Convective Heat Transfer of Radiating Magneto-Micropolar Nanofluid Flow

Nanofluid flow is widely recognized for its potential in enhancing heat transfer across industrial and engineering applications, with the conductivity of such fluids being augmented by the introduction of nanoparticles into the base fluid [1]. Investigations into mixed convective nanofluid stagnation point flow have been conducted by Ellahi et al. [2], while Ahmed et al. [3] have examined viscous dissipation and radiative heat flux over a moving wedge utilizing volume fraction. The flow of radiative nanofluids over a wedge has also been considered by Amar and Kishan [4]. Recent studies have expanded upon these aspects of nanofluids under various physical circumstances [5-7].

Fluids exhibiting microstructural properties, such as the gyration of microelements and rotary motions, are extensively utilized across numerous material and chemical engineering processes, slurry technologies, and biomechanics. Eringen [8] proposed the micropolar fluid theory to model these effects. This model has gained substantial popularity due to its ability to accurately replicate a variety of substances, from biological fluids like blood and synovial fluids to smart fluids and various colloidal suspensions [9-11].

Despite its usefulness, the nanofluid model's applicability is restricted due to its inability to explain a class of fluids that exhibit distinctive microscopic characteristics driven by the local structure of fluid particles and microrotation. To address such limitations, Bourantas and Loukopoulos [12] investigated micropolar Al₂O₃-water based nanofluid flow under magnetic field effects using the Moving Least Squares method. Türk and Tezer-Sezgin [13] employed the Finite Element Method (FEM) to examine micropolar nanofluid free convection flow. Further investigations into micropolar nanofluid models, considering factors such as inclined surfaces, have been carried out by Rafique et al. [14], among others [15-17].

Flows involving chemical reactions are prevalent across numerous fields of engineering research, including cooling towers, nuclear reactor cooling, drying, food processing, and geophysics [18, 19]. Investigations into chemical reactions with convective flows of electrically conducting fluid have garnered significant attention due to their varied applications, ranging from hybrid MHD power generators and geophysics to plasma studies, astrophysical flows, magnetic materials processing, chemical engineering, and nuclear fusion. An understanding of both fluid dynamics and the chemistry of the nanoparticles and base fluid is fundamental to nanofluid research. Chemical reactions can influence the stability, dispersion, thermal characteristics, and catalytic behaviour of nanofluids, thereby playing a pivotal role in understanding and optimizing magneto-micropolar nanofluid flow. The boundary layer theory has been employed by numerous authors [20-23] to investigate various aspects of transport phenomena.

Heat and mass transfer often occur simultaneously, involving multiple species diffusion across fluid. Due to its vast range of applications in materials fabrication, mechanical, nuclear, and chemical processes, significant interest has been directed towards mathematical models that simulate the collective impact of chemical reaction and mass transport in electromagnetic fields [24]. Several authors have conducted comprehensive studies of the convective heat and mass exchange in MHD flows [25-27].

This study delves into the examination of heat and mass transport in mixed convective magneto-micropolar nanofluid flow over a wedge, incorporating the influence of convective boundary conditions, thermal radiation, and chemical reaction. A novel model is proposed herein, integrating Eringen's model for microstructural rheological properties, Rosseland's flux model for thermal radiation effects, and the Tiwari-Das model for simulating nano-scale properties. A first-order chemical reaction is introduced into the conservation of species diffusion equation to formulate the non-linear model. The proposed model provides insights into the influence of various factors determining the flow regime, achieved through the application of a novel finite element code. This research contributes to the existing body of engineering science literature on magneto-micropolar nanofluid by introducing a flow model that has not been previously presented in the literature.

The method utilized here is described in detail by Reddy [31]. The finite-element modeling stages are given in flow chart Figure 2.

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Figure 2. FEM flow chart

3.1 Finite element modeling

By employing FEM, nonlinear ODE, Eqs. (11), (14), (15), (18) subject to boundary condition Eq. (17) are evaluated as follows

let:

$f^{\prime}=q$               (22)

The system gets reduced to

$\begin{aligned} & \left(\phi_7+K\right) q^{\prime \prime}+\phi_1 f q^{\prime}+\phi_2 M(1-q) -\frac{2 m}{1+m}\left(q^2-1\right)+K g^{\prime}+\left(\phi_3 \lambda \theta+\phi_4 \lambda_c \varphi\right) \frac{2}{1+m} \sin \frac{\Omega}{2}=0,\end{aligned}$               (23)

$\phi_1\left(f g^{\prime}-\phi_8 q g\right) f^{\phi_6}-K\left(2 g+q^{\prime}\right)+\left(g^{\prime} f^{\phi_6}\right)^{\prime}\left(\phi_7+\frac{K}{2}\right)=0$               (24)

$\left(\frac{k_{n f}}{k_f}+\frac{4}{3} R d\right) \theta^{\prime \prime}+\phi_5 \operatorname{Pr} f \theta^{\prime}=0$               (25)

$(1-\phi) \varphi^{\prime \prime}+S c f \varphi^{\prime}-\frac{2}{1+m} \operatorname{Sc} C h \varphi=0$               (26)

Now, boundary conditions become:

$\begin{aligned} & q(0)=S, f(0)=s_w, \varphi(0)=1, \\ & g(0)=-n q^{\prime}(0), g(\infty)=0, \\ & \theta^{\prime}(0)=-a \frac{k_f}{k_{n f}}[1-\theta(0)], \\ & \varphi(\infty)=0, q(\infty)=1, \theta(\infty)=0 .\end{aligned}$               (27)

3.2 Variational form

Variational form linked with Eqs. (22)-(26), over element $\left(\eta_e, \eta_{e+1}\right)$ is:

$\int_{\eta_{\varepsilon}}^{\eta_{e+1}} w_1\left(f^{\prime}-q\right) d \eta=0$               (28)

$\int_{\eta_e}^{\eta_{e+1}} w_2\left(\begin{array}{l}\left(\phi_7+K\right) q^{\prime \prime}+\phi_1 f q^{\prime}+\phi_2 M(1-q) \\ -\frac{2 m}{1+m}\left(q^2-1\right)+K g^{\prime} \\ +\left(\phi_3 \lambda \theta+\phi_4 \lambda_c \varphi\right) \frac{2}{1+m} \sin \frac{\Omega}{2}\end{array}\right) d \eta=0$               (29)

$\int_{\eta_e}^{\eta_{e+1}} w_3\left(\begin{array}{l}\phi_1\left(f g^{\prime}-\phi_8 q g\right) f^{\phi_6}-K\left(2 g+q^{\prime}\right) \\ +\left(g^{\prime} f^{\phi_6}\right)^{\prime}\left(\phi_7+\frac{K}{2}\right)\end{array}\right) d \eta=0$               (30)

$\int_{\eta_e}^{\eta_{e+1}} w_4\left(\left(\frac{k_{n f}}{k_f}+\frac{4}{3} R d\right) \theta^{\prime \prime}+\phi_5 \operatorname{Pr} f \theta^{\prime}\right) d \eta=0$               (31)

$\int_{\eta_{\varepsilon}}^{\eta_{c+1}} w_5\left((1-\phi) \varphi^{\prime \prime}+\operatorname{Sc} f \varphi^{\prime}-\frac{2}{1+m} \operatorname{Sc} C h \varphi\right) d \eta=0$               (32)

Here, variation in f,q,g,θ and φ are given by weight functions w₁, w₂, w₃, w₄ and w₅ respectively.

3.3 Finite element form

Taking ψ_i=w₁=w₂=w₃=w₄=w₅, with:

$\begin{aligned} f & =\sum_{j=1}^2 f_j \psi_j, q=\sum_{j=1}^2 q_j \psi_j, g=\sum_{j=1}^2 g_j \psi_j, \\ \theta & =\sum_{j=1}^2 \theta_j \psi_j, \varphi=\sum_{j=1}^2 \varphi_j \psi_j,\end{aligned}$               (33)

and linear interpolation functions:

$\begin{aligned} & \psi_1=\left(\eta_{e+1}-\eta_e\right)^{-1}\left(\eta_{e+1}-\eta\right), \\ & \psi_2=\left(\eta_{e+1}-\eta_e\right)^{-1}\left(\eta-\eta_e\right), \quad \eta_e \leq \eta \leq \eta_{e+1}\end{aligned}$               (34)

We substitute Eq. (33) and Eq. (34) into Eqs. (28)-(32), to obtain the finite element model as:

$\left.\begin{array}{llllll}{\left[K^{11}\right]} & {\left[K^{12}\right]} & {\left[K^{13}\right]} & {\left[K^{14}\right]} & {\left[K^{15}\right]} \\ {\left[K^{21}\right]} & {\left[K^{22}\right]} & {\left[K^{23}\right]} & {\left[K^{24}\right]} & {\left[K^{25}\right]} \\ {\left[K^{31}\right]} & {\left[K^{32}\right]} & {\left[K^{33}\right]} & {\left[K^{34}\right]} & {\left[K^{35}\right]} \\ {\left[K^{41}\right]} & {\left[K^{42}\right]} & {\left[K^{43}\right]} & {\left[K^{44}\right]} & {\left[K^{45}\right]} \\ {\left[K^{51}\right]} & {\left[K^{52}\right]} & {\left[K^{53}\right]} & {\left[K^{54}\right]} & {\left[K^{55}\right]}\end{array}\right]\left[\begin{array}{l}\{f\} \\ \{q\} \\ \{g\} \\ \{\theta\} \\ \{\varphi\}\end{array}\right]=\left[\begin{array}{l}\left\{r^1\right\} \\ \left\{r^2\right\} \\ \left\{r^3\right\} \\ \left\{r^4\right\} \\ \left\{r^5\right\}\end{array}\right]$               (35)

Here $\left[K^{m n}\right]_{2 \times 2}$ and $\left\{r^m\right\}_{2 \times 1}$ are matrices given by:

$\begin{gathered}K_{i j}^{11}=\int_{\eta_e}^{\eta_{e+1}} \psi_i \frac{d \psi_j}{d \eta} d \eta, K_{i j}^{12}=-\int_{\eta_e}^{\eta_{e+1}} \psi_i \psi_j d \eta, \\ K_{i j}^{13}=0, K_{i j}^{14}=0, K_{i j}^{15}=0, \\ K_{i j}^{21}=0, K_{i j}^{23}=K \int_{\eta_e}^{\eta_{e+1}} \psi_i \frac{d \psi_j}{d \eta} d \eta, \\ K_{i j}^{24}=\left(\frac{2}{1+m}\right) \phi_3 \lambda \sin \frac{\Omega^{\eta_{e+1}}}{2} \int_{\eta_e}^{\eta_i} \psi_i \psi_j d \eta, \\ K_{i j}^{22}=-\left(\phi_7+K\right) \int_{\eta_e}^{\eta_{e+1}} \frac{d \psi_i}{d \eta} \frac{d \psi_j}{d \eta} d \eta+\phi_1 \int_{\eta_{l_e+1}}^{\eta_{\eta_e}} \bar{f} \psi_i \frac{d \psi_j}{d \eta}d \eta-\phi_2 M \int_{\eta_e}^{\eta_{e+1}} \psi_i \psi_j d \eta \\ K_{i j}^{25}=\left(\frac{2}{1+m}\right) \phi_4 \lambda_c \sin \frac{\Omega^{\eta_{e+1}}}{2} \int_{\eta_e}^{\eta_i} \psi_i \psi_j d \eta, \\ K_{i j}^{31}=0, K_{i j}^{34}=0, K_{i j}^{35}=0, K_{i j}^{32}=-K \int_{\eta_e}^{\eta_{e+1}} \psi_i \frac{d \psi_j}{d \eta} d \eta,\end{gathered}$

$\begin{aligned} & K_y^{33}=-\left(\phi_7+\frac{K}{2}\right) \int_{\eta_e}^{\eta_{e+1}}(\bar{f})^{\phi_6} \frac{d \psi_i}{d \eta} \frac{d \psi_j}{d \eta} d \eta \\ & +\phi_1 \int_{\eta_e}^{\eta_{\eta_i}}(\bar{f})^{\frac{3-m}{1+m}} \psi_i \frac{d \psi_j}{d \eta} d \eta \\ & -\phi_1 \phi_8 \int_{\eta_e}^{\eta_{e+1}}(\bar{f})^{\phi_i} \bar{q} \psi_i \psi_j d \eta-2 K \int_{\eta_e}^{\eta_{e+1}} \psi_i \psi_j d \eta, \\ & K_{i j}^{41}=0, \quad K_{i j}^{42}=0, \quad K_{i j}^{43}=0, \quad K_{i j}^{45}=0, \\ & K_y^{44}=-\left(\frac{k_{\eta f}}{k_f}+\frac{4}{3} R d\right) \int_{\eta_e}^{\eta_{k+1}} \frac{d \psi_i}{d \eta} \frac{d \psi_j}{d \eta} d \eta+\operatorname{Pr} \phi_5 \int_{\eta_e}^{\eta_{k+1}} \bar{f} \psi_i \frac{d \psi_j}{d \eta} d \eta \\ & K_{i j}^{51}=0, \quad K_i^{52}=0, \quad K_i^{53}=0, \quad K_{i j}^{54}=0 \text {, } \\ & K_{i j}^{55}=-(1-\phi) \int_{\eta_e}^{\eta_{e+1}} \frac{d \psi_i}{d \eta} \frac{d \psi_j}{d \eta} d \eta+S c \int_{\eta_e}^{\eta_{e+1}} \bar{f} \psi_i \frac{d \psi_j}{d \eta} d \eta \\ & -\frac{2}{m+1} S c \cdot C h \int_{\eta_e}^{\eta_{e+1}} \psi_i \psi_j d \eta \text {, } \\ & r_i^1=0, \\ & r_i^2=-\left(\frac{2 m}{1+m}+\phi_2 M\right) \int_{\eta_e}^{\eta_{e+1}} \psi_i d \eta-\left(\phi_7+K\right)\left[\psi_i \frac{d q}{d \eta}\right]_{\eta_e}^{\eta_{c+1}} \text {, } \\ & r_i^3=-\left(\phi_7+\frac{K}{2}\right)\left[\psi_i f^{\phi_6} \frac{d g}{d \eta}\right]_{\eta_e}^{\eta_{c+1}}, \\ & r_i^4=-\left(\frac{k_{n f}}{k_f}+\frac{4}{3} R d\right)\left[\psi_i \frac{d \theta}{d \eta}\right]_{\eta_e}^{\eta_{e+1}}, \\ & r_i^5=-(1-\phi)\left[\psi_i \frac{d \varphi}{d \eta}\right]_{\eta_{\tau_c}}^{\eta_{\ell+1}} . \\ & \end{aligned}$

where, $\bar{f}=\sum_{i=1}^2 \bar{f}_i \psi_i, \bar{q}=\sum_{i=1}^2 \overline{q_i} \psi_i$ assumed to be known.

3.4 Convergence criteria

For higher values of η(>8) no substantial alteration in the outcome was seen. Thus, $\infty$ is taken as 8. The domain is divided into n₁-linear elements with $h_e=\left(\eta_{e+1}-\eta_e\right)=8 / n_1$. Calculations are run for 50, 100, 150, 200, 240, 320, 400, and 500 elements. The convergence of result is shown in Table 1. It is observed that accuracy is not affected for higher values of n₁ (number of elements). Thus, n₁=320 is fixed. The order of the element matrix Eq. (35) is 10. If we split domain into 320 equal linear elements, we get matrix of order 1605 after assembling all element equations. The FEM model is non-linear and linearized by using $\bar{f}, \bar{q}$. Subsequently, system of 1599 equations remain after applying the assumed boundary conditions which are solved for $f, q, g, \theta$ and $\varphi$ using iterative scheme. The iterative process is terminated when $\sum_{i, j} \mid \Delta^{n^*+1}-$ $\Delta^{n^*} \mid \leq 10^{-6}$ is satisfied. Here $n^* \Delta=f, q, g, \theta, \varphi$ and iterative steps are denoted by.

Table 1. Convergence of FEM

The significance of nano-particle concentration (ϕ), material parameter (K), wedge parameter (m) and convective boundary parameter (a), radiation parameter (Rd), Schmidt number (Sc) and chemical reaction parameter (C_h) are explored one at a time while keeping fixed other parameters such as $\operatorname{Pr}=6.2$, $\phi=0.04$, $K=2$, $M=1$, $a=1$, $\lambda=5$, $\lambda_c=1$, $S c=0.5$, $C_h=1$, $n=1 / 2$, $s_w=1$, $R d=1$ and $m=1 / 7$. Also, skin friction, couple stress, Nusselt and Sherwood numbers are reported in Table 5.

Table 5. Numerical values of $f^{\prime \prime}(0)$, $g^{\prime}(0)$, $-\theta^{\prime}(0)$ and $-\varphi^{\prime}(0)$

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Figure 3. Impact of ϕ on temperature, microrotation, concentration and velocity

Figure 3 displays the temperature, microrotation, concentration and velocity distribution for various ϕ. Throughout the boundary layer, retardation in the velocity and species concentration profiles are marked by increasing nanoparticles volume concentration. This retardation is caused by heavy density of nanoparticles near wedge surface. The temperature rises as concentration increases. This is due to boosting of heat transfer rate by adding high thermal conductivity nanoparticles. Further, it is seen that, at η=1.15 point of inflection is marked closer to the wedge surface. To achieve the appropriate boundary condition, rise in volume concentration causes the microrotation profiles to boost in the first region (η<1.15) and opposite effect can be seen in second region (η>1.15). Therefore, volume fraction is significant in enhancing the thermal conductivity of the liquid.

Figure 4 illustrates the temperature, microrotation, concentration and velocity distribution for different K. As K increases, the velocity and microrotation declines while opposite behavior occurs for temperature and concentration. As K increases, fluid motion experiences additional resistive force in linear direction which deteriorate values of $f^{\prime}$. The thermal profiles are enhanced as a result of heating effect that takes place within thermal boundary layer by increasing K. 

Figure 5 exhibits the impact of temperature, microrotation, concentration and velocity for different m($0 \leq m \leq 1$). Increasing m boosts the concentration, micro-rotation, velocity and have opposite effect on temperature. Thus, hydrodynamic boundary layer decreases and velocity profile gets closer to wedge surface.

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Figure 4. Influence of K on temperature, microrotation, concentration and velocity

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Figure 5. Influence of m on temperature, microrotation, concentration and velocity

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Figure 6. Influence of a on temperature

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Figure 7. Effect of Rd temperature

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Figure 8. Impact of Sc on species concentration

Figure 6 illustrates the impact of a on temperature function. The temperature increases in correlation with a. These results are consistent with the work by Dawar et al. [35]. The magneto-micropolar nanofluid density decreases by a stronger convective heating of the wedge. This produces favorable pressure gradient that simulates the microelements to spin faster and speeds up the flow. It is seen that, at the wedge surface temperature reaches its maxima as h_f and thermal resistance are inversely proportional. Further, it is seen from the figure that $\theta(0)=1$ as $a \rightarrow \infty$. The concept of convective boundary parameter is used in physics and engineering to understand and design heat transfer processes in various systems.

Figure 7 depicts the temperature profiles subject to various Rd. For stronger thermal radiation, the temperature of nanofluid rises. The mean absorption coefficient declines if Rd is increased. Due to this radiative heat flux divergence increases, which results in rise in the fluid temperature. Similar trends are observed by Ahmed et al. [3]

Figure 8 depicts impact of Sc on concentration profiles. These profiles become thinner under the influence of Sc. Similar trends are observed by Abdal et al. [36]. The Sc is the ratio of mass to momentum diffusion. Therefore, slower diffusion process is encountered with improvement in Sc which significantly decreases the concentration field and suppress rate of nanoparticles diffusion.

Figure 9 displays the effect of C_h on temperature, microrotation, concentration and velocity profiles. The diffusive transport and flow momentum are delayed by chemical reaction. Hence, as C_h increases velocity decreases. We observe that increasing chemical reaction parameter microrotation increases towards wedge surface whereas it decreases away from surface. A higher chemical reaction parameter causes more effective conversion of chemical energy to thermal energy, this heats fluid and raises the temperature. Concentration profiles declines with surge in chemical reaction parameter. As simulated in Eq. (6), chemical reaction is 1^st order destructive and homogeneous. In this reaction nanoparticles undergo a transformation into another species. Consequently, from a physical perspective, it is expected that for more intense chemical reaction the concentration of initial nanoparticles will decrease. The results published by Ramzan et al. [37], Bhatti et al. [38] were found to support these trends. According to Krishnamurthy et al. [39] consumption of chemicals during chemical reactions lowers species concentration.

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Figure 9. Influence of C_h on temperature, microrotation, concentration and velocity

Finally, Table 5 presents the impact of ϕ, C_h, a, K, m, Rd, and Sc on Nusselt and Sherwood numbers, skin friction, couple stress. Rising values of a, m, and Rd boost skin friction causing the flow to accelerate, while reverse behavior is observed with the surge in ϕ, C_h, K and Sc. Increment in a, m, and Rd leads to enhance couple stress, whereas it reduces with surge in ϕ, C_h, K and Sc. The Nusselt number is boosted with rise in a, m whereas it reduces with the increase of ϕ, C_h, K, Rd, and Sc. The Sherwood number is elevated with rise in all parameters except wedge angle and material parameter, as K and m have tendency to decrease Sherwood number. Sherwood number is more significantly impacted by the C_h and Sc compared to the other parameters.