Exploring the Impact of Nanomaterials on Heat- and Mass-Transfer Properties of Carreau-Yasuda Fluid with Gyrotactic Bioconvection Peristaltic Phenomena

Abstract

This research aims to investigate the impact of nanomaterials on the heat and mass transfer properties of fluids, with a particular focus on exploring the bioconvection phenomena. To achieve this, the study considers Carreau-Yasuda (CY) fluid, which is known for its shear thickening and thinning nature. The effects of a porous medium, radiation, and viscous dissipation are also considered to analyze heat-transfer rates. Velocity and thermal slip constraints are applied to the wall, while zero-mass flux conditions explain the concentration behavior of nanomaterials at the wall. The governing equations and conditions are simplified using a lubrication approach, and a numerical approach is used to solve the final equations with the help of constraints. The velocity, temperature, and concentration of nanomaterials and gyrotactic microorganisms are analyzed through graphs. The study finds that increasing the thermophoresis parameter leads to an increase in the concentration of nanomaterials. However, the opposite trend is noticed for the concentration of motile microorganisms. The results suggest that the addition of nanomaterials to fluids can significantly impact heat- and mass-transfer properties, and may have implications for biological processes.

1. Introduction

In the current era, environmental and energy sustainability have become major issues across the word due to massive increases in population. Coal, oil and natural gas fulfill the need for energy for decades ahead. These caused global climate change, which is alarming for human survival. Further, these resources are decreasing with the passage of time. Therefore, scientists work on improving the heat-transfer efficiency of the system. Choi [1] gave the idea of a nanofluid which effectively overcomes this problem. Nanofluids are the suspension of a nanomaterial in ordinary liquids. The thermal efficiency of the nanomaterials is higher as compared to ordinary material; therefore, the addition of nanoparticles intensifies the thermal efficiency of the system. The novel concept of nanofluids greatly motivates scientists, due to its better thermal features, excellent stability, physical strength and minimal clogging. Different types of nanomaterials are used, such as metals (Au, Al, Cu and Fe), oxide ceramics (CuO, TiO₂, ZnO and Al₂O₃) and carbon nanotubes etc. [2,3,4]. The use of nanofluids has become prevalent in various engineering and industrial applications, to control energy losses and improve the life span of devices.

There are several fluid models that are applied to illustrate the behavior of fluids in various applications. For instance, a few of these models include the Newtonian model, non-Newtonian model, power law model and Buongiorno model. The Buongiorno model is a mathematical model that is employed to define the behavior of nanofluids, which are fluids that contain suspended nanoparticles. The model considers the effects of nanoparticle concentration and size on fluid properties, such as viscosity, thermal conductivity and heat capacity. The Buongiorno model is widely used in the study of nanofluids and has been shown to provide good agreement with experimental data. The model can be used to optimize the design of nanofluid-based heat-transfer systems, such as heat exchangers and electronic cooling devices. This model provides a more accurate description of the behavior of nanofluids than traditional fluid models, and is more effective for optimizing the design of nanofluid-based heat-transfer systems. In biological systems, nanomaterials are used in nanoscopy, subcellular fractionation, cancer therapy, biosensor, drug delivery, artificial organ generation, tissue engineering, bio-imaging, cell tracking and Omic data generation. The viscous and thermal characteristics of a fluid can be significantly influenced by the size, shape and temperature of the nanomaterials, thereby emphasizing their key role in such processes. Seifikar et al. [5] synthesized the stable metallic nanomaterials using a laser ablation technique. Further, in shifting light to heat, the efficiency of Ag nanomaterials were examined in detail to improve solar performance. Chen et al. [6] experimentally studied the N-doped carbon quantum dot nanomaterials for heat-transport characteristics. The empirical relation for calculating the viscous and thermal features of N-CQD nanofluid was purposed. Qu et al. [7] studied hybrid nanofluids to improve the performance of solar systems. Results indicated that evaporation efficiency of the system increased in the presence of hybrid nanomaterials. Balakin et al. [8] examined nanofluids for the direct absorption of solar collectors. Nisar et al. [9] examined the peristaltic activity of a chemically reactive Carreau-Yasuda nanomaterial with modified Darcy law. The heat-transport process intensified for higher Brownian motion and thermophoresis characteristics. By increasing the Weissenberg number, the moving capacity of fluid also intensified. Gopal et al. [10] explored the Soret and Dufour effects on Carreau nanomaterial, and discussed the chemical reaction process in detail. By increasing the thermal buoyancy and Weissenberg parameter, the velocity profile increased. The concentration gradient of nanomaterial decays for the Deborah number, thermophoresis and Dufour number. Abbas et al. [11] explored the features of power law nano-liquids over a variable extended surface. Sedki et al. [12] described the impact of convection and radiation on the MHD flow of nanomaterials via a porous medium. The heat-transport process of nanofluids increased by increasing the Brownian motion and surface permeability factors, but the opposite effect was noticed for radiation, mixed-convection and Hartman numbers [13,14,15,16,17,18].

Peristalsis is a mechanism that transfers material by retrenchment and enlargement of walls of the peristaltic channel. The phenomena of peristalsis occur in many biological procedures, such as chyme movement, blood flow, bile ducts, sperm motion, digestive system, sanitary pump and nuclear production etc. Further, roller pumps and heart lung dialysis machines also follow these mechanisms. Akram et al. [19] performed electroosmotic activity of nanomaterials in the occurrence of mixed-convection and zero-mass flux constraints. Saba et al. [20] elaborated the peristaltic wave of magneto-nanofluids in a curled conduit. By increasing the curvature parameter, the peristaltic pumping region decreased. The results direct that the enhancement of nanomaterials decays the fluid velocity and temperature profiles. Tahir et al. [21] discussed the pseudoplastic and dilatant nature of nanofluids in the peristaltic activity. As the fluid parameter velocity was augmented, there was a reduction in the velocity of the nanomaterial proximate to the channel wall, whereas a converse trend was discerned in the vicinity of the medium’s center.

Bioconvection explores the convection phenomenon that occurs due to the swimming of microorganisms in the liquid. Commonly, microorganisms are flowing near the upper surface; therefore, the density fluctuates between the upper and lower surfaces of the system. Microorganisms play an important role in human life and maintain the environment stability. The bioconvection phenomena are used in biotechnology and biological engineering. Zhang et al. [22] theoretically elaborated the bioconvection flow of Williamson nanofluid with activation energy. Hussain et al. [23] elaborated the influence of thermal radiation on bioconvection flow of nanomaterials via a porous cavity. Bisht et al. [24] investigated the thermal bioconvection of nanomaterials. Results indicate that the system is more stable by controlling the diffusivity of the microorganisms [25,26,27,28,29,30]. In addition, studying the flow behavior of non-Newtonian nanofluids is crucial and has been the focus of several previous studies, as referenced in [31,32,33,34,35].

The current study is motivated by the lack of research focused on the peristaltic motion of Carreau-Yasuda nano-liquids with gyrotactic microorganisms. The novelty of this study lies in its focus on the interaction between nanomaterials and biological processes, specifically gyrotactic microorganisms. The study employs a lubrication approach and numerical methods to solve the governing equations, and analyzes the velocity, temperature, and concentration of nanomaterials and microorganisms through graphs. Overall, the underlying study contributes to an improved understanding of the impact of nanomaterials on fluid characteristics and their potential applications in biological processes. The study employs Darcy’s law to investigate the inspiration of porous media on the system, while taking into consideration the viscous dissipation and thermal radiation impacts. Wall constraints based on velocity and thermal slip are also taken into account, and the impact of relevant coefficients on the heat- and mass-transport properties is analyzed. The study highlights the potential impact of nanomaterials on biological processes, as evidenced by the analysis of gyrotactic microorganisms. It further highlights the potential benefits and challenges associated with incorporating nanomaterials into fluids, and provides insights into the complex interactions between nanomaterials and biological systems. These findings could have implications for various engineering and medical applications, such as thermal management, energy storage, heat exchangers, targeted drug delivery, and cancer treatment. Eventually, the underlying investigation addresses the research questions, including, but not limited to: What is the impact of nanomaterials on the heat- and mass-transfer properties of fluids, and how does the presence of nanomaterials affect bioconvection phenomena in Carreau-Yasuda fluid? How do velocity and thermal slip constraints at the wall affect the behavior of nanomaterials and microorganisms in the fluid? What is the concentration behavior of nanomaterials at the wall, and how does it affect the overall heat- and mass-transfer properties of the fluid? What are the implications of adding nanomaterials to fluids for biological processes that involve heat and mass transfer? Thus, we aim to find answers to these questions through underlying investigations.

2. Problem Formulation

The peristaltic motion of an incompressible nano-liquid, following the Carreau-Yasuda model, was examined in a symmetric channel. Nanofluid propagated in a porous medium with motile gyrotactic microorganisms. The waves propaged along the X direction, due to the peristaltic phenomena with wavelength $\lambda$ and speed $c$. The $X$-axis was along the centerline of the channel and the $Y$-axis was in the transverse direction. The mathematical form of the geometry wall is described as (see Figure 1):

$$\pm \overline{H}\left(\overline{X},\overline{t}\right)=\pm a_1\pm b\mathit{cos}(\frac{2\pi }{\lambda }\left(\overline{X}-c\overline{t}\right)),$$

(1)

where $b$ indicates the amplitude of the waves and $a_1$ depicts the half-width of the medium.

The configuration of nanomaterial and ordinary liquid are assumed in the equilibrium state. In the case of assuming the turbulence, peristaltic flow would change the behavior of the system compared to the assumption of the equilibrium state. Turbulence can increase mixing and mass transfer, affect stability, induce shear stresses, and require different models and simulations. The temperature, $T_0$, and nanomaterial concentration, $C_0$, were considered on the walls. The velocity profile in two dimensions was denoted as $\overline{\mathbf{V}}=[\overline{U}(\overline{X},\overline{Y},\overline{t}),\overline{V}(\overline{X},\overline{Y},\overline{t}),0]$.

The governing equations of the problem in the presence of bioconvection and radiation effects are described as follows [23,25]:

$$\frac{\partial \overline{U}}{\partial \overline{X}}+\frac{\partial \overline{V}}{\partial \overline{Y}}=0,$$

(2)

$$\begin{array}{l}{\rho }_{nf}\left(\frac{\partial }{\partial \overline{t}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\overline{U}\frac{\partial }{\partial \overline{X}}\right)\overline{U}=\frac{\partial {\overline{S}}_{\overline{x}\overline{x}}}{\partial \overline{X}}-\frac{\partial \overline{P}}{\partial \overline{X}}+\frac{\partial {\overline{S}}_{\overline{x}\overline{y}}}{\partial \overline{Y}}+{\overline{R}}_{\overline{X}}\\ +g\left[{\rho }_f\alpha \left(T-T_0\right)+\left({\rho }_p-{\rho }_f\right){\alpha }^{\ast }\left(C-C_0\right)-\left({\rho }_m-{\rho }_f\right){\alpha }^{\ast \ast }\left(N-N_0\right)\right],\end{array}$$

(3)

$${\rho }_{nf}\left(\frac{\partial }{\partial \overline{t}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\overline{U}\frac{\partial }{\partial \overline{X}}\right)\overline{V}=\frac{\partial {\overline{S}}_{\overline{y}\overline{x}}}{\partial \overline{X}}+\frac{\partial {\overline{S}}_{\overline{y}\overline{y}}}{\partial \overline{Y}}-\frac{\partial \overline{P}}{\partial \overline{y}}+{\overline{R}}_{\overline{Y}},$$

(4)

$$\begin{array}{l}{(\rho C)}_{nf}\left(\frac{\partial }{\partial \overline{t}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\overline{U}\frac{\partial }{\partial \overline{X}}\right)T=K_{nf}\left[\frac{{\partial }^{2}T}{\partial {\overline{Y}}^2}+\frac{{\partial }^{2}T}{\partial {\overline{X}}^2}\right]+\overline{S}.\overline{L}+\frac{\partial q_r}{\partial \overline{Y}}\\ +{(\rho C)}_{np}\left[D_B\left(\frac{\partial C}{\partial \overline{X}}\frac{\partial T}{\partial \overline{X}}+\frac{\partial C}{\partial \overline{Y}}\frac{\partial T}{\partial \overline{Y}}\right)+\frac{D_T}{T_m}\left({\left(\frac{\partial T}{\partial \overline{X}}\right)}^2+{\left(\frac{\partial T}{\partial \overline{Y}}\right)}^2\right)\right],\end{array}$$

(5)

$$\overline{V}\frac{\partial C}{\partial \overline{Y}}+\overline{U}\frac{\partial C}{\partial \overline{X}}+\frac{\partial C}{\partial \overline{t}}=D_B\left(\frac{{\partial }^{2}C}{\partial {\overline{X}}^2}+\frac{{\partial }^{2}C}{\partial {\overline{Y}}^2}\right)+\frac{D_T}{T_m}\left(\frac{{\partial }^{2}T}{\partial {\overline{X}}^2}+\frac{{\partial }^{2}T}{\partial {\overline{Y}}^2}\right),$$

(6)

$$\frac{\partial N}{\partial \overline{t}}+\overline{U}\frac{\partial N}{\partial \overline{X}}+\overline{V}\frac{\partial N}{\partial \overline{Y}}+\frac{b{W}_c}{C_0}\left[\nabla \left(N.\nabla C\right)\right]=D_m\left(\frac{{\partial }^{2}N}{\partial {\overline{X}}^2}+\frac{{\partial }^{2}N}{\partial {\overline{Y}}^2}\right).$$

(7)

Here, $\overline{P}$ indicates the pressure, ($\rho C$)${}_{np}$ the heat capacity of the nanomaterials, $N$ concentration of motile microorganisms, ${\overline{S}}_{ij}$ the components of extra stress tensor, $\overline{S}.\overline{L}$ the viscous dissipation, $D_B$ the mass diffusivity, $K_{nf}$ the thermal conductivity, ${\rho }_{nf}$ density, $C_{nf}$ specific heat, $D_m$ microorganism diffusion coefficient and $D_T$ thermophoretic diffusion parameter. Darcy’s law [18] for porous spaces is expressed as:

$$\overline{R}=-\frac{\epsilon }{k}\mu ({\gamma }^{\prime })\overline{V},$$

(8)

Here, $k$ indicates the permeability and $\epsilon$ describe the porosity of the porous space. Radiative heat flux, $q_r$, after using the assumption, is specified as [25]:

$$q_r=-\frac{4{\sigma }^{\ast }{T}_0^3}{3k^{\ast }}\frac{\partial T}{\partial y}.$$

(9)

The Carreau-Yasuda fluid model is defined as [18]:

$$\overline{\mathbf{S}}=\mu ({\gamma }^{\prime }){\mathbf{A}}_1,$$

(10)

Here, $A_1$ depicts the first Rivilin–Ericksen tensor and $\mu ({\gamma }^{\prime })$ depicts the viscosity of fluid that are described as:

$$\mu ({\gamma }^{\prime })={\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){\left[1+{\left(\Gamma {\mathsf{\gamma }}^{\prime }\right)}^a\right]}^{\frac{n-1}{a}},$$

(11)

and

$${\mathsf{\gamma }}^{\prime }=\sqrt{\frac{1}{2}tr({\mathbf{A}}_1{}^2)}.$$

Here, ${\mu }_{\infty }$ and ${\mu }_0$ stand for infinite and zero-shear rate viscosity of the fluid. Further, $n$ indicates the power law index that describes the shear thickening and thinning nature of the fluid. The CY model reduces to viscous fluid for $n=1$. In this problem only zero-shear rate viscosity is studied. The following transformation is used in this problem [2]:

$$\overline{x}=\overline{X}-c\overline{t},\overline{v}=\overline{V},\overline{p}(\overline{x},\overline{y})=\overline{P}(\overline{X},\overline{Y},\overline{t}),\overline{u}=\overline{U}-c,\overline{y}=\overline{Y}.$$

(12)

It is relevant to mention that the analysis conducted by Ying-Qing et al. [36] and Cao et al. [37] examined the mechanics of a hybrid nanofluid that contained both the alumina and copper nanoparticles, by utilizing a two-phase model which factored in the thermophysical properties of the base fluid and nanoparticles. On the other hand, this investigation on the transport of nanofluids through peristalsis, using Buongiorno’s nanofluid model, considered both the Browning motion and thermophoresis effects. Thus, taking this into account, we have the following equations:

$$\frac{\partial \overline{u}}{\partial \overline{x}}+\frac{\partial \overline{v}}{\partial \overline{y}}=0,$$

(13)

$$\begin{array}{c}{\rho }_{nf}\left(\overline{v}\frac{\partial }{\partial \overline{y}}+(\overline{u}+c)\frac{\partial }{\partial \overline{x}}\right)\left(\overline{u}+c\right)=-\frac{\partial \overline{p}}{\partial \overline{x}}+\frac{\partial {\overline{s}}_{\overline{x}\overline{x}}}{\partial \overline{x}}+\frac{\partial {\overline{s}}_{\overline{x}\overline{y}}}{\partial \overline{y}}-\frac{\epsilon }{k}\mu ({\gamma }^{\prime })(c+\overline{u})\\ +g\left[{\rho }_f\alpha \left(T-T_0\right)+\left({\rho }_p-{\rho }_f\right){\alpha }^{\ast }\left(C-C_0\right)-\left({\rho }_m-{\rho }_f\right){\alpha }^{\ast \ast }\left(N-N_0\right)\right],\end{array}$$

(14)

$${\rho }_{nf}\left(\overline{v}\frac{\partial }{\partial \overline{y}}+(\overline{u}+c)\frac{\partial }{\partial \overline{x}}\right)\overline{v}=-\frac{\partial \overline{p}}{\partial \overline{y}}+\frac{\partial {\overline{s}}_{\overline{y}\overline{x}}}{\partial \overline{x}}+\frac{\partial {\overline{s}}_{\overline{y}\overline{y}}}{\partial \overline{y}}-\frac{\epsilon }{k}\mu ({\gamma }^{\prime })\overline{v},$$

(15)

$$\begin{array}{l}{(\rho C)}_{nf}\left(\overline{v}\frac{\partial }{\partial \overline{y}}+(\overline{u}+c)\frac{\partial }{\partial \overline{x}}\right)T=K_{nf}\left[\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}+\frac{{\partial }^{2}T}{\partial {\overline{x}}^2}\right]+\overline{s}.\overline{L}-\frac{16\sigma T_0^3}{3k}\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}\\ +{(\rho C)}_{np}\left[D_B\left(\frac{\partial C}{\partial \overline{y}}\frac{\partial T}{\partial \overline{y}}+\frac{\partial C}{\partial \overline{x}}\frac{\partial T}{\partial \overline{x}}\right)+\frac{D_T}{T_m}\left({\left(\frac{\partial T}{\partial \overline{x}}\right)}^2+{\left(\frac{\partial T}{\partial \overline{y}}\right)}^2\right)\right],\end{array}$$

(16)

$$\overline{v}\frac{\partial C}{\partial \overline{y}}+(\overline{u}+c)\frac{\partial C}{\partial \overline{x}}=D_B\left(\frac{{\partial }^{2}C}{\partial {\overline{y}}^2}+\frac{{\partial }^{2}C}{\partial {\overline{x}}^2}\right)+\frac{D_T}{T_m}\left(\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}+\frac{{\partial }^{2}T}{\partial {\overline{x}}^2}\right),$$

(17)

$$(\overline{u}+c)\frac{\partial N}{\partial \overline{x}}+\overline{v}\frac{\partial N}{\partial \overline{y}}+\frac{b{W}_c}{C_0}\left[\nabla \left(N.\nabla C\right)\right]=D_m\left(\frac{{\partial }^{2}N}{\partial {\overline{x}}^2}+\frac{{\partial }^{2}N}{\partial {\overline{y}}^2}\right).$$

(18)

The non-dimensional parameters that are used in this problem are described as:

$$\begin{array}{c}y=\frac{\overline{y}}{a_1},x=\frac{\overline{x}}{\lambda },h=\frac{\overline{H}}{a_1},d=\frac{b}{a_1},\delta =\frac{{\overline{d}}_1}{\lambda },p=\frac{a_1{}^2\overline{p}}{c\lambda {\mu }_f},u=\frac{\overline{u}}{c}Da=\frac{k}{\epsilon a_1{}^2},v=\frac{\overline{v}}{c\delta },\\ \mathrm{Re}=\frac{{\rho }_{f}c{a}_1}{{\mu }_f},E=\frac{c^2}{C_f{T}_0},\mathrm{Pr}=\frac{{\mu }_f{C}_f}{K_{nf}},N_b=\frac{\tau D_B{C}_0}{\nu },N_t=\frac{\tau D_T{T}_0}{\nu T_m},R_d=\frac{4{\sigma }^{\ast }{T}_0^3}{k^{\ast }{K}_{nf}},\\ \upsilon =\frac{{\mu }_f}{{\rho }_f},\theta =\frac{T-T_0}{T_0},\phi =\frac{C-C_0}{C_0},\chi =\frac{N-N_0}{N_0},Pe=\frac{b{W}_c}{D_m},Br=\mathrm{Pr}E,u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}.\end{array}$$

(19)

Here, $\mathit{Re},Ec,Br,\mathit{Pr},\theta ,\delta ,\varphi ,\chi ,Pe,N_t, R_d$ and $N_b$ indicate the Reynolds number, Eckert number, travelling wave speed, Brinkman number, Prandtl number, temperature, wave number, concentration of nanomaterial, concentration of gyrotactic microorganism, Peclet number, thermophoresis number, thermal radiation and Brownian motion parameters, respectively. The dimensionless forms of the above equations are:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(20)

$$\begin{array}{c}\mathrm{Re}\delta \left(v\frac{\partial u}{\partial y}+(u+1)\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x}+\frac{\partial s_{xy}}{\partial y}+\delta \frac{\partial s_{xx}}{\partial x}\\ +G_T\theta +G_C\phi -G_n\chi -\frac{1}{Da}\frac{\mu ({\gamma }^{\prime })}{\mu }(u+1),\end{array}$$

(21)

$$\mathrm{Re}{\delta }^3\left(v\frac{\partial v}{\partial y}\delta +(1+u)\frac{\partial v}{\partial x}\right)=\delta \frac{\partial s_{yy}}{\partial y}-\frac{\partial p}{\partial y}+{\delta }^2\frac{\partial s_{yx}}{\partial x}-{\delta }^2\frac{1}{Da}\frac{\mu (\gamma )}{\mu }v,$$

(22)

$$\begin{array}{c}\mathrm{Pr}\delta \mathrm{Re}\left(v\frac{\partial \theta }{\partial y}+(1+u)\frac{\partial \theta }{\partial x}\right)=\left(\frac{{\partial }^2\theta }{\partial y^2}+{\delta }^2\frac{{\partial }^2\theta }{\partial x^2}\right)+Br.\mathsf{\Phi }+\frac{4R_d}{3}\frac{{\partial }^2\theta }{\partial y^2}\\ +\mathrm{Pr}{N}_t\left({\delta }^2{\left(\frac{\partial \theta }{\partial x}\right)}^2+{\left(\frac{\partial \theta }{\partial y}\right)}^2\right)+\mathrm{Pr}{N}_b\left(\frac{\partial \phi }{\partial y}\frac{\partial \theta }{\partial y}+\frac{\partial \phi }{\partial x}\frac{\partial \theta }{\partial x}\right),\end{array}$$

(23)

$$\mathrm{Re}.\delta \left(v\frac{\partial \phi }{\partial y}+(1+u)\frac{\partial \phi }{\partial x}\right)=N_b\left(\frac{{\partial }^2\phi }{\partial y^2}+{\delta }^2\frac{{\partial }^2\phi }{\partial x^2}\right)+N_t\left({\left(\frac{\partial \theta }{\partial y}\right)}^2+{\delta }^2{\left(\frac{\partial \theta }{\partial x}\right)}^2\right),$$

(24)

$$\mathrm{Re}.Lb.\delta \left(v\frac{\partial \chi }{\partial y}+(1+u)\frac{\partial \chi }{\partial x}\right)+Pe\left({\delta }^2\frac{\partial }{\partial x}\left((\chi +1)\frac{\partial \chi }{\partial x}\right)+\frac{\partial }{\partial y}\left((\chi +1)\frac{\partial \chi }{\partial y}\right)\right)=\frac{{\partial }^2\chi }{\partial y^2}+{\delta }^2\frac{{\partial }^2\chi }{\partial x^2}.$$

(25)

By using the lubrication approach, the above equations yield:

$$\frac{\partial p}{\partial y}=0,$$

(26)

$$\frac{\partial p}{\partial x}=\frac{\partial s_{xy}}{\partial y}+G_T\theta +G_C\phi -G_n\chi -\frac{1}{Da}\frac{\mu ({\gamma }^{\prime })}{{\mu }_0}\left(1+\frac{\partial \psi }{\partial y}\right),$$

(27)

$${\theta }_{yy}+Br\mathsf{\Phi }+\mathrm{Pr}{N}_b{\phi }_y{\theta }_y+\mathrm{Pr}{N}_t{\left({\theta }_y\right)}^2+\frac{4R_d}{3}\frac{{\partial }^2\theta }{\partial y^2}=0,$$

(28)

$$N_b{\phi }_{yy}+N_t{\theta }_{yy}=0,$$

(29)

$$Pe\left({\chi }_y{\phi }_y+(\chi +1){\phi }_{yy}\right)={\chi }_{yy}.$$

(30)

Here, the viscous dissipation, $\mathsf{\Phi }$, and the extra stress tensor, $s_{xy}$, are described as:

$$s_{xy}=s_{yx}=\left[1+\frac{(1-\beta )(n-1)W{e}^a}{a}{({\psi }_{yy})}^a\right]{\psi }_{yy}.$$

(31)

Here, $We=\frac{\mathsf{\Gamma }c}{a}$ depicts the Weissenberg number and the continuity equation is satisfied automatically. The simplified form of Equations (26)–(30) are:

$$\begin{array}{c}\frac{{\partial }^2}{\partial y^2}\left[1+\frac{(1-\beta )(n-1)W{e}^a}{a}{({\psi }_{yy})}^a\right]{\psi }_{yy}+G_T\theta +G_C\phi -G_n\chi \\ -\frac{1}{Da}\frac{\partial }{\partial y}\left[1+\frac{(1-\beta )(n-1)W{e}^a}{a}{({\psi }_{yy})}^a\right]\left(1+\frac{\partial \psi }{\partial y}\right)=0,\end{array})$$

(32)

$$\begin{array}{c}{\theta }_{yy}+Br\left[1+\frac{(1-\beta )(n-1)W{e}^a}{a}{({\psi }_{yy})}^a\right]{\psi }_{yy}^2+\frac{4R_d}{3}\frac{{\partial }^2\theta }{\partial y^2}\\ +\mathrm{Pr}{N}_b{\phi }_y{\theta }_y+\mathrm{Pr}{N}_t{\left({\theta }_y\right)}^2=0,\end{array})$$

(33)

$$N_b{\phi }_{yy}+N_t{\theta }_{yy}=0.$$

(34)

$$Pe\left({\chi }_y{\phi }_y+(\chi +1){\phi }_{yy}\right)={\chi }_{yy}.$$

(35)

The flow rates in the moving $F(=\overline{q}/ca)$ and laboratory $\eta (=\overline{Q}/ca)$ frames are specified as:

$$\eta =F+1.$$

(36)

Further, $F$ is described as:

$$F={\int }_0^h\frac{\partial \psi }{\partial y}dy.$$

(37)

It is relevant to note that the laboratory and the moving frames are considered to find the fluid flow rates, because the choice of frame of reference can affect the measured values of fluid flow rates. Using Galilean transformation, one can convert the measured fluid flow rates from one frame of reference to another. This allows us to better understand the behavior of fluid flows in different situations and make more accurate predictions and decisions in practical applications.

The zero-mass flux and slip constraints are specified as [25]:

$$\begin{array}{c}\psi =0,{\psi }_{yy}=0,{\theta }_y=0,{\phi }_y=0,{\chi }_y=0\mathrm{at}y=0,\\ \psi =F,{\psi }_y+\zeta s_{xy}=-1,\theta +\gamma {\theta }_y=0,N_b{\phi }_y+N_t{\theta }_y=0,\chi +{\gamma }_2{\chi }_y=0\mathrm{at}y=h,\end{array}$$

(38)

where $h=1+d\cos (2\pi x)$. Here, $\zeta ,\gamma ,{\gamma }_2$ indicate the velocity slip coefficients, thermal slip and bioconvection slip parameter. To solve the system of equations with boundary conditions, a numerical approach is employed. The ND solve method, which employs a shooting method and ensures the accuracy of the solution for very small step sizes of “x” and “y”, is utilized. Therefore, in this problem, 0.001 step size is considered. The subsequent section provides a detailed discussion of the obtained results concerning the heat and mass transport properties.

3. Results and Discussion

This section is focused on presenting and analyzing graphical representations of the results obtained for the velocity, concentration and temperature properties. Heat-transport characteristics are analyzed via a table. It is important to state that the parameter values were chosen arbitrarily, but not in contradiction with existing theoretical and experimental investigations. The references [18,19,20,25,33,35] provide more information on the chosen values. Moreover, reference values for different parameters can be seen in

Table 1. Effect of embedded parameters on the heat-transport process.

${\mathit{R}}_{\mathit{d}}$	${\mathit{G}}_{\mathit{T}}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{N}}_{\mathit{b}}$	We	$-{\mathit{\theta }}_{\mathit{y}}\left[\mathbf{h}\right]$
0.0	1.0	0.5	0.5	0.4	0.34847
1.0					0.14928
2.0					0.09988
1.0	1.0				0.14928
	1.5				0.14929
	2.0				0.14931
	1.0	0.5			0.14928
		1.0			0.14931
		1.5			0.14932
		0.5	0.5		0.14928
			1.0		0.14927
			1.5		0.14926
			0.5	0.3	0.15328
				0.4	0.14932
				0.5	0.14375

, as well. To examine the effect of significant coefficients on the velocity field, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 have been generated. Figure 2 depicts the influence of non-Newtonian fluid coefficients “$a$” on the velocity. Increasing the value of “$a$” slowed down the fluid motion. Figure 3 is sketched to explore the impact of the Darcy’s parameter, “$Da$” coefficient, on the velocity. The graph indicates the increasing trend of velocity for higher “$Da$” coefficients. Figure 4 highlights the effect of $\zeta$ coefficient on velocity. In the presence of slip coefficients, velocity increased near the boundary walls. Figure 5 provides information about velocity for different values of “$We$”. By increasing the “$We$”, the capacity of fluid motion decreased. Figure 6 presents the impact of “$n$” on the velocity.

Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 are generated to analyze the temperature changes in the system for varying values of relevant parameters. Figure 7 is plotted to express the impact of $R_d$ on the temperature. The higher values of $R_d$ facilitated the heat-transport process, and therefore the temperature of the fluid rapidly decayed. Figure 8 describes that the temperature of the fluid can be reduced by minimizing the values of the Brinkman number. Figure 9 signifies the impact of “$We$” on temperature. The temperature profile decreased by increasing the “$We$” coefficients. Figure 10 is generated to observe the impact of the thermal jump coefficient on temperature. It is noted that as the thermal jump parameter ($\gamma$) increased, the temperature exhibited an upward trend. Figure 11 indicates that the temperature slightly increased by increasing the thermal Grashoof number. Figure 12 expresses the effect of “$a$” on temperature. The fluid temperature decayed by increasing the fluid parameter “$a$”.

The impact of $N_t$, $N_b$, $R_d$, $We$ and $n$ on the concentration of nanomaterials is studied through Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. Figure 13 depicts the impact of $N_t$ on the concentration of nanomaterials. The concentration of nanomaterial increased by increasing the thermophoresis parameter. Figure 14 is sketched to highlight the impact of the Brownian motion coefficient on the concentration of nanomaterials. The concentration of nanomaterial decayed near the channel wall by increasing the $N_b$. The concentration of nanomaterial decayed near the wall by enhancing the $R_d$ (see via Figure 15). Figure 16 indicates that $\phi$ decayed near the channel wall for higher values of $We$. Figure 17 is outlined to notice the effect of $n$ on the $\phi$. The concentration of nanomaterial intensified for higher values of the non-Newtonian parameter, $n$.

Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 are plotted to elaborate the effect of $Pe$, $G_n$, $N_t$, $N_b$ and $\zeta$ on the concentration of motile microorganism ($\chi$). Figure 18 reveals the outcome of the Peclet number on the $\chi$. It can be visualized that $\chi$ improved for higher values of $Pe$. Figure 19 highlights the impact of $G_n$ on the $\chi$. The density of motile microorganism decayed by increasing the value of the bioconvection Rayleigh parameter. The variation in $\chi$ via $N_t$ is captured in Figure 20. It is noticed that the concentration of microorganisms was improved for higher values of $N_t$. Further, an opposite effect is noticed for higher values of $N_b$ (as can be seen in Figure 21). Figure 22 highlights the influence of the velocity slip coefficient on the $\chi$. It is concluded that $\chi$ decayed in the presence of slip constraints.

The impact of pertinent parameters on heat transpose features are discussed via tabular values, shown below, in Table 1. By increasing the $R_d$ parameter, the heat-transfer process intensified; therefore, the temperature of the system was maintained. By increasing the $G_t$ and $N_t$, the heat-transfer processes were slightly increased. However, the heat-transfer process was decreased by increasing the $N_b$ and $We$ parameters.

It is evident from the above discussion that the increase in the fluid coefficients of the Carreau-Yasuda model caused the viscosity of the fluid to increase, leading to a decrease in fluid velocity. By increasing the Darcy’s parameter and velocity slip parameter, the velocity profile of the nanomaterial near the walls intensified due to a higher resistance to flow. The temperature of the fluid decreased with an increase in the fluid coefficients of the Carreau-Yasuda model, which increased the rate of viscous dissipation. The concentration of nanomaterials decreased with an increase in the Brownian motion parameter, because of greater diffusion and dispersion, but increased with higher thermophoresis coefficients, because the particles moved towards higher-temperature regions. The motion of gyrotactic microorganisms was enhanced for higher Peclet numbers due to greater advection caused by fluid flow. Additionally, the heat-transport process of the nanofluid increased with increasing the thermal radiation parameter.

Furthermore, the numerical values obtained for heat transport at the walls were compared to the results reported by Hayat et al. [25] (refer to

Table 2. Comparison of heat-transfer rate at the upper wall to the published work [25], when $G_T=G_C=G_N=Pe=0$.

$${\mathit{R}}_{\mathit{d}}$$	$${\mathit{N}}_{\mathit{t}}$$	$${\mathit{N}}_{\mathit{b}}$$	Hayat [25]	Special Case of Current Study
0.2	0.5		0.918494	0.91849378
0.3			0.851924	0.85192356
0.4			0.816411	0.81641076
0.2	0.5		0.918494	0.91849378
	1.0		1.721720	1.72172016
	1.5		2.502060	2.50206001
	0.5	0.5	0.918494	0.91849378
		1.0	0.452167	045216715
		1.5	0.296788	0.29678792

). This comparison demonstrates a good agreement between the two studies. When a fluid flows through a channel, it generates frictional forces along the channel walls. This results in the transfer of heat between the fluid and the channel walls, which is known as wall heat transfer. The rate of wall heat transfer depends on several factors, including fluid velocity, temperature, particle concentration and other physical parameters, as stated in Table 2. Overall, the effects of embedded parameters on the heat-transport process can be physically explained by considering the underlying mechanisms of heat transfer and fluid flow.

4. Conclusions

The main findings of peristaltic activity of nanomaterial, as discussed above, are as follows:

• Increasing fluid coefficients in the CY model leads to a decrease in fluid velocity due to the model’s shear-thickening behavior.
• The velocity profile of nanomaterials intensifies near the walls when the Darcy’s parameter and velocity slip parameter are enhanced, due to the interaction between nanomaterials and the wall surface.
• Increasing the fluid coefficients in the CY model leads to a decrease in the fluid temperature due to the increased viscous dissipation.
• The concentration of nanomaterials decreases with increasing Brownian motion parameter, while increasing the thermophoresis coefficient leads to an opposite trend, due to the combined effects of thermal and diffusion forces on nanomaterials.
• The motion of gyrotactic microorganisms improves for a higher Peclet number due to the increased diffusion rate.
• Increasing the thermal radiation parameter enhances the heat-transport process of nanofluids.
• The study’s results are in close agreement with a previous study, validating the solution and providing confidence in the findings.

The above stated concluding remarks suggest several future recommendations, including investigating nanomaterial behavior in different fluid models and coefficients, exploring the impact of various parameters on nanomaterial behavior for more efficient drug delivery and sensing systems, studying gyrotactic microorganism motion in nanofluids for medical applications, understanding the thermal behavior of nanofluids for heat-transfer systems, and validating the study’s solution through experiments or simulations.