Combined Impact of Joule Heating, Activation Energy, and Viscous Dissipation on Ternary Nanofluid Flow over Three Different Geometries

The present analysis considers three different geometries plate, cone, and wedge in which a two-dimensional, steady, incompressible ternary nanofluid is examined under the influence of MHD, porous media, and activation energy. Additionally, joule heating and viscous dissipation are considered. The current problem is considered in such a way that $x$ axis is parallel to body's surface and $y$ being normal. The physical model of the flow problem is shown in Figure 1. Let the half-angle of the cone/wedge, radius of the cone, and full angle of the wedge is symbolized as $(\gamma, r, \Theta)$ respectively. The temperature nearby surface i.e., $(y=0)$ and far-field temperature i.e., $(y \rightarrow \infty)$ is represented as $\left(T_w, T_{\infty}\right) .\left(C_w, C_{\infty}\right)$ represents the concentration near the surface i.e., $(y=0)$ and at far-field i.e., $(y \rightarrow \infty)$ representing boundary conditions.

1.png

Figure 1. Flow geometry of the problem

Under the aforementioned assumptions, for the current flow problem the governing equations are as follows [25-27]:

$\frac{\partial\left(r^m u\right)}{\partial x}+\frac{\partial\left(r^m v\right)}{\partial y}=0$       (1)

$u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=v_{t n f} \frac{\partial^2 u}{\partial y^2}+\frac{g \rho_f\left(T-T_{\infty}\right) \varepsilon_T \cos \gamma}{\rho_{t n f}}-v_{t n f} \frac{u}{K^*}-\frac{\sigma_{t n f}}{\rho_{t n f}} B_0^2 u$   (2)

$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\frac{k_{t n f}}{\left(\rho C_p\right)_{t n f}} \frac{\partial^2 T}{\partial y^2}+\frac{\mu_{t n f}}{\left(\rho C_p\right)_{t n f}}\left(\frac{\partial u}{\partial y}\right)^2+\frac{\sigma_{t n f}}{\left(\rho C_p\right)_{t n f}} B_0^2 u^2+\frac{Q}{\left(\rho C_p\right)_{t n f}}\left(T-T_{\infty}\right)$        (3)

$u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_B \frac{\partial^2 C}{\partial y^2}-k_r^2\left(\frac{T}{T_{\infty}}\right)^n-e^{-\frac{E a}{K T}}\left(C-C_{\infty}\right)$     (4)

The relevant boundary conditions are [28]:

At $y=0$ :

$u=u_{\infty}=\frac{v_f x}{l^2}$        (5)

$T=T_W, v=0, C=C_w$,

and,

$u \rightarrow 0, T \rightarrow T_{\infty}, C \rightarrow C_{\infty}$. as $y \rightarrow \infty$       (6)

For the proposed problem, alternate geometries are obtainable through following cases:

Case 1: Plate- $m=1=0$ and $\gamma=0$;

Case 2: Cone- $m=1$ and $\gamma \neq 0$;

Case 3: Wedge- $m=0$ and $\gamma \neq 0$;

The similarities introduced are as follows:

$\begin{gathered}\eta=\frac{y}{l}, u=\frac{v_f x}{l^2} f^{\prime}, v=-\frac{v_f(m+1)}{l} f, \\ T=\left(T_w-T_{\infty}\right) \theta+T_{\infty}, \chi=\frac{C-C_{\infty}}{C_w-C_{\infty}}\end{gathered}$        (7)

Here, $(u, v)$ are the velocity components along $(x, y)$ directions. $K^*$ signifies permeability of porous medium, $\varepsilon_T$ is volumetric thermal expansion coefficient, $g$ denotes acceleration due to gravity, $(\nu, \mu)$ represents fluid's kinematic viscosity and dynamic viscosity, the density and thermal conductivity of the fluid is $(k, \rho), B_0$ is a uniform magnetic field, the reaction rate is $K_r^2$, the modified Arrhenius function is denoted by $\left(T / T_{\infty}\right)^n e^{-\frac{E a}{K T}}$ and $(E a, n)$ is the activation energy and fitted rate constant.

The thermophysical expressions for thermal and electrical conductivity, dynamic viscosity, density and specific heat capacity for ternary nanofluid are as follows [29]:

$\begin{aligned} & k_{t n f}=\left(\frac{k_{S 1}+2 k_{h n f}-2 \phi_1\left(k_{h n f}-k_{S 1}\right)}{k_{S 1}+2 k_{h n f}+\phi_1\left(k_{h n f}-k_{S 1}\right)}\right) k_{h n f}, \\ & k_{h n f}=\left(\frac{k_{S 2}+2 k_{n f}-2 \phi_2\left(k_{n f}-k_{S 2}\right)}{k_{S 2}+2 k_{n f}+\phi_2\left(k_{n f}-k_{S 2}\right)}\right) k_{n f}, \\ & k_{n f}=k_f\left(\frac{k_{S 3}+2 k_f-\left(k_f-k_{S 3}\right) 2 \phi_3}{k_{S 3}+2 k_f+\left(\phi_3 k_f-\phi_3 k_{S 3}\right)}\right), \\ & \sigma_{t n f}=\left(\frac{\sigma_{S 1}\left(1+2 \phi_1\right)+\sigma_{h n f}\left(1-2 \phi_1\right)}{\sigma_{S 1}\left(1-\phi_1\right)+\sigma_{h n f}\left(1+\phi_1\right)}\right) \sigma_{h n f}, \\ & \sigma_{h n f}=\left(\frac{\sigma_{S 2}\left(1+2 \phi_2\right)+\sigma_{n f}\left(1-2 \phi_2\right)}{\sigma_{S 2}\left(1-\phi_2\right)+\sigma_{n f}\left(1+\phi_2\right)}\right) \sigma_{n f}, \\ & \sigma_{n f}=\left(\frac{\sigma_{S 3}\left(1+2 \phi_3\right)+\sigma_{n f}\left(1-2 \phi_3\right)}{\sigma_{S 3}\left(1-\phi_3\right)+\sigma_{n f}\left(1+\phi_3\right)}\right) \sigma_f . \\ & \mu_{t n f}=\frac{\mu_f}{\left(1-\left(\phi_1+\phi_2+\phi_3\right)\right)}{ }^{2.5},\end{aligned}$

$\rho_{t n f}=\left(1-\phi_1\right)\left(\left(1-\phi_2\right)\left[\left(1-\phi_3\right) \rho_f+\rho_{S 3} \phi_3\right]+\rho_{S 2} \phi_2\right)+\rho_{S 1} \phi_1$,

$\left(\rho C_p\right)_{t n f}=\left(\frac{\left(\rho C_p\right)_{S_1 \phi_1}}{\left(\rho C_p\right)_f}+\left(1-\phi_1\right)\left[\left(1-\phi_2\right)\left(\left(1-\phi_3\right)+\frac{\left(\rho C_p\right)_{S 3} \phi_3}{\left(\rho C_p\right)_f}\right)+\frac{\left(\rho C_p\right)_{S 2} \phi_2}{\left(\rho C_p\right)_f}\right]\right)\left(\rho C_p\right)_f$.

In the above expressions, $\left(\phi_1, \phi_2, \phi_3\right)$ are solid volume fractions of copper, SWCNT and MWCNT nanoparticles respectively. And subscripts are $f, n f, h n f, t n f$ denotes base fluid, nanofluid, hybrid nanofluid, and ternary nanofluid, and $s_1, s_2, s_3$ are solid particles of copper, SWCNT and MWCNT. The thermophysical properties of base fluid and nanoparticles are given in Table 1 [30].

Table 1. Thermophysical properties base fluid and ternary nanoparticles

The abbreviated equations are obtained as:

$\frac{f^{\prime \prime \prime}}{B_1 B_2}+(1+m) f^{\prime \prime} f-\left(f^{\prime}\right)^2+\frac{G r \theta \cos \gamma}{B_2}-\frac{\lambda f^{\prime}}{B_1 B_2}-M f^{\prime}=0$        (8)

$\frac{B_4}{B_3} \frac{\theta^{\prime \prime}}{\operatorname{Pr} \frac{N i}{B_3} \frac{1}{B_1 B_2}} \frac{{ }^2}{\frac{\left(\sigma_{t n f} / \sigma_f\right)^2}{B_3}}$    (9)

$\chi^{\prime \prime}+S c(m+1) f \chi^{\prime}-R c S c(1+\delta \theta)^n e^{-\frac{E}{(1+\delta \theta)}} \chi=0$  (10)

and

$f^{\prime}=1, f=0, \theta=1, \chi=1$ at $\eta=0$,   (11)

$f^{\prime}(\infty)=0, \theta(\infty)=0, \chi(\infty)=0$ as $\eta \rightarrow \infty$        (12)

where,

$\begin{aligned} & B_1=\left(1-\phi_1-\phi_2-\phi_3\right)^{2.5}, \\ & B_2=\left(1-\phi_1\right)^{2.5}\left(\left(1-\phi_2\right)\left[\left(1-\phi_3\right)+\frac{\rho_{S 3}}{\rho_f} \phi_3\right]+\frac{\rho_{S 2}}{\rho_f} \phi_2\right)+\frac{\rho_{S 1}}{\rho_f} \phi_1,\end{aligned}$

$B_3=\left(\frac{\left(\rho C_p\right)_{S 1} \phi_1}{\left(\rho C_p\right)_f}+\left(1-\phi_1\right)\left[\left(1-\phi_2\right)\left\{\left(1-\phi_3\right)+\frac{\left(\rho C_p\right)_{S 3} \phi_3}{\left(\rho C_p\right)_f}\right\}+\frac{\left(\rho C_p\right)_{S 2} \phi_2}{\left(\rho C_p\right)_f}\right]\right)$,

$B_4=k_{t n f} / k_f$,

and the dimensionless parameters are

$M=\frac{\sigma_f B_0^2}{\rho_f v_f} l^2$ is magnetic field parameter,

Grashof number is $G r=\frac{g \varepsilon_T\left(T-T_{\infty}\right)}{u_w v_f} l^2$,

$\lambda=\frac{l^2}{K^*}$ is the permeable parameter,

$\operatorname{Pr}=\frac{v_f(\rho C p)}{k}$ is the Prandtl number,

$N i=\frac{Q l^2}{(\rho C p) v_f}$ is the HSS parameter,

$E c=\frac{u_W^2}{C p\left(T-T_{\infty}\right)}$ is the Eckert number,

$S c=\frac{v_f}{D_B}$ is the Schimdt number,

$R c=\frac{k_r^2 l^2}{v_f}$ is the reaction rate parameter,

With suitable similarity transformations, the significant physical parameters such as $C f_x$-Skin friction, $N u_x$-Nusselt number, and $S h_x$-Sheerwood number are given as follows:

$\begin{gathered}C f_x=\frac{\tau_w}{\rho_f u_w^2}, N u_x=\frac{l q_w}{k_f\left(T_w-T_{\infty}\right)}, \text { and } \\ S h_x=\frac{l r_w}{D_B\left(C_w-C_{\infty}\right)} \end{gathered}$       (13)

where,

$\begin{gathered}\tau_w=\left.\mu_{t n f} \frac{\partial u}{\partial y}\right|_{y=0}, q_w=-\left.k_{t n f} \frac{\partial T}{\partial y}\right|_{y=0}, \text { and } \\ r_w=-\left.D_B \frac{\partial C}{\partial y}\right|_{y=0}\end{gathered}$      (14)

Upon replacing Eq. (14) into Eq. (13), we possess

$\begin{gathered}\frac{l}{x} C f=f^{\prime \prime}(0), N u=-\left(k_{\text {tnf }} / k_f\right) \theta^{\prime}(0), \text { and } \\ S h=-\chi^{\prime}(0) \end{gathered}$      (15)

This section discusses the primary outcome of the current effort. The resulting solutions for various non-dimensional parameters are shown using appropriate profiles. Several parameter values were chosen to describe the behaviours of the heat transfer and flow dynamics characteristics. In this case, the current problem includes various relevant parameters such as magnetic field $M$, permeability parameter $\lambda$, Grashof number $G r$, Eckert number $E c$, HSS parameter $N i$, reaction rate parameter $R c$, and activation energy parameter $E$. In addition to this, the effects of some of these parameters on Skin-friction, Nusselt number, and Sherwood number, are depicted in graphs and tables. All the graphs are plotted for cone (Dash lines), wedge (solid lines), and plate (dot lines) for better comparison of results. Figures 2-4 represent the deviations of $M, \lambda, G r$ over velocity profile for three different fluid flow cases. Figure 2 explains that on increasing $M$ there is reduction in the movement of fluid particles resulting in the decreasing of fluid velocity in all three cases.

2.png

Figure 2. $f^{\prime}(\eta)$ curves for varying $M$

3.png

Figure 3. $f^{\prime}(\eta)$ curves for varying $\lambda$

4.png

Figure 4. $f^{\prime}(\eta)$ curves for varying $\mathrm{Gr}$

From the Figure 2 it is evident that $f^{\prime}$ for flow across cone declines faster than in the remaining cases. In Figure 3 it is seen that rise in the value of $\lambda$ diminishes $f^{\prime}$ for all flow cases. It is due to fact the rise in the $\lambda$ increases system's resistance.

As a result of developed force of friction, the flow velocity is reduced. Further, curves of $f^{\prime}(\eta)$ for flow past cone lessens faster than in remaining cases. The escalate in the $f^{\prime}(\eta)$ profile for upsurge values of $G r$ for all flow cases is displayed in the Figure 4. From physical standpoint increased Gr values reduce thickness of boundary layer due to buoyant forces produced by fluctuations in temperature. Besides, the $f^{\prime}(\eta)$ for the fluid flow over plate proliferates quicker than in any other cases.

5.png

Figure 5. $\theta(\eta)$ curves for different $M$ values

6.png

Figure 6. $\theta(\eta)$ curves for different $E c$ values

Figure 5 displays the escalating nature in thermal gradient profile for $M$ in all fluid flow cases. Since increasing magnetic field creates fluid resistance to flow which intern facilitates upsurge in thermal distribution. It is witnessed that thermal distribution is more in plate when compared with left cases. The effect of Eckert number over $\theta(\eta)$ is exhibited in Figure 6.

Here, an increase in the $E c$ promotes rise in $\theta(\eta)$. Physically, a spike in the $E c$ thickens thermal boundary layer and moreover this effect is more pronounced in the presence of magnetic field. The behaviours of $N i$ parameter over $\theta(\eta)$ for all flow cases is studied in Figure 7. Furthermore, the fluid flow past wedge unveils better heat transference than in left over cases. Internal heat sink/source (HSS) hinders or facilitates heat transfer. As we magnify impact of $N i$ thickness of $\theta(\eta)$ increases. Better heat transfer will be seen when we rise HSS values from negative to positive and also the occurrence of a heat source stimulates the fluid's temperature. As a result, as heat is absorbed, and with buoyancy's effect boosts the flow and enhances heat transmission. Figure 8 demonstrate the influence of $R c$ over $\chi(\eta)$ for all flow cases. In all flow cases, it is evident that increase in $R c$ descends $\chi(\eta)$. This is because, when the reaction rate parameter is raised, the concentration of the entire fluid medium diminutions due to lower mass diffusivity and also due to the fact that lot of chemical consumption takes place during chemical reaction. A lower mass transfer rate is perceived in fluid flow across cone. Figure 9 shows the effect of $E$ on $\chi(\eta)$ for all flow cases. The growth in the $E$ value stimulates $\chi(\eta)$. Increasing $E$ values reduce the Arrhenius process, thereby elevating the rate of the generative chemical process that generates concentration. More enhancement in the concentration is observed in fluid flow case over plate.

7.png

Figure 7. $\theta(\eta)$ curves for different $Q$ values

8.png

Figure 8. $\chi(\eta)$ curves for distinct $R c$ values

9.png

Figure 9. $\chi(\eta)$ curves for distinct $E$ values

Figure 10 shows the variations in the values of $C f x$ for various values of $\lambda$ and $G r$ for all three flow cases. In general, the existence of a porous media increases skin friction. This is because the flow through the porous media involves viscous dissipation within the pores, resulting in greater shear stress at the solid surface, and skin friction doubles with a surge in the Grashof number due to a raise in the velocity distribution. Hence it is portrayed that skin-friction increases for $\lambda$ and $G r$. Additionally, compared to the other circumstances, skin friction is improved in the fluid flow case through cone. Effect of skin friction is studied in the design of aircrafts, ships, and other automobiles that move through fluids. Figure 11 illustrates how the increasing value of $\phi$ over $N u_x$ versus $Q$ for all three fluid flow cases.

10.png

Figure 10. Influence of $(\lambda, G r)$ over Ski-friction $C f x$

11.png

Figure 11. Influence of $(\phi, Q)$ over Nusselt number $N u_x$

In general, a rise in the solid volume fraction corresponds to an increase in the Nusselt number. This is due to the fact that as the solid volume fraction increases, so does the heat transfer coefficient and so does the Nusselt number of nanoparticles and also presence of $Q$ promotes heat transfer across boundary layer. However, in the case of flow across wedge, an enriched heat transmission rate is observed. Analysis of Nusselt number is a significant due to its wide range of industrial applications such as cooling systems, chemical and aerospace engineering. Figure 12 presents the decreasing behaviour Sherwood number under the impacts of $E$. The activation energy may impact the concentration distribution and mass transfer rate, which in turn can affect the Sherwood number. A higher activation energy may result in a lower mass transfer rate, and thus a lower Sherwood number. In biomedical engineering analysis of Sherwood number is important to design and optimize drug delivery systems and much more.

12.png

Figure 12. Influence of $(E, \phi)$ over Sherwood number $S h_x$