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Interactions due to Hall current and photothermal effect in a magneto-thermoelastic medium with diffusion and gravity

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Abstract

Aim of the present study is to investigate the effects of mass diffusion, gravity and magnetic field with Hall current on a photothermoelastic semiconducting medium in the context of Lord-Shulman (L-S) theory. The normal mode analysis method is used to obtain the expressions of physical quantities such as stress, temperature, displacement, concentration and carrier density. A moving mechanical load is applied at the outer free surface of the medium and a complete solution of the physical field distributions is obtained. The numerical results of physical quantities for a particular material are obtained using MATLAB software and illustrated graphically. The nature of variations of the physical quantities with spatial co-ordinate is analyzed. The theory and numerical computations are found to be in close agreement. Some special cases of interest have been inferred from the present study for validation.

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Acknowledgements

One of the authors, Parmender Sheokand is thankful to the University Grants Commission, New Delhi for the financial support Vide D.O. No. F.14-34/2011(CPP-II).

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  1. Department of Mathematics, Guru Jambheshwar University of Science & Technology, Hisar, Haryana, 125001, India

    Sunita Deswal, Parmender Sheokand & Baljit Singh Punia

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Appendices

Appendix A

$$\begin{aligned}{} & {} A^*=\frac{ A_{99}}{E_{10}},\;\;B^*=\frac{ B_{99}}{E_{10}},\;\; E^*=\frac{ E_{99}}{E_{10}},\\{} & {} F^*=\frac{ F_{99}}{E_{10}},\;\; L^*=\frac{ L_{99}}{E_{10}},\\{} & {} A_{99}=-E_8+(Z_1-R_3)a_4+A_{13}(Z_1-R_3),\\{} & {} B_{99}=E_5+a_4E_8-E_7+E_8 A_{13}-(Z_1-R_3)A_4 A_{13}+\beta ^2 A_{12}^2 R_3,\\{} & {} E_{99}=E_4-A_{13} E_5+E_7 a_4-E_9-a_4 E_8 A_{13}+A_{13} E_7-\beta ^2 E_1 A_{12},\\{} & {} F_{99}=E_6-A_{13} E_4+a_4 E_9-E_7 A_4 A_{13}+E_9 A_{13}-\beta ^2 A_{12} E_2,\\{} & {} L_{99}=-A_{13} E_6-A_{13} A_4 E_9-\beta ^2 A_{12} E_3,\\{} & {} E_1=A_{12}(a_5+a_3R_3)+A_{12}a_4 R_3,\\{} & {} E_2=\epsilon _{2}\epsilon _{3}R_3 A_{12}-A_{12}a_3R_3-a_4A_{12}(a_5+a_3 R_3),\\{} & {} E_3=-\epsilon _3 A_{12} \epsilon _2 a_5+a_3a_4R_3 A_{12},\\{} & {} E_4=\epsilon _3(\epsilon _2 a_5+\epsilon _2 R_3 A_{11}-a_2 a_5-a_2R_3a^2-2a^2a_1-2a^2\epsilon _2 Z_1),\\{} & {} E_5=\epsilon _3(-\epsilon _2 R_3+a_2R_3+a_1+Z_1 \epsilon _2),\\{} & {} E_6=\epsilon _3(-\epsilon _2 a_5 A_{11}+a_2 a_5 a^2+a^4 a_1+\epsilon _2 Z_1 a^4),\\{} & {} E_7=-a_3 a_5-a_1 R_1 a^2-A_{11} a_5-a_3 R_3 A_{11}-a_1R_1 A_{11}\\{} & {} \hspace{11mm}-a_2 a_5-a_2 a^2 R_3-2a^2 a_1-2a_2 R_1 a^2 Z_1+Z_1 a^4+2a^2a_3Z_1,\\{} & {} E_8=a_5+a_3 R_3+a_1 R_1+A_{11} R_3+a_2 R_3+a_1+a_2 R_1 Z_1-2 a^2 Z_1-Z_1 a_3,\\{} & {} E_9=a_3a_5 A_{11}+a_1 R_1 a^2 A_{11}+a_2 a^2 a_5+a_1 a^4+a_2 R_1 Z_1 a^4-a_3 a^4 Z_1,\\{} & {} E_{10}=Z_1-R_3. \end{aligned}$$

Appendix B

$$\begin{aligned}{} & {} H_{1n}=\frac{-\beta ^2 A_{12}}{k_n^2-A_{13}},\;\;H_{2n}=\frac{Q_{1n}+Q_{2n}+Q_{3n}H_{1n}}{Q_{4n}(k_n^2-a_4)Z_1-\epsilon _3(a_1+\epsilon _2 Z_1)},\\{} & {} H_{3n}=\frac{-\epsilon _3 H_{2n}}{D^2-a_4},\;\; H_{4n}=\frac{(k_n^4+a^4-2a^2k_n^2)+R_1(k_n^2-a^2)H_{2n}}{R_3 k_n^2-A_8},\\{} & {} H_{5n}=\iota a-k_nH_{1n},\;\; H_{6n}=-k_n-\iota aH_{1n}, \end{aligned}$$
$$\begin{aligned}{} & {} S_{1n}=\beta ^2(-a^2+k_n^2-H_{2n}-H_{3n})-Z_1 H_{4n}-2(k_n^2-\iota ak_n H_{1n}),\\{} & {} S_{2n}=\beta ^2(-a^2+k_n^2-H_{2n}-H_{3n})-Z_1 H_{4n}+2(a^2+\iota ak_n H_{1n}),\\{} & {} S_{3n}=-2\iota ak_n+(k_n^2-a^2)H_{1n},\;\;Q_{1n}=a_1(k_n^2-A_{11})(k_n^2-a_4), \\{} & {} Q_{2n}=a_2 Z_1 (k_n^2-a^2)(k_n^2-a_4),\;\;Q_{3n}=-a_1 A_{12}(k_n^2-a_4),\\{} & {} Q_{4n}=a_1+(k_n^2-a_3). \end{aligned}$$

Appendix C

$$\begin{aligned}{} & {} \Delta = -\left[ S_{21}\frac{\Delta _1}{Q_0v_0}+S_{22}\frac{\Delta _2}{Q_0v_0}+ S_{23}\frac{\Delta _3}{Q_0v_0}+S_{24}\frac{\Delta _4}{Q_0v_0}+S_{25}\frac{\Delta _5}{Q_0v_0}\right] ,\\{} & {} \Delta _1= -Q_0v_0[S_{32}(P_3P_9P_{15}-P_3P_{10}P_{14}-P_4P_8P_{15}+P_4P_{13}P_{10}+P_5P_8P_{14}-P_5P_{13}P_9)\\{} & {} \hspace{1.1cm} -S_ {33}(P_2P_9P_{15}-P_2P_{10}P_{14}-P_4P_7P_{15}+P_4P_{12}P_{10}+P_5P_7P_{14}-P_5P_{12}P_9)\\{} & {} \hspace{1.1cm} +S_{34}(P_2P_8P_{15}-P_2P_{10}P_{13}-P_3P_7P_{15}+P_3P_{12}P_{10}+P_5P_7P_{13}-P_5P_{12}P_8)\\{} & {} \hspace{1.1cm} -S_{35}(P_2P_8P_{14}-P_2P_{13}P_9-P_3P_7P_{14}+P_3P_{12}P_9+P_4P_7P_{13}-P_4P_{12}P_8)],\\{} & {} \Delta _2=Q_0v_0[S_{31}(P_3P_9P_{15}-P_3P_{10}P_{14}-P_4P_8P_{15}+P_4P_{13}P_{10}+P_5P_8P_{14}-P_5P_{13}P_9)\\{} & {} \hspace{1.1cm} -S_{33}(P_1P_9P_{15}-P_1P_{10}P_{14}-P_4P_6P_{15}+P_4P_{11}P_{10}+P_5P_6P_{14}-P_5P_{11}P_9)\\{} & {} \hspace{1.1cm} +S_{34}(P_1P_8P_{15}-P_1P_{10}P_{13}-P_3P_6P_{15}+P_3P_{11}P_{10}+P_5P_6P_{13}-P_5P_{11}P_8)\\{} & {} \hspace{1.1cm}-S_{35}(P_1P_8P_{14}-P_1P_{13}P_9-P_3P_6P_{14}+P_3P_{11}P_9+P_4P_6P_{13}-P_4P_{11}P_8)],\\{} & {} \Delta _3=-Q_0v_0[S_{31}(P_2P_9P_{15}-P_2P_{10}P_{14}-P_4P_7P_{15}+P_4P_{12}P_{10}+P_5P_7P_{14}-P_5P_{12}P_9)\\{} & {} \hspace{1.1cm} -S_{32}(P_1P_9P_{15}-P_1P_{10}P_{14}-P_4P_6P_{15}+P_4P_{11}P_{10}+P_5P_6P_{14}-P_5P_{11}P_9)\\{} & {} \hspace{1.1cm} +S_{34}(P_1P_7P_{15}-P_1P_{10}P_{12}-P_2P_6P_{15}+P_2P_{11}P_{10}+P_5P_6P_{12}-P_5P_{11}P_7)\\{} & {} \hspace{1.1cm} -S_{35}(P_1P_7P_{14}-P_1P_{12}P_9-P_2P_6P_{14}+P_2P_{11}P_9+P_4P_6P_{12}-P_4P_{11}P_7)],\\ \Delta _4= & {} Q_0v_0[S_{31}(P_2P_8P_{15}-P_2P_{10}P_{13}-P_3P_7P_{15}+P_3P_{12}P_{10}+P_5P_7P_{13}-P_5P_{12}P_8)\\{} & {} \hspace{1.1cm} -S_{32}(P_1P_8P_{15}-P_1P_{10}P_{13}-P_3P_6P_{15}+P_3P_{11}P_{10}+P_5P_6P_{13}-P_5P_{11}P_8)\\{} & {} \hspace{1.1cm} +S_{33}(P_1P_7P_{15}-P_1P_{10}P_{12}-P_2P_6P_{15}+P_2P_{11}P_{10}+P_5P_6P_{12}-P_5P_{11}P_7)\\{} & {} \hspace{1.1cm}-S_{35}(P_1P_7P_{13}-P_1P_{12}P_8-P_2P_6P_{13}+P_2P_{11}P_8+P_3P_6P_{12}-P_3P_{11}P_7)], \\ \Delta _5= & {} -Q_0v_0[S_{31}(P_2P_8P_{14}-P_2P_9P_{13}-P_3P_7P_{14}+P_3P_{12}P_9+P_4P_7P_{13}-P_4P_{12}P_8)\\{} & {} \hspace{1.1cm} -S_{32}(P_1P_8P_{14}-P_1P_9P_{13}-P_3P_6P_{14}+P_3P_{11}P_9+P_4P_6P_{13}-P_4P_{11}P_8)\\{} & {} \hspace{1.1cm} +S_{33}(P_1P_7P_{14}-P_1P_9P_{12}-P_2P_6P_{14}+P_2P_{11}P_9+P_4P_6P_{12}-P_4P_{11}P_7)\\{} & {} \hspace{1.1cm} -S_{34}(P_1P_7P_{13}-P_1P_{12}P_8-P_2P_6P_{13}+P_2P_{11}P_8+P_3P_6P_{12}-P_3P_{11}P_7)],\\ P_i= & {} -k_iH_{2i} (i=1,2,3,4,5),\;\;P_6=(k_1+v)H_{31},\;\;P_7=(k_2+v)H_{32},\\ P_8= & {} (k_3+v)H_{33},\;\;P_9=(k_4+v)H_{34},\;\;P_{10}=(k_5+v)H_{35},\\ P_{1i}= & {} -l_iH_{4i}(i=1,2,3,4,5),\;\;m=\frac{s C_T t^*}{ D_E}. \end{aligned}$$

Appendix D

$$\begin{aligned}{} & {} P_1^{*}=\frac{A_{00}}{E_{77}},\;\; P_2^{*}=\frac{B_{00}}{E_{77}},\;\; P_3^{*}=\frac{E_{00}}{E_{77}},\;\; P_4^{*}=\frac{F_{00}}{E_{77}},\\{} & {} A_{00}=E_{44}+A_{11} R_3+E_{22}+A_{13} R_3,\\{} & {} B_{00}=E_{55}-E_{44} A_{11}+E_{11}-A_{13} E_{44}-A_{13} E_{22}-R_3 E_6,\\{} & {} E_{00}=-A_{11} E_{55}+E_{33}-A_{13} E_{55}-E_{44} E_{66}-A_{13} E_{11},\\{} & {} F_{00}=E_{44} E_{55}-A_{13} E_{33},\\{} & {} E_{11}=(-a_2 a_5-2a_2 R_1 a^2 Z_1-a_2 a^2 R_3-2A^2 a_1),\\{} & {} E_{22}=a_2 R_3+a_2 R_1 Z_1+a_1,\\{} & {} E_{33}=a_2 a^2 a_5+a_1 a^4+a_2 R_1 Z_1 a^4,\\{} & {} E_{44}=a_5+a_3 R_3+a_1 R_1,\\{} & {} E_{55}=-a_3 a_5-a_1 R_1 a^2,\\{} & {} E_{66}=A_{11} A_{13}+\beta ^2 A_{12}^2,\\{} & {} E_{77}=-R_3, \\{} & {} B_{1n}=\frac{-\beta ^2 A_{12}}{l_n^2-A_{13}},\;\;B_{2n}=\frac{U_{1n}+U_{2n}+U_{3n}B_{1n}}{U_{4n}},\\{} & {} B_{3n}=\frac{(l_n^4+a^4-2a^2l_n^2)+R_1(l_n^2-a^2)B_{2n}}{R_3 l_n^2-A_8},\\{} & {} B_{4n}=\iota a-l_nB_{1n},\;\; B_{5n}=-l_n-\iota aB_{1n},\\{} & {} B_{6n}=\beta ^2(-a^2+l_n^2-B_{2n})-Z_1B_{3n}-2(l_n^2-\iota al_n B_{1n}),\\{} & {} B_{7n}=\beta ^2(-a^2+l_n^2-B_{2n})-Z_1B_{3n}+2(a^2+\iota al_n B_{1n}),\\{} & {} B_{8n}=-2\iota al_n+(l_n^2-a^2)B_{1n},\;\; U_{1n}=a_1(l_n^2-A_{11}),\\{} & {} U_{2n}=a_2 Z_1 (l_n^2-a^2),\;\;U_{3n}=-a_1 A_{12},\;\; U_{4n}=a_1+(l_n^2-a_3)Z_1,\\{} & {} \Delta _{0}=-B_{71}\frac{\Delta _{11}}{Q_0V_0}+B_{72}\frac{\Delta _{12}}{Q_0V_0}-B_{73}\frac{\Delta _{13}}{Q_0V_0}+B_{74}\frac{\Delta _{14}}{Q_0V_0},\\{} & {} \Delta _{11}=-Q_0 v_0[B_{82}(T_3 T_8-T_7 T_4)-B_{83}(T_2 T_8-T_6 T_4)+B_{84}(T_2 T_7-T_6 T_3)],\\{} & {} \Delta _{12}=-Q_0 v_0[B_{81}(T_3 T_8-T_7 T_4)-B_{83}(T_1 T_8-T_5 T_4)+B_{84}(T_1 T_7-T_5 T_3)],\\{} & {} \Delta _{13}=-Q_0 v_0[B_{81}(T_2 T_8-T_6T_4)-B_{82}(T_1 T_8-T_5T_4)+B_{84}(T_1 T_6-T_5T_2)],\\{} & {} \Delta _{14}=-Q_0 v_0[B_{81}(T_2 T_7-T_6 T_3)-B_{82}(T_1 T_7-T_5 T_3)+B_{83}(T_1 T_6-T_5 T_2)],\\{} & {} T_1=-l_1 B_{21},\;\;T_2=-l_2 B_{22},\;\;T_3=-l_3B_{23},\;\;T_4=-l_4B_{24},\\{} & {} T_5=-l_1B_{31},\;\;T_6=-l_2B_{32},\;\;T_7=-l_3B_{33}, \;\;T_8=-l_4B_{34}. \end{aligned}$$

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Deswal, S., Sheokand, P. & Punia, B.S. Interactions due to Hall current and photothermal effect in a magneto-thermoelastic medium with diffusion and gravity. Acta Mech 235, 235–254 (2024). https://doi.org/10.1007/s00707-023-03748-3

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  • DOI: https://doi.org/10.1007/s00707-023-03748-3

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