Heat transfer analysis of steady laminar 2D flow of CNTs-blood-based nanofluid over a moving permeable plate with viscous dissipation and thermal radiation

1. Introduction

Due to the numerous uses of heat transfer in various industries, improving thermal device efficiency is currently one of the top concerns for industry designers of units. Increasing thermal efficiency and enhancing the functionality of thermal devices can save energy, reduce device dimensions, and save material and manufacturing costs. Nanofluids serve as an innovative medium for heat transfer, and advancements in nanotechnology over the last two decades have created new avenues for research exploration. Nanoparticle suspensions in non-Newtonian fluids are called non-Newtonian nanofluids (nNNFs). In these fluids, the fluid matrix and suspended nanoparticles interact to produce complicated rheological behavior. Adding nanoparticles to a non-Newtonian fluid can significantly alter how the fluid flows and transfers heat, leading to better or tailored performance for various uses. Thermal management systems and heat exchangers can utilize nNNFs as coolants. Using nanoparticles helps the base fluid transfer heat better and work more efficiently by boosting its thermal conductivity (TC) and convective heat transfer coefficient. nNNFs may find use in medical diagnostics and medication delivery systems. It is possible to create controlled release of medication or contrast agents by adjusting the fluid's rheological properties with nanoparticles. Furthermore, when localized heating is required for targeted cancer therapy, these fluids can be employed in hyperthermia therapies.

Afify and El-Aziz, 2017 investigated the Lie group analysis of nNNF flow and heat transfer across a stretching surface with convective boundary conditions (BCs). Besthapu et al., 2017 checked how slip and thermal radiation (TR) affect the flow of nNNF magnetohydrodynamic (MHD) at the point when it stops flowing across a convective stretching surface. Si et al., 2017 examined how a plate's vertical stretch affects heat transmission and mixed convection flow; they used pseudo-plastic power law nanofluids to study the problem. Mahmoud, 2017 investigated the impact of slip velocity on a fluid moving over a porous surface that is actively generating heat and defies the Newtonian power law. Karimipour et al., 2015 investigated the effects of many nanoparticles, mainly Ag and Al₂O, on the flow of MHD nanofluids in a microchannel and the heat transfer. They considered aspects like temperature jump and slide velocity.

Bahiraei and Mazaheri, 2018 examined the application of a unique hybrid nanofluid (HNF) with disordered and twisted platinum and graphene nanoparticles for use in tiny devices. They examined topics related to energy efficiency and heat. The effects of employing a novel HNF that blends platinum and graphene nanoparticles on energy and thermal efficiency were investigated by Safaei et al., 2016a. They created this fluid to be used in tiny gadgets with a twisted, chaotic topology. Computer simulations of heat transfer and the flow of a water/functoinalized multi-walled carbob nanotube (FMWCNT) nanofluid via a narrowing conduit from the rear were utilized in a study by Alrashed et al., 2018. Alrashed et al., 2018 investigated the effects of four modeling types (artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), regression models and TSVD) on changes in the thermophysical characteristics of carbon-based nanofluids. Safaei et al., 2016b investigated the use of mathematical modeling for nanofluid simulation. Two methods for a nanofluid to flow in three dimensions through a stretching sheet with nonlinear permeability were investigated by Raju et al., 2016. In an MHD stagnation point flow that is not aligned, Khan et al., 2016 examined the effects of nanofluids with varying viscosities flowing over a stretching sheet that is heated externally. A nanofluid's flow over a moving plate was examined by Mabood et al., 2016; Mohamed et al., 2016; and Madhu and Kishan 2016 while taking convective BC and MHDs into consideration. They examined the impact of fluctuating density using MHDs. In a two-phase model, Eid and Mahny, 2017 examined the unequal distribution of mass and heat in an nNFF flow across a porous stretched wall and considered the meaning of heat production and absorption. In their study, Zhang et al., 2017 investigated the power-law heat transfer and unsteady flow characteristics of a thin layer of nanofluid on a stretching sheet. They also looked at how the power law and a changing magnetic field affected the velocity slip phenomenon. To produce a film that is glossy, Rehman et al., 2022 investigated how Marangoni convection affected the dynamics of heat transfer and the viscous dissipation linked to MHDs in HNFs in a system rotating between two surfaces. Rehman and Salleh, 2021 examined the shifting TC and magnetic field in a water-based carbon nanotube (CNT) nanofluid that was being tugged and pushed across a stretched surface. In their study, Rehman et al., 2019 and Rehman et al., 2022 used time-dependent MHD analysis to examine the motion of an HNF around a spinning sphere. Kumar et al., 2024a looked at how trihybrid nanofluid influences the short-term heat performance of slanted dovetail fins, especially the heat generated within the fins. Using the homotopy analysis method (HAM), Kumar et al., 2024b compare the triangular and rectangular designs to investigate the thermal profile of porous fins. Kumar et al., 2024c examined how heat produced inside the fins and the shape of the heat transfer fluid influence the short-term thermal performance and efficiency of fully wet porous longitudinal fins. Mahanthesh et al., 2016 examined the effects of heat and mass transfer on the mixed flow of a reactive nanofluid over a vertical plate that can move or remain stationary. Ramesh and Gireesha, 2017 examined whether the vertical plate is stationary or moving, and examined the effects of mass and heat transfer on the mixed convective flow of a chemically reactive nanofluid. A nonlinear, radiative nanofluid flowing across a stationary or moving Riga plate was examined by Ramesh and Gireesha, 2017 and Alhadhrami et al., 2021.

2. Problem formulation

The current study investigates a problem involving a 2D, laminar, incompressible, steady-state flow of blood-based nanofluid (BBN) containing CNTs, comprising two uniform, spherical nanoparticles of MWCNT and SWCNT on a sliding permeable plate. Blood serves as the base fluid in this research. Moreover, both the flat plate and the fluid flow are progressing in a unified direction. It is worth noting that a no-slip BC is enforced on nanoparticles and base fluid, which are in thermal equilibrium. Fig. 1 illustrates the schematic and model of the problem being investigated. Table 1 presents the thermophysical properties (TPP) of nanofluids and the viscous characteristics of base fluids. The governing equations for the considered problem can be expressed as follows, employing the Boussinesq approximation and boundary layer (BL) assumptions.

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Fig. 1.

Geometry of 2D steady laminar flow of BBN with CNTs over a plate.

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where the velocity components in $x\text{ and }y-\text{directions}$ are represented by $u\text{ and }v$ , kinematic viscosity is denoted by $\nu$ , $q_r$ denotes the radiative heat flux, $c_p$ represent the specific heat, TC of the fluid is denoted by $k$ and $T$ represents fluid temperature. Rosseland's approximation for radiative heat flux is expressed as $q_r=-\left(\frac{4{\sigma }_s}{3k_1}\right)\frac{\partial T^4}{\partial y}$ , where ${\sigma }_s$ represents the Stefan-Boltzmann constant and $k_1$ signifies the absorption coefficient. The application of Taylor's series facilitates the expansion of the temperature variation $T^4$ around $T_{\infty }$ , yielding $T^4=4T_{\infty }^{3}T-3T_{\infty }^4$ . Through this expansion, the radiative heat flux can be articulated as $q_r=-\frac{16{\sigma }_s{T}_{\infty }^3}{3k_1}\frac{\partial T}{\partial y}$ [%%]. As a result, equation (3) is transformed into:

The BCs for the specified problem are as follows:

Alsagri et al., 2019 provide an analysis of the TPP associated with nanofluids.

To convert the established PDEs (1) – (4) into ODEs along with the associated BCs (5), we apply a similarity transformation utilizing dimensionless variables, specifically Ψ (referred to as the stream function) and T (representing temperature):

where $\eta$ is a similarity parameter and defined as $\eta ={\left(\frac{U^{2-n}}{\nu x}\right)}^{\frac{1}{n+1}}y$ . Using the definition of stream function, we have. Equation (7) satisfied equation identically, and equations (6) and (7) reduce equations (2–4) to the following nonlinear ODEs,

the BCs (5) are:

$\gamma =\frac{{\eta }_{0}a}{{\nu }^2{}_{bf}{\rho }_{bf}}$ is the couple stress parameter (CSP), $f_w=\left(n+1\right){\left(\frac{x^n}{\nu U^{2n-1}}\right)}^{\frac{1}{n+1}}{v}_w$ represents local suction injection parameter, $Ec=\frac{U^2}{\left(\rho C_p\right)\left(T_w-T_{\infty }\right)}$ shows the local Eckert number (EN), $P_r=\frac{U}{\alpha }{\left(\frac{U^{2-n}}{\nu }\right)}^{\frac{-2}{n+1}}$ shows the local non-Newtonian Prandtl number (PN), $R=\frac{k{k}_1}{4{\sigma }_s{T}_{\infty }^3}$ shows the TR parameter, $n$ represent power law index parameter, and velocity ratio parameter is $\lambda =\frac{U_w}{U}$ .

The problem under consideration also involves the examination of the flow and wall heat transfer physical quantities.

Eqs. (11) – (12) in the dimensionless form under similarity (6) are:

where ${\mathrm{Re}}_x$ represents the local non-Newtonian Reynolds’ number, which primary controlling parameter of viscous fluid flow.

3. Solution by HAM

A potent semi-analytical method for resolving nonlinear ODEs, such as those involving fluid dynamics, heat transfer, and nanofluid flow, is the HAM. Engineering and science recognize the series solution as an effective strategy for addressing nonlinear challenges. The latest versions of HAM, namely the BVPh. 1.0 and BVPh. 2.0 packages, have been designed to enhance the convergence rates of the problems being investigated. While the BVPh. 2.0 package presents obstacles in achieving the 100th iteration; it is nonetheless highly advantageous for promoting rapid convergence. The authors use HAM, a method for approximating results, on the nonlinear ODE system shown by Eqs. (8, 9) and the BC that goes with them. The convergence of the approximate solution relies on the inclusion of an auxiliary parameter (AP), which is of utmost importance. Since it offers freedom in modifying the convergence of the series solution, the AP ℏ is an essential component of HAM. A small problem with a known solution is gradually transformed into the original, more complex problem by the HAM's construction of a homotopy. The appropriate selection of ℏ determines the series's convergence. The series converges quickly for appropriate choices, but it may diverge or converge slowly for bad ones.

The initial solution is proposed as follows for the analysis:

In relation to the problem at hand, the linear operator is denoted as $L_F$ and ${\text{L}}_{\theta }$ :

Having the subsequent properties:

The operators $N_F$ and $N_{\theta }$ are characterized as non-linear entities, as detailed in the following sections.

The basic concept of HAM, as outlined in references (Liao 2012; Liao 2033; Abbasbandy 2006; Abbasbandy 2007) for Eqs. (18–19) is:

Where $l\in \left[0,1\right]$ , is inserting a factor, when $l=0\text{ and }l=1$ we take:

Expanding $F(\eta ,l)\text{and}\theta (\eta ,l)$ in Taylor’s series about $l=0$

4. Discussion

Fluid stability and good heat transfer are essential for improving energy conversion in nano-biofuel cells, and the issue at hand is very significant. Gyrotactic bacteria promote the mixing of nanoparticles, increasing the stability and TC of the nanofluid. The impacts of various parameters, including the couple stress nanoparticles volume friction, EN, TR, PN, suction/injection parameter, and power-law index (PLI), on the examined problem's fluid heat and flow transfer characteristics. We use two types of nanoparticles: MWCNT and SWCNT, with blood serving as the base fluid. It's important to note that, because similar patterns have been seen in other nanofluids, the study of how different factors affect TP and VP is mainly focused on the SWCNT/MWCNT/blood nanofluid, which helps to simplify and summarize the results shown in Figs. 2–9. Nevertheless, Tables 2 and 3 introduce and summarize local skin friction and NN values for various nanomaterials. Eqs. (8, 9)'s governing parameters were computed over a large range of values, and the Mathematica function bvp2 was used to determine BCs (10) semi-numerically. Given the transform BC (10), we can find semi-numerical solutions to this complex nonlinear system using a well-known analytical method called HAM. Increasing the rate of heat transport is the aim of the suggested model. The major results derived from the data are presented both mathematically and graphically. We used blood as the foundation fluid and combined SWCNT and MWCNT nanoparticles to create two distinct nanofluids. SWCNTs are referred to as SWCNTs, and multi-wall CNTs as MWCNTs. As shown in Fig. 1, the geometry of the flow problem centers around its physical shape.

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Fig. 2.

Variation of fluid velocity with respect to the couple stress parameter (CSP). The colors represent different CSP values, facilitating comparison of their effect on fluid velocity.

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Fig. 3.

Effect of nanoparticle volume fraction on fluid velocity. The colors depict the distinct effects of varying nanoparticle volume fractions on fluid velocity, aiding clearer visualisation and comparison.

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Fig. 4.

The impact of suction and injection conditions on the speed of fluid movement. The colors represent different suction and injection conditions, clearly illustrating their effects on fluid velocity.

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Fig. 5.

Impact of the power law index parameter on fluid velocity. The colors represent different power-law index parameters, facilitating comparison of their effects on fluid velocity.

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Fig. 6.

The influence of the TR parameter on the temperature of the fluid. The colors highlight the impact of the thermal radiation (TR) parameter on fluid temperature, aiding visual comparison of temperature variations.

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Fig. 7.

The relationship between the power law index parameter and the temperature of fluids. The colors differentiate the effects of the power-law index (PLI) on fluid temperature, making it easier to compare temperature variations across different PLI values.

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Fig. 8.

The role of the EN in determining the temperature behavior of the fluid.

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Fig. 9.

The comparison of the current work with Alhadhrami et al., 2021. The colors represent different EN values, clearly illustrating their effects on fluid temperature distribution.

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5. Future Work

To improve the model's suitability for industrial settings, this study could investigate the incorporation of extra physical events, such as chemical reactions or nonlinear TR effects. We could learn more about how to make heat transfer work better by looking at how different shapes of porous materials or different mixes of nanoparticles (like HNFs and ternary HNFs) affect it. Predictions under dynamic operating environments could be further improved by expanding the analysis to include unsteady flow conditions or sophisticated turbulence models. The theoretical framework would be improved by testing the numerical results with experiments, particularly for flows with high Reynolds numbers or significant temperature changes.

6. Limitations

There are some restrictions and difficulties to be mindful of while applying the HAM to solve nanofluid flow problems. We outline the primary constraints below, which fall into three categories: methodological, computational, and physical modeling. APs used in HAM include the auxiliary linear operator, starting guess, and convergence-control parameter. It’s important to choose these carefully because bad decisions can result in divergence or pointless outcomes. The best way to choose parameters depends on the situation and may include trial-and-error or optimization methods because there is no general guideline.

7. Conclusions

This study examines the heat transfer of a non-Newtonian couple stress BBN flow across a dynamic absorbent flat plate alongside TR and viscous dissipation. Research has been done on absorption and generation in BLs. (SWCNT + Blood) and (MWCNT + Blood) demonstrate the two different nanofluids. When compared to Newtonian nanofluids, couple stress nNNFs generally exhibit superior thermal performance. NN can be increased by 3–35 times, particularly in suction cases. In contrast to Newtonian nanofluids, nNNFs exhibit a small rise in skin friction factor values.

The key findings of the current research are outlined as follows:

• 1.

  The volume friction of nanoparticles, PLI, and CSP is negatively correlated with the field velocity.

• 2.

  Suction and injection parameters are directly related to the velocity field.

• 3.

  The relationship between the EN and temperature distribution, PLI, and TR parameters is direct.

• 4.

  Temperature distribution is inversely related to PN.

• 5.

  There exists a direct relationship between the EN, the PLI parameter, and TR with the NN.

• 6.

  There exists a direct relationship between the PLI, couple stress, and the phenomena of suction and injection with respect to skin friction.

• 7.

  An increase in iterations is associated with a decrease in the residual error.