1.4296	2.2983
		0.2	1.3403	2.2581	1.3370	2.2697	1.3310	2.2674
		0.3	1.2724	2.2249	1.2652	2.2377	1.2578	2.2360

Figure 2(a) portrays impression of curvature factor on dimensionless velocity profile. It is observed that velocity profile amplifies with an increase in values of curvature factor. Parametric representation of $K=\sqrt{a/{\nu }_f}R$ implies that larger values of $K$ correspond to reduced kinematic viscosity and therefore the nanoliquid will be subjected to lesser viscous resistance to its motion thus leading to an increase in speed of fluid. It is also worth noting that pressure effects are significant on curved surface which are usually ignored on flat surface. Figure 3(a)–(d) shows the contour plots for increasing values of curvature factor. It can be observed from the graphs that increase in values of curvature parameter results in thickening the boundary layer. We know that pressure force is significant on curved surface which is negligible in case of flat surface and is generally ignored in studying fluid flow behaviour on flat sheet. Physically, when radius of curvature is increased, the curvature decreases and surface tends to flatten, which results in decrease in of pressure force. We can see the contour lines widening, which means that velocity boundary layer is increasing with increasing numerical values of curvature parameter. Additionally, the increase of velocity can be justified from numerical values of surface drag shown in Table 4. It can be noted that as surface tends to flatten by increasing radius of curvature, the surface drag force recedes thus allowing the fluid to move smoothly and speedy on the surface. The impression of porosity parameter $\left(\lambda \right)$ on dimensionless velocity is pictured in Figure 2(b). It is clear that greater porosity factor decreases velocity in the boundary layer at curved surface. Physically, this reduction in flow velocity is the result of additional resistance introduced by the porous media. Figure 4 demonstrates the impression of increasing curvature factor on temperature and concentration panels. It is noted that higher curvature values have inverse relation with temperature in the boundary layer. The concentration boundary layer profile shows opposite results for active and passive control of ternary nanoparticles at surface. Here decrease in temperature panel is the result of increase in velocity of flow which also takes away the nanoparticles, thus allowing less contact time with the boundary. Figure 4(a) also reveals that the impact of curvature is larger in case of passive control of nanoparticles. From concentration panel, it is seen that passive control of nanoparticles has positive impact on nanoparticles concentration in the boundary layer. The numerical values for heat transfer rate for varying $K$ shown in Table 5 indicate an inverse relationship which substantiate the trend shown in Figure 4(a). The nanoparticles concentration decreases for increasing the curvature factor in case of active control of nanoparticles, however, it shows an improving impact in case of passive control of the tiny particles at solid fluid interface as is illustrated in Figure 4(b).

Figure 2.

Dimensionless curvature (K) and porosity parameter (λ) on dimensionless velocity.

Figure 3.

Variations of streamlines for various values of dimensionless curvature $\left(K\right)$ .

Figure 4.

The impression of dimensionless curvature factor $\left(K\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

The effects of porous media parameter (λ) on temperature and concentration profiles are depicted in Figure 5. It is evident from the portrayal in Figure 5(a) that porosity parameter increases the temperature panel for both active and passive controls. This increase in temperature is greater for active control of nanoparticles in comparison to passive control. Physically, this rise of temperature is due to increased interaction of nanoparticles with porous surface area which enhances heat transfer rate from nanoparticles to surface.

Figure 5.

The effect of porosity parameter (λ) on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

Also, the numerical values in Table 4 for frictional force at curved surface as result of increasing the porosity parameter show a rising trend. This resistive force also contributes in augmentation of temperature in boundary layer region. Figure 5(b) indicates that porosity factor augments the nanoparticles concentration in boundary layer for active control of nanoparticles whereas reverse effects is observed for passive control. Figures 6 and 7 elucidate the impression of Brownian motion parameter $\left(\mathrm{Nb}\right)$ and thermophoresis factor $\left(\mathrm{Nt}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ resp. It is perceived that heightening Brownian and thermophoresis parameters have encouraging impact on the temperature. Physically, the increased zigzag motion of nanoparticles enhances the heat convection rate. The larger thermal gradient will stimulate intense thermophoresis process which consequently augments the temperature field. Figure 6(a) indicates a stagnant impact on temperature in case of passive control of nanoparticles. Figure 7(a) also depicts that this rising effect is larger for active control in comparison to passive control. Figure 6(b) implies that Brownian motion improves concentration of particles for passive control while declining trend is noted in case of active control. Here an interesting scenario catches our attention. We notice an increase in mass diffusion becoming lucid near $\eta =1$ and after this point a sudden cross over is observed which wipes out after small period. The same pattern is also noted for $\left(\mathrm{Nt}\right)$ graph, however, the behaviour of thermophoresis on concentration in boundary layer is exactly opposite to that of Brownian motion as depicted in Figure 7(b). The numerical data shown in Table 5 shows that Nusselt number decreases for both said parameters while in case of Sherwood number, the impact is opposite to each other.

Figure 6.

The effect of Brownian parameter $\left(\mathrm{Nb}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

Figure 7.

The effect of thermophoresis parameter $\left(\mathrm{Nt}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

Thermal radiations influence on temperature and concentration is portrayed in Figure 8. It can be seen that temperature elevates by intensifying the radiations factor. Additionally, the radiations give better heat transfer rate for active control of tiny particles at surface rather than the passive control. Figure 7(b) communicates that results are opposite for passive and active control strategies inside $0\le \eta 1$ . Active control provides enormous mass transport for higher radiations whereas the reverse is true for passive strategy.

Figure 8.

The effect of thermal radiations constant $\left(\mathrm{Rd}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

Figures 9 and 10 demonstrate impressions of Eckert number and Lewis number on thermal and concentration boundary layers. Figure 9(a) depicts the inspiration of Eckert number on temperature profile. The increase of Eckert number increases temperature for both active and passive controls. On the other hand, decrease in mass diffusivity is noted for both active and passive strategies. Eckert number is interpreted as ratio of kinetic energy and heat enthalpy. Augmentation of $\mathrm{Ec}$ values boost process of converting mechanical energy into heat energy which causes thermal profile to rise. Alternatively, improved temperature is the result of dissipative heat generation due to frictional aspects of adjacent layers and this resistance leads to declining the concentration panel as shown in Figure 9(b).

Figure 9.

The effect of Eckert Number $\left(\mathrm{Ec}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

Figure 10.

The effect of Lewis number $\left(\mathrm{Le}\right)$ on $\theta \left(\eta \right)\mathrm{and}\varphi \left(\eta \right)$ .

We receive distinct consequences on thermal boundary for active and passive control strategies in Figure 10(a). Temperature rises for passive control whereas converse is observed for active control. We know that Lewis number relates thermal diffusivity to the Brownian diffusivity. Hence augmentation of Lewis number values dominates the thermal diffusion leading to elevated thermal profile. However, heat transport gets lower owing to high er $\left(\mathrm{Le}\right)$ in case of active control. From Figure 10(b) smooth decrease in concentration profile is witnessed for rising Lewis number values for active control. Here, we do not perceive any flipping trajectory of mass distribution boundary layer as was the case of Brownian and thermophoresis. Parametrization of $\left(\mathrm{Le}\right)$ indicates an inverse relationship with $\left(D_B\right)$ , declaring lower molecular diffusivity for higher Lewis number values leading to decline of concentration panel. The numerical values shown in Table 6 reflect an amplification in mass transport rate for both active and passive control strategies.

Conclusion

This dynamic study explores thermo-rheological properties of ternary nanofluid flow by use of active and passive control mechanisms on nanoparticles navigating a curved geometry. We presumed that a porous surface is coiled in a circular region of radius $R$ and texture of upper curved portion is stretching with uniform velocity. The nanometer sized particles in spherical shape of three different metals $(\mathrm{Co},\mathrm{Ag},\mathrm{Zn})$ are homogenously dispersed in liquid dihydrogen monoxide as the host fluid. A strong magnetic field of strength $B_{°}$ acts along radial direction of surface. Thermal radiation factor in heat equations suggests a strong heat source in vicinity. Our unique approach which has not been discussed so far by any researcher incorporates Brownian motion and thermophoresis to construct the hydrothermal framework. Moreover, we introduce passive control of particles on surface influenced by Brownian movement and thermophoresis. This passive flow is considered more realistic in comparison to constant concentration hypothesis (active control). The acquired mathematical equations are solved numerically by bvp4c in MATLAB 2023a. The salient of study are summarized below.

• Curvature and porosity factors have opposite effects on velocity profile. Increase in radius of curvature reduces pressure and shear stress in boundary layer blowing velocity while porosity factor acts as damping force declining velocity.

• All involved parameters augment temperature profile in both active and passive controls except curvature factors. In passive control flow, a static perspective is maintained by temperature panel when Brownian factor is varied. In all cases, active control assures higher temperature profiles over passive control.

• Concentration of nanoparticles decreases for higher radiations and Eckert number For all other pertinent parameters, dual consequences have been noted for active and passive controls. Moreover, under action of passive control, a sudden cross over is observed only in $\mathrm{Nb}\mathrm{and}\mathrm{Nt}$ profiles which diminishes after a small interval. However, smooth increase and decrease are observed for all other relevant factors.

• Skin friction at surface decreases for increase in curvature factor while porosity and Lorentz force cause frictional force to decrease. Also, an increasing trend in skin friction is prominent as we add nanoparticles of metals which can be ordered as $C_{f(\mathrm{mono})}<C_{f(\mathrm{hybrid})}<C_{f(\mathrm{ternary})}$ .

• Heat transport rate is a decreasing function of $M,\mathrm{Nb},\mathrm{Nt},\lambda$ and $K$ . However, Nusselt number is found increasing for increasing nanoparticles concentration with ternary nanofluid having maximum heat transfer rate.

• Brownian motion and Lewis number improve the nanoparticles concentration for active control of tiny particles, however the impact of former for passive control is stagnant. Thermophoresis reduces the mass transport rate for both active and passive controls. Here the rate of reduction is better at active control than passive strategy.

This study is helpful in optimizing thermal energy in industrial applications by advancing understanding of heat and mass transport phenomena in ternary hybrid nanofluids on curved porous surface, having implications for both research and practical applications. This study can be expanded to include nanoparticles of different metals, metal oxides or nanotubes, different base fluids with changing volume fractions, boundary constraints, geometries and noval active and passive controls.

Limitations of the study

This study is a theoretical in nature. Experimental investigations may be carried out to validate the theoretical outcomes of this study and its prospective applications in heat and mass transfer domains.