Electrothermal analysis in two-layered couple stress fluid flow in an asymmetric microchannel via peristaltic pumping

Abstract

Being motivated from the recent developments in biomicrofluidics, a mathematical model is presented to analyze the two-layered electrothermal flow via peristaltic propulsion of couple stress fluids caused by velocity and thermal slip conditions. An asymmetric microchannel is considered for the flow regime with different zeta potential moving with wave velocity. Couple stress fluid has been taken for aqueous solution to represent the non-Newtonian characteristics of physiological fluids. A lubrication approach with Debye H$\ddot{u}$ckel linearization is taken to obtain the analytical solution. Furthermore, heat transfer analysis is performed to analyze the thermal characteristics and variations in Nusselt number in the presence of thermal radiation. The study shows that the temperature reduces with increasing the electrical double layer thickness, thermal radiation, couple stress parameter, and magnitude of Brinkman number, however, it enhances with increasing the Joule heating and thermal slip effects. The rate of heat transfer enhances with Joule heating effects, while the trend is reversed due to the presence of two-layered electroosmotic flow and couple stress effect.

Appendix

The expressions that appear in Sect. 2 are listed as follows:

$$\begin{aligned} j_{1}= &\, {} \left( \frac{m_1^2 \gamma _1^2}{m_1^4-\gamma _1^2 m_1^2}\right) ,\\ j_{2}= &\, {} \left( \frac{m_2^2\gamma _2^2 \epsilon }{\mu \left( m_2^4-\gamma _2^2 m_2^2\right) }\right) ,\\ j_{3}= &\, {} \left( h_1 + \beta _1\right) ,\\ j_{4}= &\, {} \left[ \sinh \left( \gamma _1 h_1\right) +\left( \beta _1 \gamma _1\right) \cosh \left( \gamma _1 h_1\right) \right] ,\\ j_{5}= &\, {} \left[ \cosh \left( \gamma _1 h_1\right) +\left( \beta _1 \gamma _1\right) \sinh \left( \gamma _1 h_1\right) \right] ,\\ j_{6}= &\, {} -\left( \frac{h_1^2}{2}+\beta _1 h_1\right) ,\\ j_{7}= &\, {} \left[ \left( j_1 A_1\right) \left( \cosh \left( m_1 h_1\right) \right. \right. \\&\left. \left. +\left( \beta _1 m_1\right) \sinh \left( m_1 h_1\right) \right) \right. \\&\left. +\left( j_1 B_1\right) \left[ \sinh \left( m_1 h_1\right) +\left( \beta _1 m_1\right) \cosh \left( m_1 h_1\right) \right] \right] ,\\ j_{8}= &\, {} \left( h_2-\beta _2\right) ,\\ j_{9}= &\, {} \left[ \sinh \left( \gamma _2 h_2\right) -\left( \beta _2 \gamma _2\right) \cosh \left( \gamma _2 h_2\right) \right] ,\\ j_{10}= &\, {} \left[ \cosh \left( \gamma _2 h_2\right) \right. \\&\left. -\,\left( \beta _2 \gamma _2\right) \sinh \left( \gamma _2 h_2\right) \right], \end{aligned}$$

$$\begin{aligned}j_{11}= &\, {} -\left( \frac{h_2^2}{2\mu }-\frac{h_2\beta _2}{\mu }\right) \\ j_{12}= &\, {} \left[ \left( j_2 A_2\right) \left( \cosh \left( m_2 h_2\right) -\left( \beta _2 m_2\right) \sinh \left( m_2 h_2\right) \right) \right. \\&\left. +\,\left( j_2 B_2\right) \left[ \sinh \left( m_2 h_2\right) -\left( \beta _2 m_2\right) \cosh \left( m_2 h_2\right) \right] \right] ,\\ j_{13}= &\, {} \left( \gamma _1^2 \sinh \left( \gamma _1 h_1\right) \right) ,\\ j_{14}= &\, {} \left( \gamma _1^2 \cosh \left( \gamma _1 h_1\right) \right) ,\\ j_{15}= &\, {} j_1\left[ \left( A_1 m_1^2\right) \left( \cosh \left( m_1 h_1\right) +\left( B_1 m_1^2\right) \sinh \left( m_1 h_1\right) \right) \right] ,\\ j_{16}= &\, {} \left( \gamma _2^2 \sinh \left( \gamma _2 h_2\right) \right) ,~\\ j_{17}= &\, {} \left( \gamma _2^2 \cosh \left( \gamma _2 h_2\right) \right) ,\\ j_{18}= &\, {} j_2\left[ \left( A_2 m_2^2\right) \left( \cosh \left( m_2 h_2\right) +\left( B_2 m_2^2\right) \sinh \left( m_2 h_2\right) \right) \right] ,\\ j_{19}= &\, {} \left( j_1 A_1 - j_2 A_2\right) ,\\ j_{20}= &\, {} \left[ \left( j_1 B_1\right) \left( m_1-\frac{m_1^3}{\gamma _1^2}\right) \right. \\&\left. - \left( j_2 \mu B_2 \right) \left( m_2-\frac{m_2^3}{\gamma _2^2}\right) + \left( m_1^2 A_1 - \epsilon m_2^2 A_2 \right) \right] ,\\ j_{21}= &\, {} \left( j_1 B_1 m_1 - j_2 B_2 m_2\right) ,\end{aligned}$$

$$\begin{aligned} j_{22}= &\, {} \left( \frac{j_1 A_1 m_1^2}{\gamma _1^2}-\frac{j_2 A_2 m_2^2}{\gamma _2^2}\right) ,\\ j_{23}= &\, {} \left( j_5-1\right) ,\\ j_{24}= &\, {} \left( j_7-j_{19}\right) ,\\ j_{25}= &\, {} \left( j_{21}-j_{20}\right) ,~\\ j_{26}= &\, {} \left( j_{24}-j_3 j_{20}\right) ,\\ j_{27}= &\, {} \left( \mu j_3 - j_8\right) ,\\ j_{28}= &\, {} \left( 1-j_{10}\right) ,~\\ j_{29}= &\, {} \left( j_{26}-j_{12}\right) ,~\\ j_{30}= &\, {} \left( j_4 j_{14} - j_{23} j_{13}\right) ,\\ j_{31}= &\, {} \left( j_{14} j_{27}\right) ,~\\ j_{32}= &\, {} -\left( j_9 j_{14}\right) ,~\\ j_{33}= &\, {} \left( j_{14} j_{28}\right) ,\\ j_{34}= &\, {} \left( j_6 j_{14} + j_{23}\right) ,~\\ j_{35}= &\, {} -\left( j_{11} j_{14}\right) ,~\\ j_{36}= &\, {} \left( j_{14} j_{29} - j_{23} j_{15}\right) ,\\ j_{37}= &\, {} \left( \frac{j_{14}}{\gamma _1^2}-1\right) ,~\\ j_{38}= &\, {} -\left( \frac{j_{14}}{\mu \gamma _2^2}\right) ,~\\ j_{39}= &\, {} \left( j_{15}-j_{14} j_{22}\right) ,\\ j_{40}= &\, {} \left( j_{16} j_{30}\right) ,~\\ j_{41}= &\, {} \left( j_{16} j_{31}\right) ,\\ j_{42}= &\, {} \left( j_{16} j_{33} -j_{32} j_{17}\right) ,\\ j_{43}= &\, {} \left( j_{16} j_{34}\right) ,\\ j_{44}= &\, {} \left( j_{16} j_{35}+ \frac{j_{32}}{\mu }\right) ,\\ j_{45}= &\, {} \left( j_{16} j_{36}- j_{18} j_{32} \right) ,\\ j_{46}= &\, {} \left( \gamma _2 j_{30}+\gamma _1 j_{32}\right) ,\\ j_{47}= &\, {} \left( \gamma _2 j_{31}+j_{32}(\mu - 1)\right), \end{aligned}$$

$$\begin{aligned} j_{48}= &\, {} \left( \gamma _2 j_{33}\right) ,\\ j_{49}= &\, {} \left( \gamma _2 j_{34}\right) ,\\ j_{50}= &\, {} \left( \gamma _2 j_{35}\right) ,\\ j_{51}= &\, {} \left( \gamma _2 j_{36}+j_{25} j_{32}\right) ,\\ j_{52}= &\, {} \left( j_{40} j_{47}-j_{41} j_{46}\right) ,\\ j_{53}= &\, {} \left( j_{42} j_{47}-j_{48} j_{41}\right) \\ j_{54}= &\, {} \left( j_{43} j_{47}-j_{49} j_{41}\right) ,\\ j_{55}= &\, {} \left( j_{44} j_{47}-j_{50} j_{41}\right) ,\\ j_{56}= &\, {} \left( j_{45} j_{47}-j_{51} j_{41}\right) ,\\ j_{57}= &\, {} \left( j_{52} j_{14}-j_{53} j_{13}\right) ,\\ j_{58}= &\, {} \left( j_{52} j_{37}-j_{54} j_{13}\right) ,\\ j_{59}= &\, {} \left( j_{52} j_{38}-j_{55} j_{13}\right) ,\\ j_{60}= &\, {} \left( j_{52} j_{39}-j_{56} j_{13}\right) ,\\ j_{61}= &\, {} -\left( \frac{j_{60}}{j_{57}}\right) ,\\ j_{62}= &\, {} -\left( \frac{j_{58}}{j_{57}}\right) ,\\ j_{63}= &\, {} -\left( \frac{j_{59}}{j_{57}}\right) ,\\ j_{64}= &\, {} -\left( \frac{\left( j_{14}j_{61}+j_{39}\right) }{j_{13}}\right) ,\\ j_{65}= &\, {} -\left( \frac{\left( j_{14}j_{62}+j_{37}\right) }{j_{13}}\right) ,\\ j_{66}= &\, {} -\left( \frac{\left( j_{14}j_{63}+j_{38}\right) }{j_{13}}\right) ,\\ j_{67}= &\, {} -\left( \frac{\left( j_{40}j_{64}+j_{42}j_{61}+j_{45}\right) }{j_{41}}\right) ,\\ j_{68}= &\, {} -\left( \frac{\left( j_{40}j_{65}+j_{42}j_{62}+j_{43}\right) }{j_{41}}\right) ,\\ j_{69}= &\, {} -\left( \frac{\left( j_{40}j_{66}+j_{42}j_{63}+j_{44}\right) }{j_{41}}\right), \end{aligned}$$

$$\begin{aligned} j_{70}= &\, {} \left( \mu j_{67} - j_{20}\right) ,\\ j_{71}= &\, {} \left( \mu j_{68} \right) ,\\ j_{72}= &\, {} \left( \mu j_{69} \right) ,\\ j_{73}= &\, {} \left( j_{61} - j_{22}\right) ,\\ j_{74}= &\, {} \left( j_{62} + \frac{1}{\gamma _1^2}\right) ,\\ j_{75}= &\, {} \left( j_{63} - \frac{1}{\mu \gamma _2^2}\right) ,\\ j_{76}= &\, {} -\left( \frac{j_{17}j_{61}+j_{18}}{j_{16}}\right) ,\\ j_{77}= &\, {} -\left( \frac{j_{17}j_{62}}{j_{16}}\right) ,\\ j_{78}= &\, {} -\left( \frac{j_{17}j_{63}-\frac{1}{\mu }}{j_{16}}\right) ,\\ j_{79}= &\, {} -\left( j_8 j_{67}+ j_9 j_{76} +j_{10} j_{61} +j_{12}\right) ,\\ j_{80}= &\, {} -\left( j_8 j_{68}+ j_9 j_{77} +j_{10} j_{62} \right) ,\\ j_{81}= &\, {} -\left( j_8 j_{69}+ j_9 j_{78} +j_{10} j_{63} +j_{11}\right) ,\\ j_{82}= &\, {} \left( -j_{73}+j_{79}+j_{61}-j_{19}\right) ,\\ j_{83}= &\, {} \left( -j_{74}+j_{80}+ j_{62}\right) ,\\ j_{84}= &\, {} \left( -j_{75}+j_{81}+j_{63}\right) ,\\ j_{85}= &\, {} j_{82}h_1+ j_{70} \frac{h_1^2}{2}+\frac{j_{64}}{\gamma _1}\left[ \cosh (\gamma _1 h_1)-1\right] \\&+\,\frac{j_{73}}{\gamma _1}\sinh (\gamma _1 h_1)+\frac{j_1 A_1}{m_1}\sinh (m_1 h_1)\\&+\,\frac{j_1B_1}{m_1}\left[ \cosh (m_1 h_1)-1\right] ,\end{aligned}$$

$$\begin{aligned} j_{86}= &\, {} \left( j_{83}h_1+ j_{71} \frac{h_1^2}{2}+\frac{j_{65}}{\gamma _1}\left[ \cosh (\gamma _1 h_1)-1\right] \right. \\&\left. +\,\frac{j_{74}}{\gamma _1}\sinh (\gamma _1 h_1)-\frac{h_1^3}{6}\right) ,\\ j_{87}= &\, {} \left( j_{84}h_1+ j_{72} \frac{h_1^2}{2}+\frac{j_{66}}{\gamma _1}\left[ \cosh (\gamma _1 h_1)-1\right] \right. \\&\left. +\,\frac{j_{75}}{\gamma _1}\sinh (\gamma _1 h_1)\right) ,\\ j_{88}= &\, {} \left( -j_{79}h_2 - j_{67} \frac{h_2^2}{2}+\frac{j_{76}}{\gamma _2}\left[ 1-\cosh (\gamma _2 h_2)\right] \right. \\&\left. -\frac{j_{61}}{\gamma _2}\sinh (\gamma _2 h_2)-\frac{j_2 A_2}{m_2}\sinh (m_2 h_2) \right. \\&\left. +\,\frac{j_2 B_2}{m_2}\left[ 1-\cosh (m_2 h_2)\right] \right), \end{aligned}$$

$$\begin{aligned} j_{89}= &\, {} \left( -j_{80}h_2 - j_{68} \frac{h_2^2}{2}+\frac{j_{77}}{\gamma _2}\left[ 1-\cosh (\gamma _2 h_2)\right] \right. \\&\left. -\,\frac{j_{62}}{\gamma _2}\sinh (\gamma _2 h_2)\right) ,\\ j_{90}= &\, {} \left( -j_{81}h_2 - j_{69} \frac{h_2^2}{2}+\frac{j_{78}}{\gamma _2}\left[ 1-\cosh (\gamma _2 h_2)\right] \right. \\&\left. -\,\frac{j_{63}}{\gamma _2}\sinh (\gamma _2 h_2)+\frac{h_2^3}{6 \mu }\right) ,\\ G_1= &\, {} \left( \frac{j_{90}\left( q_1 - j_{85}\right) -\left( q_2 - j_{88}\right) j_{87}}{j_{90}j_{86}-j_{87}j_{89}}\right) ,\\ G_2= &\, {} \frac{\left( q_2-j_{88}-j_{89}G_1\right) }{j_{90}},\\ j_{91}= &\, {} \left( -{{Br}}_1\left[ \frac{c_2^2}{2}-\frac{j_1^2 m_1^2}{4}\left( A_1^2-B_1^2\right) +\left( \frac{j_1^2 A_1^2 m_1^4}{4 \gamma _1^2}\right) \right. \right. \\&\left. \left. -\left( \frac{j_1^2 B_1^2 m_1^4}{4 \gamma _1^2}+\frac{G_1^2}{2\gamma _1^2}\right) \right] -\frac{S_1}{2}\right) ,\\ j_{92}= &\, {} \left( \frac{c_2G_1{{Br}}_1}{3}\right) ,\\ j_{93}= &\, {} -\left( \frac{G_1^2 {{Br}}_1}{12}\right) ,\\ j_{94}= &\, {} -{{Br}}_1\left( \frac{c_3^2}{4}+\frac{c_4^2}{4}\right) ,\\ j_{95}= &\, {} -{{Br}}_1\left( \frac{c_3 c_4}{2}\right) ,\\ j_{96}= &\, {} -{{Br}}_1\left( \frac{j_1^2 A_1^2}{8}+\frac{j_1^2 B_1^2}{8}+\frac{j_1^2 A_1^2 m_1^2}{8\gamma _1^2}+\frac{j_1^2 B_1^2 m_1^2}{8 \gamma _1^2}\right) ,\\ j_{97}= &\, {} -{{Br}}_1\left( \frac{j_1^2 A_1 B_1}{4}+\frac{j_1^2 A_1 B_1 m_1^2}{4 \gamma _1^2}\right) ,\\ j_{98}= &\, {} -{{Br}}_1\left( 2\left( \frac{c_3c_4}{\gamma _1}-\frac{c_4 G_1}{\gamma _1^2}\right) + \frac{4 c_4 G_1}{\gamma _1^2}\right), \end{aligned}$$

$$\begin{aligned}j_{99}= &\, {} -{{Br}}_1\left( 2\left( \frac{c_2c_4}{\gamma _1}-\frac{c_3 G_1}{\gamma _1^2}\right) + \frac{4 c_3 G_1}{\gamma _1^2}\right) ,\\ j_{100}= &\, {} -{{Br}}_1\left( \frac{2}{m_1}\left( c_2 j_1 B_1-\frac{j_1A_1m_1G_1}{\gamma _1^2}\right) +\frac{4j_1A_1G_1}{m_1^2}\right) ,\\ j_{101}= &\, {} -{{Br}}_1\left( \frac{2}{m_1}\left( c_2 j_1 A_1-\frac{j_1B_1m_1G_1}{\gamma _1^2}\right) +\frac{4j_1B_1G_1}{m_1^2}\right) ,\\ j_{102}= &\, {} -{{Br}}_1\left( (c_3\gamma _1j_1A_1m_1+c_4j_1B_1m_1^2)\right. \\&\left. +\,\frac{1}{(m_1+\gamma _1)^2}(c_4\gamma _1j_1B_1m_1+c_3j_1A_1m_1^2)\right) ,\\ j_{103}= &\, {} -{{Br}}_1\left( (c_3\gamma _1j_1A_1m_1+c_4j_1B_1m_1^2)\right. \\&\left. -\,\frac{1}{(m_1-\gamma _1)^2}(c_4\gamma _1j_1B_1m_1+c_3j_1A_1m_1^2)\right), \end{aligned}$$

$$\begin{aligned} j_{104}= &\, {} -{{Br}}_1\left( (c_3\gamma _1j_1B_1m_1+c_4j_1A_1m_1^2)\right. \\&\left. +\,\frac{1}{(m_1+\gamma _1)^2}(c_4\gamma _1j_1A_1m_1+c_3j_1B_1m_1^2)\right) ,\\ j_{105}= &\, {} -{{Br}}_1\left( (c_3\gamma _1j_1B_1m_1+c_4j_1A_1m_1^2)\right. \\&\left. -\,\frac{1}{(m_1-\gamma _1)^2}(c_4\gamma _1j_1A_1m_1+c_3j_1B_1m_1^2)\right) ,\\ j_{106}= &\, {} {{Br}}_1\left( \frac{2 j_1 A_1 G_1}{m_1}\right) ,\\ j_{107}= &\, {} {{Br}}_1\left( \frac{2 j_1 B_1 G_1}{m_1}\right) ,\\ j_{107a}= &\, {} {{Br}}_1\left( \frac{2 c_4 G_1}{\gamma _1}\right) ,\\ j_{107b}= &\, {} {{Br}}_1\left( \frac{2 c_3 G_1}{\gamma _1}\right) ,\\ j_{108}= &\, {} \left( -{{Br}}_2\left[ \frac{c_6^2}{2}-\frac{j_2^2 m_2^2}{4}\left( A_2^2-B_2^2\right) +\left( \frac{j_2^2 m_2^4}{4 \gamma _2^2}\right) \left( A_2^2-B_2^2\right) \right. \right. \\&\left. \left. +\frac{G_2^2}{2\mu ^2 \gamma _2^2}\right] -\frac{S_2}{2}\right) ,\\ j_{109}= &\, {} \left( \frac{c_6G_2{{Br}}_2}{3 \mu }\right), \end{aligned}$$

$$\begin{aligned}j_{110}= &\, {} -\left( \frac{G_1^2 {{Br}}_1}{12}\right) ,\\ j_{111}= &\, {} -{{Br}}_2\left( \frac{c_7^2}{4}+\frac{c_8^2}{4}\right) ,\\ j_{112}= &\, {} -{{Br}}_2\left( \frac{c_7 c_8}{2}\right) ,\\ j_{113}= &\, {} -{{Br}}_2\left( \frac{j_2^2 A_2^2}{8}+\frac{j_2^2 B_2^2}{8}+\frac{j_2^2 A_2^2 m_2^2}{8\gamma _2^2}+\frac{j_2^2 B_2^2 m_2^2}{8 \gamma _2^2}\right) ,\\ j_{114}= &\, {} -{{Br}}_2\left( \frac{j_2^2 A_2 B_2}{4}+\frac{j_2^2 A_2 B_2 m_2^2}{4 \gamma _2^2}\right) ,\\ j_{115}= &\, {} -{{Br}}_2\left( 2\left( \frac{c_6c_7}{\gamma _2}-\frac{c_8 G_2}{\mu \gamma _2^2}\right) + \frac{4 c_8 G_2}{\mu \gamma _2^2}\right),\\j_{116}= &\, {} -{{Br}}_2\left( 2\left( \frac{c_6c_7}{\gamma _2}-\frac{c_7 G_2}{\mu \gamma _2^2}\right) + \frac{4 c_7 G_2}{\mu \gamma _2^2}\right) ,\\ j_{117}= &\, {} -{{Br}}_2\left( \frac{2}{m_2}\left( c_6 j_2 B_2-\frac{j_2A_2m_2G_2}{\mu \gamma _2^2}\right) +\frac{4j_2A_2G_2}{\mu m_2^2}\right), \end{aligned}$$

$$\begin{aligned} j_{118}= &\, {} -{{Br}}_2\left( \frac{2}{m_2}\left( c_2 j_2 A_2-\frac{j_2B_2m_2G_2}{\gamma _2^2}\right) +\frac{4j_2B_2G_2}{\mu m_2^2}\right) ,\\ j_{119}= &\, {} -{{Br}}_2\left( (c_7\gamma _2j_2A_2m_2+c_8j_2B_2m_2^2)\right. \\&\left. +\,\frac{1}{(m_2+\gamma _2)^2}(c_8\gamma _2j_2B_2m_2+c_7j_2A_2m_2^2)\right) ,\\ j_{120}= &\, {} -{{Br}}_2\left( (c_7\gamma _2j_2A_2m_2+c_8j_2B_2m_2^2)\right. \\&\left. -\,\frac{1}{(m_2-\gamma _2)^2}(c_8\gamma _2j_2B_2m_2+c_7j_2A_2m_2^2)\right) ,\\ j_{121}= &\, {} -{{Br}}_2\left( (c_7\gamma _2j_2B_2m_2+c_8j_2A_2m_2^2)\right. \\&\left. +\,\frac{1}{(m_2+\gamma _2)^2}(c_8\gamma _2j_2A_2m_2+c_7j_2B_2m_2^2)\right) ,\\ j_{122}= &\, {} -{{Br}}_2\left( (c_7\gamma _2j_2B_2m_2+c_8j_2A_2m_2^2)\right. \\&\left. -\,\frac{1}{(m_2-\gamma _2)^2}(c_8\gamma _2j_2A_2m_2+c_7j_2B_2m_2^2)\right) ,\\ j_{123}= &\, {} {{Br}}_2\left( \frac{2 j_2 A_2 G_2}{\mu m_2}\right) ,\\ j_{124}= &\, {} {{Br}}_2\left( \frac{2 j_2 B_2 G_2}{\mu m_2}\right) ,\\ j_{124a}= &\, {} {{Br}}_2\left( \frac{2 c_8 G_2}{\mu \gamma _2}\right),\end{aligned}$$

$$\begin{aligned} j_{124b}= &\, {} {{Br}}_2\left( \frac{2 c_7 G_2}{\mu \gamma _2}\right) ,\\ j_{125}= &\, {} -\left[ j_{91}\left( h_1^2+2 st_1 h_1\right) +j_{92}\left( h_1^3+3 st_1 h_1^2\right) \right. \\&\left. +\,\,j_{93}\left( h_1^4+4 st_1 h_1^3\right) +j_{94}\left( \cosh (2 \gamma _1 h_1)\right. \right. \\&\left. \left. +\,2st_1 \gamma _1 \sinh (2 \gamma _1 h_1)\right) +j_{95}\left( \sinh (2 \gamma _1 h_1)\right. \right. \\&\left. \left. +\,2st_1 \gamma _1 \cosh (2 \gamma _1 h_1)\right) +j_{96}\left( \cosh (2 m_1 h_1)\right. \right. \\&\left. \left. +\,2st_1 m_1 \sinh (2 m_1 h_1)\right) +j_{97}\left( \sinh (2 m_1 h_1) \right. \right. \\&\left. \left. +\,2st_1 m_1 \cosh (2 m_1 h_1)\right) + j_{98}\left( \cosh ( \gamma _1 h_1)\right. \right. \\&\left. \left. +st_1 \gamma _1 \sinh ( \gamma _1 h_1)\right) +j_{99}\left( \sinh ( \gamma _1 h_1)\right. \right. \\&\left. \left. +\,st_1 \gamma _1 \cosh ( \gamma _1 h_1)\right) +j_{100}\left( \cosh ( m_1 h_1)\right. \right. \\&\left. \left. +\,st_1 m_1 \sinh ( m_1 h_1)\right) +j_{101}\left( \sinh ( \gamma _1 h_1)\right. \right. \\&\left. \left. +\,st_1 m_1 \cosh ( m_1 h_1)\right) +j_{102}\left( \sinh ((m_1+ \gamma _1) h_1) \right. \right. \\&\left. \left. +\,st_1 (\gamma _1+h_1) \cosh ((m_1+ \gamma _1) h_1)\right) \right. \\&\left. +j_{103}\left( \sinh ((m_1- \gamma _1) h_1)\right. \right. \\&\left. \left. +\,st_1 (\gamma _1-h_1) \cosh ((m_1- \gamma _1) h_1)\right) \right. \\&\left. +j_{104}\left( \cosh ((m_1+ \gamma _1) h_1)\right. \right. \\&\left. \left. +\,st_1 (\gamma _1+h_1) \sinh ((m_1+ \gamma _1) h_1)\right) \right. \\&\left. +\,j_{105}\left( \cosh ((m_1- \gamma _1) h_1)\right. \right. \\&\left. \left. +\,st_1 (\gamma _1-h_1) \sinh ((m_1- \gamma _1) h_1)\right) \right. \\&\left. +j_{106}\left( (h_1+ st_1) \sinh (m_1 h_1)\right. \right. \\&\left. \left. +\,st_1 h_1 m_1 \cosh (m_1 h_1)\right) +j_{107}\left( (h_1+ st_1) \cosh (m_1 h_1)\right. \right. \\&\left. \left. +\,st_1 h_1 m_1 \sinh (m_1 h_1)\right) +j_{107a}\left( (h_1+ st_1) \sinh (\gamma _1 h_1)\right. \right. \\&\left. \left. +\,st_1 h_1 \gamma _1 \cosh (\gamma _1 h_1)\right) +j_{107b}\left( (h_1+ st_1) \cosh (\gamma _1 h_1)\right. \right. \\&\left. \left. +\,st_1 h_1 \gamma _1 \cosh (\gamma _1 h_1)\right) \right],\end{aligned}$$

$$\begin{aligned}j_{126}= &\, {} -\left[ j_{108}\left( h_2^2+2 st_2 h_2\right) +j_{109}\left( h_2^3+3 st_2 h_2^2\right) \right. \\&\left. +\,\,j_{110}\left( h_2^4+4 st_2 h_2^3\right) +j_{111}\left( \cosh (2 \gamma _2 h_2)\right. \right. \\&\left. \left. +\,2st_2 \gamma _2 \sinh (2 \gamma _2 h_2)\right) +j_{112}\left( \sinh (2 \gamma _2 h_2)\right. \right. \\&\left. \left. +\,2st_2 \gamma _2 \cosh (2 \gamma _2 h_2)\right) +j_{113}\left( \cosh (2 m_2 h_2)\right. \right. \\&\left. \left. +\,2st_2 m_2 \sinh (2 m_2 h_2)\right) ++j_{114}\left( \sinh (2 m_2 h_2)\right. \right. \\&\left. \left. +\,2st_2 m_2 \cosh (2 m_2 h_2)\right) + j_{115}\left( \cosh ( \gamma _2 h_2)\right. \right. \\&\left. \left. +\,st_2 \gamma _2 \sinh ( \gamma _2 h_2)\right) +j_{116}\left( \sinh ( \gamma _2 h_2)\right. \right. \\&\left. \left. +\,st_2 \gamma _2 \cosh ( \gamma _2 h_2)\right) +j_{117}\left( \cosh ( m_2 h_2)\right. \right. \\&\left. \left. +st_2 m_2 \sinh ( m_2 h_2)\right) +j_{118}\left( \sinh ( \gamma _2 h_2)\right. \right. \\&\left. \left. +\,st_2 m_1 \cosh ( m_2 h_2)\right) +j_{119}\left( \sinh ((m_2+ \gamma _1) h_2)\right. \right. \\&\left. \left. +\,st_2 (\gamma _1+h_2) \cosh ((m_1+ \gamma _2) h_2)\right) \right. \\&\left. +\,j_{120}\left( \sinh ((m_2- \gamma _2) h_2)\right. \right. \\&\left. \left. +\,st_2 (\gamma _2-h_2) \cosh ((m_2- \gamma _1) h_2)\right) \right. \\&\left. +\,j_{121}\left( \cosh ((m_2+ \gamma _2) h_2)\right. \right. \\&\left. \left. +\,st_2 (\gamma _2+h_2) \sinh ((m_2+ \gamma _1) h_2)\right) \right. \\&\left. +\,j_{122}\left( \cosh ((m_2- \gamma _2) h_2)\right. \right. \\&\left. \left. +\,st_2 (\gamma _2-h_2) \sinh ((m_2- \gamma _1) h_2)\right) \right. \\&\left. +\,j_{123}\left( (h_2+ st_2) \sinh (m_2 h_2) \right. \right. \\&\left. \left. +\,st_2 h_2 m_2 \cosh (m_2 h_2)\right) +j_{124}\left( (h_2+ st_2) \cosh (m_2 h_2)\right. \right. \\&\left. \left. +\,st_2 h_2 m_2 \sinh (m_2 h_2)\right) +j_{124a}\left( (h_2+ st_2) \sinh (\gamma _2 h_2)\right. \right. \\&\left. \left. +\,st_2 h_2 \gamma _2 \cosh (\gamma _2 h_2)\right) +j_{124b}\left( (h_2+ st_2) \cosh (\gamma _2 h_2)\right. \right. \\&\left. \left. +\,st_2 h_2 \gamma _2 \cosh (\gamma _2 h_2)\right) \right], \end{aligned}$$

$$\begin{aligned} j_{127}= &\, {} \frac{K_{\mathrm{R}}}{K_{\mathrm{R}}+N_{\mathrm{R}}}\left( j_{112}2\gamma _2 + j_{114} 2m_2 +j_{116} \gamma _2 +j_{118} m_2 \right. \\&\left. +\,j_{119} (m_2+ \gamma _2)+ j_{120} (m_2 - \gamma _2) + j_{124} + j_{124a}\right) \\&-\,\frac{1}{1+N_{\mathrm{R}}} \left( j_{95} 2 \gamma _1 + j_{97} 2 m_1 + j_{99} \gamma _1 j_{101} m_1 + j_{102} (m_1+\gamma _1)\right. \\&\left. +\, j_{103} (m_1-\gamma _1) + j_{107} + j_{107b}\right), \end{aligned}$$

$$\begin{aligned} j_{128}= &\, {} \frac{K_{\mathrm{R}}}{K_{\mathrm{R}}+N_{\mathrm{R}}}\left( j_{111}+ j_{113} +j_{115} +j_{117} + j_{121} + j_{122}\right) \\&-\,\frac{K_{\mathrm{R}}}{K_R+N_{\mathrm{R}}} \left( j_{94} + j_{96} + j_{99} + j_{98} + j_{100}+ j_{104} + j_{105}\right) ,\\ j_{129}= &\, {} \left( \frac{K_{\mathrm{R}}}{1+N_{\mathrm{R}}}\right) ,\\ j_{130}= &\, {} \left( \frac{K_{\mathrm{R}}}{K_{\mathrm{R}}+N_{\mathrm{R}}}\right) ,\\ j_{131}= &\, {} \left( \frac{1}{1+N_{\mathrm{R}}}\right) ,\\ j_{132}= &\, {} j_{129}\left( h_1+st_1\right) ,\\ j_{133}= &\, {} \left( j_{129} j_{125}- j_{128}\right) ,\\ j_{134}= &\, {} \left( j_{130} (h_2 - st_2)\right) ,\\ j_{135}= &\, {} \left( j_{133} - j_{126} j_{130}\right) ,\\ c_{1}= &\, {} j_{82}+j_{83}G_1+j_{84}G_2,\\ c_{2}= &\, {} j_{70}+j_{71}G_1+j_{72}G_2,\\ c_{3}= &\, {} j_{64}+j_{65}G_1+j_{66}G_2,\\ c_{4}= &\, {} j_{73}+j_{74}G_1+j_{75}G_2,\\ c_{5}= &\, {} j_{79}+j_{80}G_1+j_{81}G_2,\\ c_{6}= &\, {} j_{67}+j_{68}G_1+j_{69}G_2,\\ c_{7}= &\, {} j_{76}+j_{77}G_1+j_{78}G_2,\\ c_{8}= &\, {} j_{61}+j_{62}G_1+j_{63}G_2,\\ cc_1= &\, {} j_{127}+j_{130} cc_3,\\ cc_2= &\, {} j_{125}-cc_1 (h_1+st_1), \\ cc_3= &\, {} \frac{j_{135}j_{131}-j_{127}j_{132}}{j_{130}j_{132}+j_{134}j_{131}},\\ cc_4= &\, {} j_{126}-cc_3 (h_2-st_2).\end{aligned}$$