Magnetic-field- and thermal-radiation-induced entropy generation in a multiphase nonisothermal plane Poiseuille flow

The effect of radiative heat transfer on the entropy generation in a two-phase nonisothermal fluid flow between two infinite horizontal parallel plates under the influence of a constant pressure gradient and transverse noninvasive magnetic field have been explored. Both fluids are considered to be viscous, incompressible, immiscible, Newtonian, and electrically conducting. The governing equations in Cartesian coordinates are solved analytically with appropriate boundary conditions to obtain the velocity and temperature profile inside the channel. Application of a transverse magnetic field is found to reduce the throughput and the temperature distribution of the fluids in a pressure-driven flow. The temperature and fluid flow inside the channel can also be noninvasively altered by tuning the magnetic field intensity, temperature difference between the channel walls and the fluids, and several intrinsic fluid properties. The entropy generation due to the heat transfer, magnetic field, and fluid flow irreversibilities can be controlled by altering the Hartmann number, radiation parameter, Brinkmann number, filling ratio, and ratios of fluid viscosities and thermal and electrical conductivities. The surfaces of the channel wall are found to act as a strong source of entropy generation and heat transfer irreversibility. The rate of heat transfer at the channel walls can also be tweaked by the magnetic field intensity, temperature differences, and fluid properties. The proposed strategies in the present study can be of significance in the design and development of next-generation microscale reactors, micro-heat exchangers, and energy-harvesting devices.

Figure 1

Schematic diagram of a bilayer $x$-directional fluid flow under the combined influence of a transverse uniform magnetic field and radiative heat transfer from the parallel plates (not to scale). The distance between the parallel plates is $d$, and the thickness of the lower layer of fluid ($i$ = 2, cyan region) is $h$. The temperatures of both parallel plates are constant and denoted by $T_w$. The strength of the uniform magnetic field is $B_0$.

Figure 2

Plots (a)–(d) and (e)–(h) show the velocity $u$ and temperature $\theta$ distribution across the width $y$ of the channel for different filling ratios $a$, respectively. The other parameters are $\eta =2.0, {\mathrm{Ha}}_1=3.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 3

Plots (a)–(d), (e)–(h), and (i)–(l) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different filling ratios $a$, respectively. The other parameters are $\eta =2.0, {\mathrm{Ha}}_1=3.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 4

Plots (a) and (b) show the variation of velocity $u$ and temperature $\theta$ across the width $y$ of the channel for different viscosity ratios $\eta$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=5.0$, and $F_1=2.0$.

Figure 5

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different viscosity ratios $\eta$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=5.0$, and $F_1=2.0$.

Figure 6

The temperature $\theta$ distribution across the width $y$ of the channel for different thermal conductivity ratios $k$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, \eta =2.0, \sigma =1.0, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 7

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different thermal conductivity ratios $k$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, \eta =2.0, \sigma =1.0, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 8

The temperature $\theta$ distribution across the width $y$ of the channel for different electrical conductivity ratios $\sigma$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, \eta =2.0, k=1.5, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 9

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different electrical conductivity ratios $\sigma$, respectively. The other parameters are $a=0.4, {\mathrm{Ha}}_1=3.0, \eta =2.0, k=1.5, {\mathrm{Br}}_1=2.0$, and $F_1=2.0$.

Figure 10

Plots (a) and (b) show the variation of velocity $u$ and temperature $\theta$ distribution across the width $y$ of the channel for different Hartmann numbers ${\mathrm{Ha}}_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=5.0$, and $F_1=2.0$.

Figure 11

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different Hartmann numbers ${\mathrm{Ha}}_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Br}}_1=5.0$, and $F_1=2.0$.

Figure 12

The temperature $\theta$ distribution across the width $y$ of the channel for different Brinkman numbers ${\mathrm{Br}}_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Ha}}_1=5.0$, and $F_1=2.0$.

Figure 13

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different Brinkman numbers ${\mathrm{Br}}_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Ha}}_1=5.0$, and $F_1=2.0$.

Figure 14

The temperature $\theta$ distribution across the width $y$ of the channel for different radiation parameters $F_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Ha}}_1=3.0$, and ${\mathrm{Br}}_1=2.0$.

Figure 15

Plots (a), (b), and (c) show the variation of Bejan number Be, magnetic field irreversibility parameter I, and fluid flow irreversibility parameter J across the width $y$ of the channel for different radiation parameters $F_1$, respectively. The other parameters are $a=0.4, \eta =2.0, k=1.5, \sigma =1.0, {\mathrm{Ha}}_1=3.0$, and ${\mathrm{Br}}_1=2.0$.

Figure 16

Plots (a)–(f) show the variation of heat transfers, $h_1$ (red symbols, left side of $y$-axis) and $h_2$ (blue symbols, right side of $y$-axis) at the wall plates $y=1$ and $y=0$ for different viscosity ratios $\eta$, thermal conductivity ratios $k$, and electrical conductivity ratios $\sigma$, Hartmann numbers ${\mathrm{Ha}}_1$, Brinkman numbers ${\mathrm{Br}}_1$, and radiation parameters $F_1$, respectively. The other parameters are $a=0.4$ for (a)–(f), $\eta =2.0$ for (b)–(f), $k=1.5$ for (a) and (c)–(f), $\sigma =10.0$ for (a) and (f), $\sigma =1.0$ for (b), (d), and (e), ${\mathrm{Ha}}_1=3.0$ for (a)–(c) and (e)–(f), ${\mathrm{Br}}_1=2.0$ for (a)–(d) and (f), and $F_1=2.0$ for (a)–(e).

Figure 17

Comparison of present study (circular symbols: red, black, and blue) and the analytical results (continuous lines: red, black, and blue) of Duwairi et al. [58]. Plots show the spatial variation of (a) velocity $u$ and (b) temperature $\theta$ profile for increasing Hartmann numbers Ha.

Figure 18

Comparison of present study (discontinuous symbols: red, blue, and black) and the analytical (continuous lines: red, blue, and black) results of Haddad et al. [55]. Plots show the spatial variation of (a) velocity ($u$, blue circular symbols and continuous line, left $y$-axis) and temperature ($\theta$, red square symbols and continuous line, right $y$-axis) and (b) distribution of the local Bejan number (Be: black circular symbol and continuous line) profile for Kn = 0.001.