Analysis of Homogeneous/Heterogeneous Reactions in an Electrohydrodynamic Environment Utilizing the Second Law

The axial electric field $E_x$ and transverse magnetic field $B_y$ in Figure 1 influence the incompressible Eyring–Powell fluid’s unsteady electro-osmotic peristaltic flow through a two-dimensional asymmetric channel. The rectangular coordinates $\overline{X}$ and $\overline{Y}$ are taken into account along the flow and perpendicular to flow, respectively. The constant temperature and the potential of the lower and upper walls are represented by $\left(\overline{{\zeta }_1}, T_1\right)$ and $\left(\overline{{\zeta }_2}, T_0\right)$, respectively. The channel is inclined at an angle of ${\alpha }_0$.

Following Merkin and Chaudhary [1], a homogeneous reaction can be written for cubic autocatalysis as and the first order heterogeneous reaction associated with a catalyst surface is defined as in which $\overline{\acute{a}}$ and $\overline{\acute{b}}$ represent the concentrations of chemical species $A$ and $B$, respectively, and $K_1$ and $K_2$ denote the rate constants. Further, these two reactions are isothermal. Mathematically, peristaltic asymmetric channel walls are written as: where ${\overline{H}}_1\left(\overline{X},\overline{t}\right), {\overline{H}}_2\left(\overline{X}, \overline{t}\right), l_1+l_2, \lambda , {\overline{b}}_1, {\overline{b}}_2, c, \overline{t}$, and $\mathsf{\Phi }$ represent the lower wall, upper wall, channel width, wave length, amplitudes of lower and upper waves, wave speed, time, and phase difference in the laboratory frame. Furthermore, the amplitude, channel width and phase difference satisfy the condition ${\overline{b}}_1^2+{\overline{b}}_2^2+2{\overline{b}}_1{\overline{b}}_2\cos \left(\mathsf{\Phi }\right)\le {(l_1+l_2)}^2$.

The Cauchy stress tensor ${\overline{\tau }}_{ij}$ for considered fluid is defined as [14,17,18] where $\mu$ is the dynamic viscosity, ${\beta }^{\ast }$ and $c^{\ast }$ are the Eyring–Powell fluid constants, and $\overline{\mathit{V}}$ is the velocity field. Considering

The velocity field $\overline{\mathit{V}}$ having components $\overline{U}$ and $\overline{V}$ along $\overline{X}$ and $\overline{Y}$ is defined as

Using Equations (6) and (7), Equation (5) in component form is written as

The electric potential distribution $\overline{\varphi }$ in the EDL of a microchannel is given by the Poisson–Boltzmann equation [20] where ${\rho }_e$ is the total charge density, $\epsilon$ is the relative permittivity of medium, and ${\epsilon }_0$ is the permittivity of vacuum. For a symmetric electrolyte, the net charge density ${\rho }_e$ obeys Boltzmann distribution and can be defined as [20] where $z, e, {\overline{\acute{n}}}_{+}$, and ${\overline{\acute{n}}}_{-}$ represent valance of ionic species, charge on proton, cations, and anions, respectively. The charge number distribution of an individual species is defined by the Nernst–Planck equation [20] in which $D, T_{av}$ and $K_B$ illustrates ionic species diffusivity, $T_{av}$ is the average temperature, and $K_B$ is the Boltzmann constant.

We considered the Galilean transformation to shift coordinates and velocities from laboratory frame $\left(\overline{X}, \overline{Y}, \overline{t}\right)$ to moving frame $\left(\overline{x}, \overline{y}\right)$ as

Hence, Equations (3), (4) and (14) become

This introduces the dimensionless quantities: where $\varphi ,n_0, \delta , {\zeta }_1, {\zeta }_2, l, u$, and v depict the dimensionless electric potential, bulk ionic concentration, wave number, dimensionless surface potential at the lower and upper walls, semi channel width, and dimensionless velocity components along $x$ and $y$ directions, respectively. Moreover, $m_e, Pe$, and ${\lambda }_L$ are electroosmotic parameter, Peclet number, and Debye length, respectively.

Utilizing Equation (19) in Equation (16), we have

The preceding Equation (20) is simplified by considering a long wavelength and a small ionic Peclet number i-e $Pe, \delta \ll 1,$ yielding

Boundary conditions are:

Using the above boundary conditions Equation (22) and (23), the general solution of Equation (21) is

Substitute Equation (13) in Equation (12), the Poisson–Boltzmann equation has the form in wave frame

The dimensional boundary conditions for $\overline{\varphi }$ are

Using dimensionless variables in Equation (19), the Equations (17), (18) and (25) become

Using lubrication [12,13] and Debye–Huckel linearization [20] assumptions, the above Equation (26) simplify to the form

Solving Equation (28) with respect to boundary conditions Equation (27), the analytical solution of $\varphi$ is

The equations of continuity, momentum, energy, and homogeneous–heterogeneous reaction for Eyring–Powell fluid in the presence of axially applied electric field, inclined channel, transverse magnetic field, and mixed convection in a laboratory frame are [14,16]

${\beta }_T$ is the thermal expansion coefficient, ${\beta }_a$ is the concentration expansion coefficient, $a_0$ is the uniform concentration of reactant $A$, ${\alpha }_0$ is the angle of inclination,$c_p$ is the specific heat, $D_A$ and $D_B$ are the coefficients of mass diffusivity for homogeneous and heterogeneous reactions.

The stationary frame and moving frame are related by the following transformations:

By employing the above transformations Equation (36), Equations (30)–(35) yield

Let us now introduce the non-dimensional parameters listed below as: where $\theta , Ha, u_{HS,} U_{HS}, Re, \gamma , Sc, Pr, Br, D_R, f, \acute{g}, G_t, G_a, M_1, M_2, B_1$, and $B_2$ are the dimensionless temperature, Hartmann number, Helmholtz–Smoluchowski velocity, the ratio of the Helmholtz–Smoluchowski velocity to the characteristic wave speed, Reynolds number, Joule heating parameter, Schmidt number, Prandtl number, Brinkman number, diffusion ratio coefficient, dimensionless concentration of species A, dimensionless concentration of species B, temperature Grashoff number, concentration Grashoff number, homogeneous reaction parameter, heterogeneous reaction parameter, and Eyring–Powell fluid parameters, respectively.

The dimensionless boundary conditions are [16]

It is assumed that $D_A=D_B$ i-e $D_R=1$. This assumption results in the given relation [16]

From the assumption of a low Reynolds number and a long wavelength [12,13], we have the following system of equations

Eliminating the pressure term from Equations (47) and (48), we obtain

Introducing the stream function $\psi$ with velocity components $u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$, Equation (46) is automatically satisfied and Equations (49), (50) and (52) become

The boundary conditions are: where is related to $\mathsf{\Theta }$ by

The dimensionless pressure rise per wavelength $\Delta p$ is written as:

Figure 2a–f shows the effects of the electroosmotic parameter $m_e,$ Hartmann number $Ha,$ Eyring–Powell fluid parameters $B_1$ and $B_2$, temperature Grashoff number $G_t$ and mass Grashoff number $G_a$ on the velocity profile $u$ as a function of transverse coordinate $y$. Figure 2a demonstrates that the Eyring–Powell fluid axial velocity $u$ decelerates in the channel’s centre and accelerates near the channel walls as the Debye–Huckel parameter $m_e$ increases. As the electroosmotic parameter $m_e$ rises, the EDL thickness decreases, and it is inferred that the electric body force will be considerably greater towards the channel’s boundary than in its centre, causing the velocity to increase in the region near the channel walls. Additionally, because the lower wall $\left({\zeta }_1=-0.9\right)$ has a smaller zeta potential than the upper wall $\left({\zeta }_2=-0.8\right)$, the velocity is higher near the lower wall. The behavior of the velocity field $u$ under the influence of an external magnetic field is depicted in Figure 2b. By raising the Hartmann number $Ha$, it is shown that fluid flow is hindered in the centre and boosted at the walls. This fact is caused by a magnetic field’s anisotropic properties, which resist motion that is normal to it while permitting motion parallel to it to continue unimpeded. Figure 2c shows that for different values of the temperature Grashoff number $G_t$, the behavior of the velocity distribution $u$ is not symmetric. The Eyring–Powell fluid velocity $u$ is observed to increase near the lower wall and decrease near the upper wall when $G_t$ enhances. It is due to the fact that the lower section of the fluid has less fluid viscosity because of the greater temperature at the lower wall $\left(T_1>T_0\right)$, which accelerates velocity. By rising the mass Grashof number $G_a$, it can be seen in Figure 2d that the velocity increases at the upper section and drops near the bottom portion of the asymmetric channel. This observation is comparable to the findings of the study [16]. According to Figure 2e, the axial velocity $u$ in the middle of the microchannel increases as the material fluid parameter $B_1$ increases, but it decreases close to the channel walls. This is because that the Eyring–Powell fluid becomes thin in the centre due to decreased viscosity, whereas viscous effects are more pronounced close to the walls. Figure 2f indicates that the behavior of Eyring–Powell fluid parameter $B_2$ on the velocity distribution is opposite to $B_1$. Physically, the Eyring–Powell fluid’s thickness is enhanced by increasing its dynamic viscosity and the velocity $u$ decelerated. It is important to note that the Newtonian fluid is represented by the curve $B_1=B_2=0$ in Figure 2e,f.

Figure 3a–h shows the effects of emerging parameters on temperature profile $\theta$ for flow in a microchannel with walls set at different constant temperatures. It is obvious that the temperature $\theta$ curves decrease as the transverse coordinate y increases in order to satisfy the boundary conditions. Figure 3a gives the effect of the EDL thickness on the temperature profile $\theta$ along the height of the channel. The fluid’s temperature $\theta$ drops due to the electroosmotic parameter $m_e$. The magnetic field causes the temperature $\theta$ of the Eyring–Powell fluid to rise throughout the channel, depicted in Figure 3b. For larger Hartmann number $Ha$, the Lorentz force boosts up, which causes more resistance to the flow. Therefore, temperature $\theta$ rises as a result of the conversion of fluid particle kinetic energy into heat energy. The fluid temperature $\theta$ rises as a result of the impacts of both heat Grashof number $G_t$ and mass Grashof numbers $G_a$, as seen in Figure 3c,d. This is due to the decline in viscous forces. As the relationship between temperature and viscosity is inverse, a rise in temperature $\theta$ is caused by a reduction in viscous forces. The descriptions of the Eyring–Powell fluid parameters $B_1$ and $B_2$ on the temperature distribution $\theta$ are shown in Figure 3e,f. It is observed that enhancing $B_1$ rises fluid temperature, whereas increasing $B_2$ lessens fluid temperature. Physically, the increased value of $B_1$ decreases the dynamic viscosity, which causes the temperature to rise, and the increment of $B_2$ improves the dynamic viscosity, which causes the temperature to fall. According to Figure 3g,h, the temperature field $\theta$ is an increasing function for both viscous dissipation paramater $Br$ and Ohmic dissipation parameter $\gamma$. Since both the Brinkman number $Br$ (which produces heat via shear stress) and the Joule heating parameter $\gamma$ (which produces heat through electric and magnetic fields) are related to the creation of heat, as $Br$ and $\gamma$ are boosted, more heat is generated, which causes the Eyring–Powell fluid’s temperature $\theta$ to rise.

Figure 4a–c gives the graphical results for the variation of concentration distribution $f$ with different values of Schmidt number $Sc$, homogeneous reaction parameter $M_1$, and heterogeneous reaction parameter $M_2$. The concentration distribution $f$ for the Schmidt number $Sc$, homogeneous reaction parameter $M_1$, and heterogeneous reaction parameter $M_2$ are shown in Figure 4a–c. Figure 4a reveals that concentration field $f$ is a decreasing function of Schmidt parameter $Sc$. Physically, when the Schmidt number $Sc$ rises, the density of the Eyring–Powell fluid particle diminishes. So, the less dense fluid particles are moving at a faster rate and distributing more molecules; consequently, the concentration $f$ drops. Figure 4b elucidates that the concentration $f$ diminished with the effects of the homogeneous reaction parameter $M_1$. Since the solute reaction rate is sped up by increasing the homogeneous reaction parameter $M_1$ as a result of a fall in fluid viscosity, which decreases the reactant concentration, the reduction in concentration $f$ is seen in Figure 4c for a greater heterogeneous reaction parameter $M_2$. Since the heterogeneous reaction parameter $M_2$ has a direct proportionality to the reaction rate, the concentration of species decreases as $M_2$ rises as a consequence of an increase in reaction rate and a reduction in diffusion rate.

Figure 5a–f is plotted to illustrate the impact of $Ha$, $B_1$, $B_2$ $\varrho$, $\gamma$, and $Br$ on total entropy generation $N_s$ because an irreversibility analysis is required to determine the system’s efficiency. These figures show that the entropy generation profiles $N_s$ are parabolic, with higher values at the ends of channel walls and lower values in the centre. Furthermore, the entropy generation number $N_s$ is lower in the lower portion of the channel than in the upper portion of the channel due to low zeta potential values at the lower wall and the nature of the asymmetric channel. The entropy generation number $N_s$ increases as the magnetic field strength increases, as seen in Figure 5a. The reason for this is that the thermal irreversibility $N_s$ and the Hartmann number $Ha$ are directly related. Figure 5b explored that the entropy generation number $N_s$ grows quickly close to the channel walls while remaining constant near the centre for larger values of material fluid parameter $B_1$ since fluid temperature $\theta$ rises with increase in $B_1$ and entropy is directly related to temperature also increases. With an increase in the material fluid parameter $B_2$, Figure 5c reveals that the total entropy generation $N_s$ decreases. This is related to the damping impact of $B_2$ on fluid flow, which reduces entropy in the microchannel. Figure 5d displays the change in thermal irreversibility for various values of temperature difference parameter $\varrho$. It has been shown that $N_s$ declines as $\varrho$ increases. According to Figure 5e,f, when the Ohmic dissipation parameter $\gamma$ and viscous dissipation parameter $Br$ increase, the total entropy generation $N_s$ rises along the height of the channel. Increases in the Joule heating parameter $\gamma$ and Brinkman number $Br$ imply an increase in heat generation through dissipation, which raises irreversibility within the channel. The fluctuation of the Bejan number caused by $Ha, B_1, B_2, \varrho , \gamma$, and $Br$ is examined in Figure 6a–f. These findings reveal that $Be=0,$ near the bottom wall, which denotes irreversibility caused by the combined effects of viscous dissipation, electric field, and magnetic field, is prevalent. In Figure 6a, it is obvious that as the Hartmann number $Ha$ rises, the Bejan number $Be$ diminishes in the centre and increases close to the channel’s boundaries because the irreversibility caused by the magnetic field in the channel’s core is more immense than the irreversibility caused by heat transfer. Figure 6b expresses that the magnitude of $Be$ profile increases near the lower and upper walls of the channel, while the opposite behavior is observed in the centre region. When $B_1$ increases, the fluid becomes thick and irreversibly due to viscosity of fluid near the walls is higher than the heat transfer irreversibility. From the Figure 6c, the magnitude of the $Be$ profile diminishes at the bottom and top borders of the channel, whereas the opposite behavior is seen in the centre. This is because irreversibility caused by fluid viscosity close to walls is greater than irreversibility caused by heat transfer. It is observed from Figure 6d that $Be$ decreases as the Temperature difference parameter $\varrho$ increases. The fluctuation in heat transfer irreversibility in relation to the total irreversibility caused by $\gamma$ and $Br$ is shown in Figure 6e–f. The rate of entropy formation is larger in the centre—due to Ohmic, magnetic, and viscous dissipation—than it is near to the walls, where heat transfer irreversibility predominates due to a significant temperature gradient.

In Table 1, the entropy generation numbers $N_s$ for Eyring–Powell fluid and viscous fluid are contrasted for three different values of $Ha, Br, \gamma$, and $\varrho$. Due to the decreased viscous forces, Eyring–Powell fluid (shear thinning fluid) has less entropy than viscous fluid.