Dynamics prediction using an artificial neural network for a weakly conductive ionized fluid streamed over a vibrating electromagnetic plate

Abstract

Recently, the study of weak conductor dynamics influenced by an electromagnetic (Riga) plate has garnered scholarly interest, taking into account various physical factors. Electromagnetic sensors find extensive use across engineering and industrial domains, spurring our exploration into the flow characteristics and heat–mass transfer mechanisms of a mildly conducting ionized fluid near an oscillating electromagnetic plate embedded within porous structures. The chosen flow scenario is meticulously modeled, encapsulating various physical dynamics such as radiative heat discharge, chemical interactions, Darcian porous drag effects, buoyancy forces, and velocity slippage conditions. Within this context, the Darcy model is employed to articulate drag influences in the porous domain. The resulting flow model is mathematically articulated as a set of time-dependent partial differential equations (PDEs). Leveraging the Laplace transform (LT) approach, we derive concise representations for the core variables in the model. In our study, dimensionless fluid velocity, temperature, and concentration gradients are extensively graphed, and the corresponding non-dimensional heat transfer, mass transfer, and friction rates are tabled. Key observations highlight an amplification in the velocity with an enhancement in the modified Hartmann number and diminishing with an enlargement in the Darcy number. Higher radiative heat intensities correspondingly dissipate more energy, cooling the medium further. Notably, the chemical reactions induce higher heat and mass transfer rates in the hybrid nanofluid. An artificial neural network (ANN) model is also developed based on reference datasets procured via the LT evaluation. This ANN framework exhibits commendable precision in predicting essential flow quantities, achieving an impressive 99.95% Such innovative insights from this research hold great promise for practical applications in steam generators, chemical reactors, electromagnetic sensors and gadgets, and material processing phase transitions.

Graphical Abstract

Appendix A

The following constant expressions are utilized in the results.

$$\begin{aligned} \lambda _1 &= {} \textrm{Qr},\; \lambda _2=\textrm{Sc},\\ a &= {} \frac{1}{\lambda _2-\textrm{Pr}},\; \lambda _3=a(\textrm{Ra}-\textrm{Kr} \lambda _2),\; \lambda _4=\frac{\textrm{Ra}}{\textrm{Pr}}, \;\lambda _{5}=\textrm{Pr},\\ a_1 &= {} \frac{1}{\lambda _2-1},\; \lambda _{6}=a_1\left( \frac{1}{\textrm{Da}}-\textrm{Kr} \lambda _2\right) ,\; a_2=\frac{1}{\lambda _{5}-1},\\ \lambda _{7} &= {} a_2\left( \frac{1}{\textrm{Da}}-\lambda _4 \lambda _{5}\right) ,\;A_1=a_1 \textrm{Gc}-A_2, \;A_2=\frac{a a_1 \lambda _1 \textrm{Gr}}{\lambda _{6}-\lambda _3},\\ B_1 &= {} a_2 \textrm{Gr}+B_2,\; B_2=\frac{a a_2 \lambda _1 \textrm{Gr}}{\lambda _{7}-\lambda _3},\\ \lambda _{8} &= {} S^2-\frac{1}{\textrm{Da}},\;r_0=\frac{1}{\beta },\; r=\frac{1}{r_1 \textrm{Da}},\;r_1=\frac{1}{\beta \sqrt{\lambda _2}},\;r_2=\frac{1}{\beta \sqrt{\lambda _{5}}}\\ F_1(\eta , a, \tau ) &= {} L^{-1}\left[ \frac{\textrm{e}^{-\sqrt{s+a}\,\eta }}{s}\right] \\ &= {} \frac{1}{2}\left[ \textrm{e}^{\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}+\sqrt{a\tau }\right) +\textrm{e}^{-\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}-\sqrt{a\tau }\right) \right] ,\\ F_2(\eta , a, b, \tau ) &= {} L^{-1}\Big [\frac{\textrm{e}^{-\sqrt{s+a}\,\eta }}{s(s-b)}\Big ]\\ &= {} \frac{1}{2b}\textrm{e}^{b\tau }\left[ \textrm{e}^{\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}+\sqrt{(a+b)\tau }\right\} +\textrm{e}^{-\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}-\sqrt{(a+b)\tau }\right\} \right] \\{} &\quad {} -\frac{1}{2b}\left[ \textrm{e}^{\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}+\sqrt{a\tau }\right) +\textrm{e}^{-\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}-\sqrt{a\tau }\right) \right] ,\\ F_3(\eta , \alpha _0, a, b, \tau ) &= {} L^{-1}\left[ \frac{\textrm{e}^{-\sqrt{s+a}\,\eta }}{(s-b) (\alpha _0+\sqrt{s+a})}\right] \\ &= {} -\frac{\alpha _0 \textrm{e}^{\eta \alpha _0+(\alpha _0^2-a)\tau }}{(\alpha _0^2-a-b)}\,\textrm{erfc}\left( \frac{\eta }{2\sqrt{\tau }}+\alpha _0 \sqrt{\tau }\right) \\{} &\qquad +\frac{\textrm{e}^{b \tau }}{2(\alpha _0^2-a-b)}\left[ (\alpha _0+\sqrt{a+b}) \textrm{e}^{\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}+\sqrt{(a+b)\tau }\right\} \right. \\{} &\qquad \left. +(\alpha _0-\sqrt{a+b}) \textrm{e}^{-\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}-\sqrt{(a+b)\tau }\right\} \right] ,\\ F_4(\eta , \alpha _0, a, b, \tau ) &= {} L^{-1}\Big [\frac{\textrm{e}^{-\sqrt{s+a}\,\eta }}{s(s-b) (\alpha _0+\sqrt{s+a})}\Big ]\\ &= {} -\frac{\alpha _0 \textrm{e}^{\eta \alpha _0+(\alpha _0^2-a)\tau }}{(\alpha _0^2-a-b) (\alpha _0^2-a)}\,\textrm{erfc}\left( \frac{\eta }{2\sqrt{\tau }}+\alpha _0 \sqrt{\tau }\right) \\ &\qquad +\frac{\textrm{e}^{b \tau }}{2b(\alpha _0^2-a-b)}\left[ (\alpha _0+\sqrt{a+b}) \textrm{e}^{\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}+\sqrt{(a+b)\tau }\right\} \right. \\ &\qquad {} \left. +(\alpha _0-\sqrt{a+b}) \textrm{e}^{-\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}-\sqrt{(a+b)\tau }\right\} \right] \\ &\qquad {} -\frac{1}{2b(\alpha _0^2-a)}\left[ (\alpha _0+\sqrt{a}) \textrm{e}^{\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}+\sqrt{a\tau }\right) \right. \\ &\qquad {} \left. +(\alpha _0-\sqrt{a})\textrm{e}^{-\eta \sqrt{a}}\,\textrm{erfc} \left( \frac{\eta }{2\sqrt{\tau }}-\sqrt{a\tau }\right) \right] , \end{aligned}$$

$$\begin{aligned} F_5(\eta , \alpha _0, a, b, c, \tau ) &= L^{-1}\Big [\frac{\textrm{e}^{-\sqrt{s+a}\,\eta }}{(s-b) \sqrt{s+c} (\alpha _0+\sqrt{s+a})}\Big ]\\ &= frac{\alpha ^2_0 \textrm{e}^{\eta \alpha _0+(\alpha _0^2-a)\tau }}{(\alpha _0^2-a+c) (\alpha _0^2-a-b)} \,\textrm{erfc}\left( \frac{\eta }{2\sqrt{\tau }}+\alpha _0 \sqrt{\tau }\right) \\{} &\quad {} +\frac{(a-c) \textrm{e}^{-c \tau }}{2 (b+c) (\alpha _0^2-a+c)}\left[ \left( 1+\frac{\alpha _0}{\sqrt{a-c}}\right) \textrm{e}^{\eta \sqrt{a-c}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}+\sqrt{(a-c)\tau }\right\} \right. \\{} &\quad {} \left. +\left( 1-\frac{\alpha _0}{\sqrt{a-c}}\right) \textrm{e}^{-\eta \sqrt{a-c}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}-\sqrt{(a-c)\tau }\right\} \right] \\{} &\quad {} -\frac{(a+b) \textrm{e}^{b \tau }}{2 (b+c) (\alpha _0^2-a-b)}\left[ \left( 1+\frac{\alpha _0}{\sqrt{a+b}}\right) \textrm{e}^{\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}+\sqrt{(a+b)\tau }\right\} \right. \\{} &\quad {} \left. +\left( 1-\frac{\alpha _0}{\sqrt{a+b}}\right) \textrm{e}^{-\eta \sqrt{a+b}}\,\textrm{erfc} \left\{ \frac{\eta }{2\sqrt{\tau }}-\sqrt{(a+b)\tau }\right\} \right] ,\\ F^\prime _1(0, a, \tau ) &= {} -\left[ \sqrt{a}\,\textrm{erf}(\sqrt{a \tau })+\frac{1}{\sqrt{\pi \tau }} \textrm{e}^{-a \tau }\right] ,\\ F^\prime _2(0, a, b, \tau ) &= {} \frac{1}{b}\left[ \textrm{e}^{-b \tau }\left\{ \sqrt{b+a}\,\textrm{erf} \,\sqrt{(b+a)\tau }+\frac{\textrm{e}^{-(b+a) \tau }}{\sqrt{\pi \tau }}\right\} +\sqrt{a}\,\textrm{erf}(\sqrt{a\tau })+\frac{\textrm{e}^{-a \tau }}{\sqrt{\pi \tau }}\right] ,\\ F^\prime _3(0, \alpha _0, a, b, \tau ) &= {} -\frac{\alpha ^2_0 \textrm{e}^{(\alpha _0^2-a)\tau }}{(\alpha _0^2-a-b)}\,\textrm{erfc}(\alpha _0 \sqrt{\tau })\\{} &\quad -\frac{\textrm{e}^{b \tau }}{(\alpha _0^2-a-b)}[(\alpha _0 \sqrt{a+b})\,\textrm{erf} \{\sqrt{(a+b)\tau }\}-(a+b)],\\ F^\prime _4(0, \alpha _0, a, b, \tau ){} & =-\frac{\alpha ^2_0 \textrm{e}^{(\alpha _0^2-a)\tau }}{(\alpha _0^2-a-b) (\alpha _0^2-a)}\,\textrm{erfc}(\alpha _0 \sqrt{\tau })\\{} &\qquad {} +\frac{\textrm{e}^{b \tau }}{b(\alpha _0^2-a-b)}[-\alpha _0 \sqrt{a+b}\,\textrm{erf} \{\sqrt{(a+b)\tau }\}+(a+b)]\\{} &\qquad {} -\frac{1}{b(\alpha _0^2-a)}[-\alpha _0 \sqrt{a}\,\textrm{erf} (\sqrt{a\tau })+a],\\ F^\prime _5(0, \alpha _0, a, b, c, \tau )& = \frac{\alpha ^3_0 \textrm{e}^{(\alpha _0^2-a)\tau }}{(\alpha _0^2-a+c) (\alpha _0^2-a-b)}\,\textrm{erfc}(\alpha _0 \sqrt{\tau })\\{} &\qquad -\frac{(a-c) \textrm{e}^{-c \tau }}{(b+c) (\alpha _0^2-a+c)}[{\sqrt{a-c}}\,\textrm{erf} \{\sqrt{(a-c)\tau }\}-\alpha _0]\\{} &\qquad +\frac{(a+b) \textrm{e}^{b \tau }}{(b+c) (\alpha _0^2-a-b)}[{\sqrt{a+b}}\,\textrm{erf} \{\sqrt{(a+b)\tau }\}-\alpha _0],\\ F^\prime _5(0, \alpha _0, a, 0, c, \tau ) &= \frac{\alpha ^3_0 \textrm{e}^{(\alpha _0^2-a)\tau }}{(\alpha _0^2-a+c) (\alpha _0^2-a)}\,\textrm{erfc}(\alpha _0 \sqrt{\tau })\\{} &\qquad -\frac{(a-c) \textrm{e}^{-c \tau }}{c (\alpha _0^2-a+c)}[{\sqrt{a-c}}\,\textrm{erf} \{\sqrt{(a-c)\tau }\}-\alpha _0]\\{} &\qquad +\frac{a}{c (\alpha _0^2-a)}[{\sqrt{a}}\,\textrm{erf} \{\sqrt{a\tau }\}-\alpha _0]. \end{aligned}$$