Slow motion of a porous spherical particle with a rigid core in a spherical fluid cavity

Abstract

We present an analytical study of viscous flow past a porous spherical particle composed of a rigid core inside. We consider that this particle is located inside a spherical fluid cavity filled with incompressible Newtonian fluid, under the creeping flow conditions. The governing equations inside the fluid region and the porous region are governed by Stokes equation and Brinkman equation respectively, supplemented with the corresponding mass conservation. Hydrodynamic drag and torque exerted by the fluid on particle are obtained which are later used to obtain the translational and rotational mobility of the particle inside the spherical cavity. In general the presence of cavity wall retards the particle movement as a result the mobility parameter becomes smaller than unity. The boundary effect is more pronounced when the separation distance between the particle surface and the cavity wall is less. Various limiting cases are obtained which agree with earlier existing results.

Appendix

The expressions corresponding to $D^{0}, D$ and $E^{0}, E$ are given as follows

$$\begin{aligned} D^{0}= & \, \frac{D^{0}_{1}}{D^{0}_{2}}\\ D^{0}_1= & \, \left( 2\, \zeta \,{\lambda }^{3}+\zeta \,{ \varepsilon }^{3}{\lambda }^{3}+3\, \zeta \,{\varepsilon }^{2}{ \lambda }^{2}+3\, \zeta \,\varepsilon \,\lambda \right) \cosh (\lambda ){\mathrm{e}}^\lambda -\left( \zeta \,{\varepsilon }^{3}{\lambda }^{4}+3\, \zeta \,{\varepsilon }^{2}{\lambda }^{3}+3\, \zeta \, \varepsilon \,{\lambda }^{2}+ \zeta \,{\varepsilon }^{3}{ \lambda }^{2}+3\, \zeta \,{\varepsilon }^{2}\lambda +2\, \zeta \,{\lambda }^{2}+3\, \zeta \,\varepsilon \right. \\& \left. +\,2\, \zeta \,{\lambda }^{4} \right) \sinh (\lambda ){\mathrm{e}}^\lambda +\left( {\mathrm{e}^{\lambda \,\varepsilon }} \zeta \,{\varepsilon }^{3}{ \lambda }^{2}+ \zeta \,{\varepsilon }^{3}{\lambda }^{4}{ {\mathrm{e}}^{\lambda \,\varepsilon }}+3\, \zeta \,\varepsilon \,{ \lambda }^{2}{{\mathrm{e}}^{\lambda \,\varepsilon }}+3\, \zeta \, \varepsilon \,\lambda \,{{\mathrm{e}}^{\lambda \,\varepsilon }}- \zeta \,{\lambda }^{3}\varepsilon \,{{\mathrm{e}}^{\lambda }}+ \zeta \,{\varepsilon }^{3}{\lambda }^{3}{\mathrm{e}^{\lambda \,\varepsilon }}+{ \lambda }^{2}\varepsilon \,{\mathrm{e}^{\lambda }}+2\, \zeta \, {\lambda }^{2}{\mathrm{e}^{\lambda \,\varepsilon }}\right. \\& \left. +\,2\, \zeta \,{\lambda }^{4}{\mathrm{e}^{\lambda \,\varepsilon }}+2\, \zeta \,{\lambda }^{3}{\mathrm{e}^{\lambda \,\varepsilon }}+3\,{\mathrm{e}^{ \lambda \,\varepsilon }} \zeta \,\varepsilon \right) \sinh (\lambda \varepsilon ) +\left( {\lambda }^{2}\varepsilon \,{\mathrm{e}^{\lambda }}-3\,{\mathrm{e}^{\lambda \, \varepsilon }} \zeta \,{\varepsilon }^{2}\lambda -3\, \zeta \,{\varepsilon }^{2}{\lambda }^{3}{\mathrm{e}^{\lambda \, \varepsilon }}-3\, \zeta \,{\varepsilon }^{2}{\lambda }^{2}{ \mathrm{e}^{\lambda \,\varepsilon }}- \zeta \,{\lambda }^{3} \varepsilon \,{\mathrm{e}^{\lambda }} \right) \cosh (\lambda \varepsilon )\\ D^{0}_2= & \, \left( \zeta \,{\varepsilon }^{3}{\lambda }^{4}{\mathrm{e}^{\lambda } }+3\, \zeta \,{\varepsilon }^{2}{\lambda }^{3}{\mathrm{e}^{ \lambda }}+3\, \zeta \,\varepsilon \,{\lambda }^{2}{\mathrm{e}^ {\lambda }}+3\, \zeta \,\varepsilon \,\lambda \,{\mathrm{e}^{ \lambda \,\varepsilon }}-\lambda \,{\mathrm{e}^{\lambda }}+3\, \zeta \,{\mathrm{e}^{\lambda }}+4\, \zeta \,{\lambda }^{ 2}{\mathrm{e}^{\lambda }}+2\, \zeta \,{\lambda }^{4}{ \mathrm{e}^{\lambda }} \right) \sinh (\lambda )\\& +\,\left( - \zeta \,{\varepsilon }^{3}{\lambda }^{3}{\mathrm{e}^{\lambda }}-3\, \zeta \,{\varepsilon }^{2}{\lambda }^{2}{\mathrm{e}^{ \lambda }}-3\, \zeta \,\varepsilon \,\lambda \,{\mathrm{e}^{ \lambda }}+3\, \zeta \,\varepsilon \,\lambda \,{\mathrm{e}^{ \lambda \,\varepsilon }}-2\, \zeta \,{\lambda }^{3}{\mathrm{e}^ {\lambda }}-3\, \zeta \,\lambda \,{\mathrm{e}^{\lambda }} \right) \cosh (\lambda )\\& +\,\left( 3\, \zeta \,{\varepsilon }^{2}{\lambda }^{3}{\mathrm{e}^{ \lambda \,\varepsilon }}+3\, \zeta \,{\varepsilon }^{2}{\lambda }^{2}{\mathrm{e}^{\lambda \,\varepsilon }}+ \zeta \,{\lambda }^ {3}\varepsilon \,{\mathrm{e}^{\lambda }}+3\, \zeta \,\varepsilon \,\lambda \,{\mathrm{e}^{\lambda }}-{\lambda }^{2}\varepsilon \,{\mathrm{e}^{ \lambda }} \right) \cosh (\lambda \varepsilon )+ \left( 3\, \zeta \,\varepsilon \,\lambda \,{\mathrm{e}^{\lambda }}-{ \lambda }^{2}\varepsilon \,{\mathrm{e}^{\lambda }}-3\, \zeta \, \lambda \,{\mathrm{e}^{\lambda \,\varepsilon }}-2\, \zeta \,{ \lambda }^{3}{\mathrm{e}^{\lambda \,\varepsilon }}\right. \\& \left. -\,2\, \zeta \, {\lambda }^{4}{\mathrm{e}^{\lambda \,\varepsilon }}-4\, \zeta \,{\lambda }^{2}{\mathrm{e}^{\lambda \,\varepsilon }}- \zeta \, {\varepsilon }^{3}{\lambda }^{4}{\mathrm{e}^{\lambda \,\varepsilon }}-3\, \zeta \,\varepsilon \,{\lambda }^{2}{\mathrm{e}^{\lambda \,\varepsilon }}+ \lambda \,{\mathrm{e}^{\lambda \,\varepsilon }}-3\, \zeta \,{ \mathrm{e}^{\lambda \,\varepsilon }}+ \zeta \,{\lambda }^{3} \varepsilon \,{\mathrm{e}^{\lambda }}-3\, \zeta \,\varepsilon \, \lambda \,{\mathrm{e}^{\lambda \,\varepsilon }}- \zeta \,{ \varepsilon }^{3}{\lambda }^{3}{\mathrm{e}^{\lambda \,\varepsilon }} \right) \sinh (\lambda \varepsilon ) \end{aligned}$$

$$\begin{aligned} \beta _{1}^{\prime }= & \, \frac{3D}{4}\\ D= & \, \frac{D_{1}}{D_{2}+D_{3}}\\ D_{1}= & \, 4\lambda \left[ -120\, \zeta \,\lambda \,\varepsilon \,{l}^{5}\left( f_{1}(\lambda )g_{2}(\lambda )+f_{2}(\lambda )g_{1}(\lambda )\right) +24\, \zeta \,{\lambda }^{2} \left( 1-{l}^{5}\right) \left( f_{1}(\lambda )g_{1}(\lambda \varepsilon )-f_{1}(\lambda \varepsilon )g_{1}(\lambda )\right) +\left( -4\, \zeta \,{\lambda }^{3}{\varepsilon }^{4}-8\, \zeta \,{\lambda }^{3}\varepsilon \right. \right. \\& \left. +\,120\, \zeta \, \lambda \,{\varepsilon }^{4}{l}^{5}+4\, \zeta \,{\lambda }^{ 3}{\varepsilon }^{4}{l}^{5}+8\, \zeta \,{\lambda }^{3} \varepsilon \,{l}^{5} \right) f_{1}(\lambda )g_{2}(\lambda \varepsilon )+\left( -12\,{\lambda }^{3} \zeta \,{l}^{5}+60\,{l}^{5}+60\, \lambda \, \zeta \,{l}^{5}+18\,{\lambda }^{2}{l}^{5}+12 \, \zeta \,{\lambda }^{3} \right) \left( f_{2}(\lambda )g_{1}(\lambda \varepsilon )\right. \\& +\,\left. f_{1}(\lambda \varepsilon )g_{2}(\lambda )\right) + \left( 40\,{\lambda }^{2} \zeta \,{\varepsilon }^{4}{l}^{5}-20\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5}-4\,{\lambda }^{ 4} \zeta \,\varepsilon -3\,{\lambda }^{3}{\varepsilon }^{4}{l}^ {5}-6\,{\lambda }^{3}\varepsilon \,{l}^{5}-2\,{\lambda }^{4} \zeta \,{\varepsilon }^{4}-20\,{l}^{5}\varepsilon \,\lambda +4\,{\lambda }^{4 } \zeta \,\varepsilon \,{l}^{5}+2\,{\lambda }^{4} \zeta \,{\varepsilon }^{4}{l}^{5}\right) \\& \times \, f_{2}(\lambda )g_{2}(\lambda \varepsilon )+ \left( 2\,{\varepsilon }^{3}{\lambda }^{2}-60\,{l}^{5}{\varepsilon }^{3}-60\, \zeta \,\lambda \,{l}^{5}{\varepsilon }^{3}+2\, \zeta \,{\lambda }^{3}{l}^{5}{\varepsilon }^{3}-20\,{\lambda }^{2}{l}^{5} {\varepsilon }^{3}-2\,{\varepsilon }^{3} \zeta \,{\lambda }^{3} \right) \left( f_{1}(\lambda \varepsilon )g_{2}(\lambda \varepsilon )+f_{2}(\lambda \varepsilon )g_{1}(\lambda \varepsilon )\right) \\& +\,\left( -4\, \zeta \,{\lambda }^{3}{\varepsilon }^{4}-8\, \zeta \,{\lambda }^{3}\varepsilon +120\, \zeta \, \lambda \,{\varepsilon }^{4}{l}^{5}+4\, \zeta \,{\lambda }^{ 3}{\varepsilon }^{4}{l}^{5}+8\, \zeta \,{\lambda }^{3} \varepsilon \,{l}^{5} \right) f_{2}(\lambda \varepsilon )g_{1}(\lambda )+\left( -2\,{\lambda }^{4} \zeta \,{\varepsilon }^{4}{l}^{5}-4\,{ \lambda }^{4} \zeta \,\varepsilon \,{l}^{5}+20\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5}\right. \\& -\,\left. \left. 40\,{\lambda }^{2} \zeta \,{\varepsilon }^{4}{l}^{5}+6\,{\lambda }^{3} \varepsilon \,{l}^{5}+3\,{\lambda }^{3}{\varepsilon }^{4}{l}^{5}+20\,{l}^{5} \varepsilon \,\lambda +2\,{\lambda }^{4} \zeta \,{\varepsilon }^ {4}+4\,{\lambda }^{4} \zeta \,\varepsilon \right) f_{2}(\lambda \varepsilon )g_{2}(\lambda )\right] \\ D_{2}= & \, \left( 24\, \zeta \,{\lambda }^{2}\varepsilon +108\,{l}^{5} \varepsilon \,\lambda +240\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{3}-480\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{6} +216\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5} \right) f_{1}(\lambda )g_{2}(\lambda )+\left( 72+360\,{l}^{3}-432\,{l}^{5}-240\, \zeta \,{\lambda }^{ 3}{l}^{3}\right. \\& \left. +\,216\, \zeta \,{\lambda }^{3}l+216\, \zeta \,{\lambda }^{3}{l}^{5}-360\, \zeta \, \lambda \,{l}^{3}-96\, \zeta \,{\lambda }^{3}{l}^{6}+432 \, \zeta \,\lambda \,{l}^{5}-108\,{l}^{5}{\lambda }^{2}+ 108\,{l}^{3}{\lambda }^{2}-96\, \zeta \,{\lambda }^{3}- 72\, \zeta \,\lambda \right) f_{1}(\lambda )g_{1}(\lambda \varepsilon )\\& +\,\left( 120\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{3}+32\, \zeta \,{\lambda }^{4}\varepsilon \,{l}^{6}-216\, \zeta \,{\lambda }^{2}{\varepsilon }^{4}{l}^{5}-240\, \zeta \,{\lambda }^{2}{\varepsilon }^{4}{l}^{3}-72\, \zeta \,{\lambda }^{4}\varepsilon \,l-36\, \zeta \,{ \lambda }^{4}{\varepsilon }^{4}{l}^{5}-72\, \zeta \,{ \lambda }^{4}\varepsilon \,{l}^{5}+480\, \zeta \,{\lambda }^ {2}{\varepsilon }^{4}{l}^{6}+24\, \zeta \,{\lambda }^{2} \varepsilon \right. \\& -\,24\,{\varepsilon }^{4} \zeta \,{\lambda }^{2}-36\, \zeta \,{\lambda }^{4}{\varepsilon }^{4}l+32\, \zeta \,{\lambda }^{4}\varepsilon +16\, \zeta \,{ \lambda }^{4}{\varepsilon }^{4}+144\,{l}^{5}\varepsilon \,\lambda +36\,{l}^{5} \varepsilon \,{\lambda }^{3}-36\,{l}^{3}\varepsilon \,{\lambda }^{3}+18\,{l}^{5} {\varepsilon }^{4}{\lambda }^{3}-18\,{l}^{3}{\varepsilon }^{4}{\lambda }^{3}-120 \,{l}^{3}\varepsilon \,\lambda -108\,{l}^{5}{\varepsilon }^{4}\lambda \\& \left. +\,80\, \zeta \,{\lambda }^{4}\varepsilon \,{l}^{3}+40\, \zeta \,{\lambda }^{4}{\varepsilon }^{4}{l}^{3}-144\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5}+16\, \zeta \,{\lambda }^{4}{\varepsilon }^{4}{l}^{6}-24\,\lambda \,\varepsilon \right) f_{1}(\lambda )g_{2}(\lambda \varepsilon )+\left( 24\, \zeta \,{\lambda }^{2}\varepsilon +108\,{l}^{5} \varepsilon \,\lambda +240\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{3}-480\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{6} \right. \\& \left. +\,216\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5} \right) f_{2}(\lambda )g_{1}(\lambda )+\left( -120\,{\lambda }^{4} \zeta \,{l}^{3}+108\,{\lambda }^{4} \zeta \,l+108\,{\lambda }^{4} \zeta \, {l}^{5}-300\,{\lambda }^{2} \zeta \,{l}^{3}-48\,{ \lambda }^{4} \zeta \,{l}^{6}+108\,{\lambda }^{2} \zeta \,{l}^{5}+240\,{\lambda }^{2} \zeta \,{l}^{6}-48\, \zeta \,{\lambda }^{2}\right. \\& \left. -\,108\,{ \lambda }^{3}{l}^{5}+72\,{\lambda }^{3}{l}^{6}-432\,\lambda \,{l}^{5}+240 \,\lambda \,{l}^{6}+60\,\lambda \,{l}^{3}+36\,{\lambda }^{3}{l}^{3}-48\, \zeta \,{\lambda }^{4}+24\,\lambda \right) f_{2}(\lambda )g_{1}(\lambda \varepsilon )\\& + \,\left( -8\,{\lambda }^{2}\varepsilon +16\, \zeta \,{\lambda }^{3} \varepsilon -8\,{\varepsilon }^{4} \zeta \,{\lambda }^{3}-24\,{ \lambda }^{4}\varepsilon \,{l}^{6}+18\,{\lambda }^{4}{\varepsilon }^{4}{l}^{5}+ 36\,{\lambda }^{4}\varepsilon \,{l}^{5}+8\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}+16\,{\lambda }^{5} \zeta \, \varepsilon -36\,{l}^{5}{\varepsilon }^{4}{\lambda }^{2}-80\,{\lambda }^{2} \varepsilon \,{l}^{6}-6\,{\lambda }^{4}{\varepsilon }^{4}{l}^{3}\right. \\& +\,144\,{\lambda } ^{2}\varepsilon \,{l}^{5}-12\,{\lambda }^{4}\varepsilon \,{l}^{3}-20\,{\lambda } ^{2}\varepsilon \,{l}^{3}-12\,{\lambda }^{4}{\varepsilon }^{4}{l}^{6}+160\,{ \lambda }^{3} \zeta \,{\varepsilon }^{4}{l}^{6}-36\,{ \lambda }^{3} \zeta \,\varepsilon \,{l}^{5}+100\,{\lambda }^ {3} \zeta \,\varepsilon \,{l}^{3}+40\,{\lambda }^{5} \zeta \,\varepsilon \,{l}^{3}-80\,{\lambda }^{3} \zeta \,\varepsilon \,{l}^{6}+16\,{\lambda }^{5} \zeta \,\varepsilon \,{l}^{6}\\& \left. -\,72\,{\lambda }^{3} \zeta \,{\varepsilon }^{4}{l}^{5}-80\,{\lambda }^{3} \zeta \,{ \varepsilon }^{4}{l}^{3}+20\,{\lambda }^{5} \zeta \,{ \varepsilon }^{4}{l}^{3}-18\,{\lambda }^{5} \zeta \,{ \varepsilon }^{4}{l}^{5}-36\,{\lambda }^{5} \zeta \, \varepsilon \,l+8\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}{l} ^{6}-18\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}l-36\,{ \lambda }^{5} \zeta \,\varepsilon \,{l}^{5} \right) f_{2}(\lambda )g_{2}(\lambda \varepsilon ) \end{aligned}$$

$$\begin{aligned} D_{3}= & \, \left( -72+432\,{l}^{5}-360\,{l}^{3}+72\, \zeta \,\lambda +108 \,{l}^{5}{\lambda }^{2}-108\,{l}^{3}{\lambda }^{2}+96\, \zeta \,{\lambda }^{3}+96\, \zeta \,{\lambda }^{3}{l} ^{6}-216\, \zeta \,{\lambda }^{3}{l}^{5}+240\, \zeta \,{\lambda }^{3}{l}^{3}-216\, \zeta \,{ \lambda }^{3}l-432\, \zeta \,\lambda \,{l}^{5}\right. \\& \left. +\,360\, \zeta \,\lambda \,{l}^{3} \right) f_{1}(\lambda \varepsilon )g_{1}(\lambda )+\left( -120\,{\lambda }^{4} \zeta \,{l}^{3}+108\,{\lambda }^{4} \zeta \,l+108\,{\lambda }^{4} \zeta \, {l}^{5}-300\,{\lambda }^{2} \zeta \,{l}^{3}-48\,{ \lambda }^{4} \zeta \,{l}^{6}+108\,{\lambda }^{2} \zeta \,{l}^{5}+240\,{\lambda }^{2} \zeta \,{l}^{6}-48\, \zeta \,{\lambda }^{2}\right. \\& \left. -\,108\,{ \lambda }^{3}{l}^{5}+72\,{\lambda }^{3}{l}^{6}-432\,\lambda \,{l}^{5}+240 \,\lambda \,{l}^{6}+60\,\lambda \,{l}^{3}+36\,{\lambda }^{3}{l}^{3}-48\, \zeta \,{\lambda }^{4}+24\,\lambda \right) f_{1}(\lambda \varepsilon )g_{2}(\lambda )\\& +\,\left( -240\, \zeta \,{\lambda }^{2}{l}^{6}{\varepsilon }^{3}+180 \, \zeta \,{\lambda }^{2}{\varepsilon }^{3}{l}^{3}-18\, \zeta \,{\lambda }^{4}l{\varepsilon }^{3}+20\, \zeta \,{\lambda }^{4}{l}^{3}{\varepsilon }^{3}-18\, \zeta \,{\lambda }^{4}{l}^{5}{\varepsilon }^{3}+36\, \zeta \,{\lambda }^{2}{l}^{5}{\varepsilon }^{3}+8\, \zeta \,{\lambda }^{4}{l}^{6}{\varepsilon }^{3}-8\,{\lambda }^{3}{ \varepsilon }^{3}+72\,{l}^{5}{\lambda }^{3}{\varepsilon }^{3}\right. \\& -\,\left. 80\,{l}^{6}{ \lambda }^{3}{\varepsilon }^{3}+288\,{l}^{5}{\varepsilon }^{3}\lambda +60\,{ \varepsilon }^{3}\lambda \,{l}^{3}-2\,{l}^{3}{\lambda }^{3}{\varepsilon }^{3}+18 \,l{\lambda }^{3}{\varepsilon }^{3}+8\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}-240\,{l}^{6}{\varepsilon }^{3}\lambda +24\,{ \varepsilon }^{3} \zeta \,{\lambda }^{2} \right) f_{1}(\lambda \varepsilon )g_{2}(\lambda \varepsilon )\\& +\,\left( 120\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{3}+32\, \zeta \,{\lambda }^{4}\varepsilon \,{l}^{6}-216\, \zeta \,{\lambda }^{2}{\varepsilon }^{4}{l}^{5}-240\, \zeta \,{\lambda }^{2}{\varepsilon }^{4}{l}^{3}-72\, \zeta \,{\lambda }^{4}\varepsilon \,l-36\, \zeta \,{ \lambda }^{4}{\varepsilon }^{4}{l}^{5}-72\, \zeta \,{ \lambda }^{4}\varepsilon \,{l}^{5}+480\, \zeta \,{\lambda }^ {2}{\varepsilon }^{4}{l}^{6}+24\, \zeta \,{\lambda }^{2} \varepsilon \right. \\& -\,24\,{\varepsilon }^{4} \zeta \,{\lambda }^{2}-36\, \zeta \,{\lambda }^{4}{\varepsilon }^{4}l+32\, \zeta \,{\lambda }^{4}\varepsilon +16\, \zeta \,{ \lambda }^{4}{\varepsilon }^{4}+144\,{l}^{5}\varepsilon \,\lambda +36\,{l}^{5} \varepsilon \,{\lambda }^{3}-36\,{l}^{3}\varepsilon \,{\lambda }^{3}+18\,{l}^{5} {\varepsilon }^{4}{\lambda }^{3}-18\,{l}^{3}{\varepsilon }^{4}{\lambda }^{3}-120 \,{l}^{3}\varepsilon \,\lambda -108\,{l}^{5}{\varepsilon }^{4}\lambda \\& \left. +\,80\, \zeta \,{\lambda }^{4}\varepsilon \,{l}^{3}+40\, \zeta \,{\lambda }^{4}{\varepsilon }^{4}{l}^{3}-144\, \zeta \,{\lambda }^{2}\varepsilon \,{l}^{5}+16\,\zeta \,{\lambda }^{4}{\varepsilon }^{4}{l}^{6}-24\,\lambda \,\varepsilon \right) f_{2}(\lambda \varepsilon )g_{1}(\lambda )\\& +\,\left( 8\,{\lambda }^{2}\varepsilon -16\, \zeta \,{\lambda }^{3} \varepsilon +8\,{\varepsilon }^{4} \zeta \,{\lambda }^{3}+24\,{ \lambda }^{4}\varepsilon \,{l}^{6}-18\,{\lambda }^{4}{\varepsilon }^{4}{l}^{5}- 36\,{\lambda }^{4}\varepsilon \,{l}^{5}-8\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}-16\,{\lambda }^{5} \zeta \, \varepsilon +36\,{l}^{5}{\varepsilon }^{4}{\lambda }^{2}+80\,{\lambda }^{2} \varepsilon \,{l}^{6}+6\,{\lambda }^{4}{\varepsilon }^{4}{l}^{3}\right. \\& -\,144\,{\lambda } ^{2}\varepsilon \,{l}^{5}+12\,{\lambda }^{4}\varepsilon \,{l}^{3}+20\,{\lambda } ^{2}\varepsilon \,{l}^{3}+12\,{\lambda }^{4}{\varepsilon }^{4}{l}^{6}-160\,{ \lambda }^{3} \zeta \,{\varepsilon }^{4}{l}^{6}+36\,{ \lambda }^{3} \zeta \,\varepsilon \,{l}^{5}-100\,{\lambda }^ {3} \zeta \,\varepsilon \,{l}^{3}-40\,{\lambda }^{5} \zeta \,\varepsilon \,{l}^{3}+80\,{\lambda }^{3} \zeta \,\varepsilon \,{l}^{6}-16\,{\lambda }^{5} \zeta \,\varepsilon \,{l}^{6}\\& \left. +\,72\,{\lambda }^{3} \zeta \,{\varepsilon }^{4}{l}^{5}+80\,{\lambda }^{3} \zeta \,{ \varepsilon }^{4}{l}^{3}-20\,{\lambda }^{5} \zeta \,{ \varepsilon }^{4}{l}^{3}+18\,{\lambda }^{5} \zeta \,{ \varepsilon }^{4}{l}^{5}+36\,{\lambda }^{5} \zeta \, \varepsilon \,l-8\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}{l} ^{6}+18\,{\lambda }^{5} \zeta \,{\varepsilon }^{4}l+36\,{ \lambda }^{5} \zeta \,\varepsilon \,{l}^{5} \right) f_{2}(\lambda \varepsilon )g_{2}(\lambda )\\& +\,\left( -2\,{l}^{3}{\lambda }^{3}{\varepsilon }^{3}+20\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}{l}^{3}+8\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}{l}^{6}-240\,{\varepsilon }^{3} \zeta \,{\lambda }^{2}{l}^{6}-18\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}{l}^{5}+180\,{\varepsilon }^{3} \zeta \,{\lambda }^{2}{l}^{3}+36\,{\varepsilon }^{3} \zeta \,{\lambda }^{2}{l}^{5}-18\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}l+18\,l{\lambda }^{3}{\varepsilon }^{3}\right. \\& \left. +\,60\,{l}^{3} {\varepsilon }^{3}\lambda +24\,{\varepsilon }^{3} \zeta \,{ \lambda }^{2}+8\,{\varepsilon }^{3} \zeta \,{\lambda }^{4}- 80\,{l}^{6}{\varepsilon }^{3}{\lambda }^{3}-240\,{l}^{6}{\varepsilon }^{3} \lambda +288\,{l}^{5}{\varepsilon }^{3}\lambda +72\,{l}^{5}{\varepsilon }^{3}{ \lambda }^{3}-8\,{\lambda }^{3}{\varepsilon }^{3} \right) f_{2}(\lambda \varepsilon )g_{1}(\lambda \varepsilon ) \end{aligned}$$

$$\begin{aligned} {E}^{0}= & \, {\frac{\begin{array}{l}- \zeta \,{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \right) {\mathrm{e}^{\lambda }}+ \zeta \,{\lambda } ^{2}\varepsilon \,\sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}- \zeta \,{\lambda }^{2}\cosh \left( \lambda \right) { \mathrm{e}^{\lambda }}+ \zeta \,\lambda \,\sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}+{\lambda }^{3}\varepsilon \,\sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}+3\,{\mathrm{e}^{\lambda }} \varepsilon \,\sinh \left( \lambda \right) \lambda \\ -3\,{\lambda }^{2}{ \mathrm{e}^{\lambda }}\varepsilon \,\cosh \left( \lambda \right) +\sinh \left( \lambda \right) {\lambda }^{2}{\mathrm{e}^{\lambda }}+3\,{\mathrm{e}^{ \lambda }}\sinh \left( \lambda \right) -3\,{\mathrm{e}^{\lambda }}\cosh \left( \lambda \right) \lambda + \zeta \,{\lambda }^{3} \varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \, \varepsilon }}- \zeta \,{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}\\ +\zeta \,{\lambda }^{2}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}- \zeta \, \lambda \,\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \, \varepsilon }}+{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}-{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}+3\,{\lambda }^{ 2}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \, \varepsilon }}-3\,\lambda \,\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e} ^{\lambda \,\varepsilon }}\\ +3\,{\mathrm{e}^{\lambda \,\varepsilon }}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) \lambda -3\,{\mathrm{e}^{\lambda \, \varepsilon }}\sinh \left( \lambda \,\varepsilon \right) \end{array}}{\begin{array}{l}- \zeta \,{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \right) { \mathrm{e}^{\lambda }}+ \zeta \,{\lambda }^{2}\varepsilon \, \sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}+ \zeta \,{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}+ \zeta \,{ \lambda }^{2}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) {\mathrm{e}^ {\lambda \,\varepsilon }}+\sinh \left( \lambda \right) {\lambda }^{2}{ \mathrm{e}^{\lambda }}-{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}\\ -\zeta \,{ \lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{\lambda \, \varepsilon }}- \zeta \,{\lambda }^{2}\cosh \left( \lambda \right) {\mathrm{e}^{\lambda }}+ \zeta \,\lambda \,\sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}+{\lambda }^{3}\varepsilon \, \sinh \left( \lambda \right) {\mathrm{e}^{\lambda }}-\zeta \,\lambda \,\sinh \left( \lambda \,\varepsilon \right) {\mathrm{e}^{ \lambda \,\varepsilon }}+{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \, \varepsilon \right) {\mathrm{e}^{\lambda \,\varepsilon }}\end{array}}} \end{aligned}$$

$$\begin{aligned} \delta _{1}^{\prime }= & \, E= \frac{\begin{array}{l} -\zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{3} \varepsilon \,\cosh \left( \lambda \right) + \zeta \,{ \mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{2}\varepsilon \,\sinh \left( \lambda \right) - \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{2}\cosh \left( \lambda \right) + \zeta \, {\mathrm{e}^{-\lambda \,\varepsilon }}\lambda \,\sinh \left( \lambda \right) +{ \mathrm{e}^{-\lambda }}{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \, \varepsilon \right) \\ -{\mathrm{e}^{-\lambda }}{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) +3\,{\mathrm{e}^{-\lambda }}{\lambda }^{2} \varepsilon \,\cosh \left( \lambda \,\varepsilon \right) -3\,{\mathrm{e}^{- \lambda }}\lambda \,\sinh \left( \lambda \,\varepsilon \right) +3\,{\mathrm{e}^ {-\lambda }}\lambda \,\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) - 3\,{\mathrm{e}^{-\lambda }}\sinh \left( \lambda \,\varepsilon \right) + \zeta \,{\mathrm{e}^{-\lambda }}{\lambda }^{3}\varepsilon \, \cosh \left( \lambda \,\varepsilon \right) \\ -\zeta \,{ \mathrm{e}^{-\lambda }}{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) + \zeta \,{\mathrm{e}^{-\lambda }}{\lambda }^{2} \varepsilon \,\cosh \left( \lambda \,\varepsilon \right) - \zeta \,{\mathrm{e}^{-\lambda }}\lambda \,\sinh \left( \lambda \,\varepsilon \right) +{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{3}\varepsilon \,\sinh \left( \lambda \right) +3\,{\mathrm{e}^{-\lambda \,\varepsilon }}\lambda \, \varepsilon \,\sinh \left( \lambda \right) \\ -3\,{\mathrm{e}^{-\lambda \, \varepsilon }}{\lambda }^{2}\varepsilon \,\cosh \left( \lambda \right) +{ \mathrm{e}^{-\lambda \,\varepsilon }}\sinh \left( \lambda \right) {\lambda }^{2 }+3\,{\mathrm{e}^{-\lambda \,\varepsilon }}\sinh \left( \lambda \right) -3\,{ \mathrm{e}^{-\lambda \,\varepsilon }}\lambda \,\cosh \left( \lambda \right) \end{array}}{\begin{array}{l} -{l}^{3} \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}{ \lambda }^{3}\varepsilon \,\cosh \left( \lambda \right) +{l}^{3} \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{2}\varepsilon \, \sinh \left( \lambda \right) +{l}^{3} \zeta \,{\mathrm{e} ^{-\lambda }}{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) +{l}^{3} \zeta \,{\mathrm{e}^{-\lambda }}{\lambda }^{2}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) + \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \right) \\ - \zeta \,{\mathrm{e}^{-\lambda \, \varepsilon }}{\lambda }^{2}\varepsilon \,\sinh \left( \lambda \right) - \zeta \,{\mathrm{e}^{-\lambda }}{\lambda }^{3}\varepsilon \, \cosh \left( \lambda \,\varepsilon \right) - \zeta \,{ \mathrm{e}^{-\lambda }}{\lambda }^{2}\varepsilon \,\cosh \left( \lambda \, \varepsilon \right) +{\mathrm{e}^{-\lambda }}{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) -{\mathrm{e}^{-\lambda \,\varepsilon }}\sinh \left( \lambda \right) {\lambda }^{2}+ \zeta \,{\mathrm{e}^{- \lambda \,\varepsilon }}{\lambda }^{2}\cosh \left( \lambda \right) \\ - \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}\lambda \,\sinh \left( \lambda \right) -{\mathrm{e}^{-\lambda }}{\lambda }^{3}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) + \zeta \,{\mathrm{e}^{ -\lambda }}{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) + \zeta \,{\mathrm{e}^{-\lambda }}\lambda \,\sinh \left( \lambda \,\varepsilon \right) -{\mathrm{e}^{-\lambda \,\varepsilon }}{\lambda }^{3} \varepsilon \,\sinh \left( \lambda \right) -{\mathrm{e}^{-\lambda }}{l}^{3}{ \lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) \\ -3\,{\mathrm{e}^{- \lambda }}{l}^{3}\lambda \,\sinh \left( \lambda \,\varepsilon \right) +{ \mathrm{e}^{-\lambda \,\varepsilon }}{l}^{3}\sinh \left( \lambda \right) { \lambda }^{2}-3\,{\mathrm{e}^{-\lambda \,\varepsilon }}{l}^{3}\lambda \,\cosh \left( \lambda \right) -{l}^{3} \zeta \,{\mathrm{e}^{- \lambda \,\varepsilon }}{\lambda }^{2}\cosh \left( \lambda \right) +{l}^{3} \zeta \,{\mathrm{e}^{-\lambda \,\varepsilon }}\lambda \,\sinh \left( \lambda \right) \\ -{l}^{3} \zeta \,{\mathrm{e}^{- \lambda }}{\lambda }^{2}\sinh \left( \lambda \,\varepsilon \right) -{l}^{3} \zeta \,{\mathrm{e}^{-\lambda }}\lambda \,\sinh \left( \lambda \,\varepsilon \right) +{\mathrm{e}^{-\lambda }}{l}^{3}{\lambda }^{3} \varepsilon \,\cosh \left( \lambda \,\varepsilon \right) +3\,{\mathrm{e}^{- \lambda }}{l}^{3}{\lambda }^{2}\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) +3\,{\mathrm{e}^{-\lambda }}{l}^{3}\lambda \,\varepsilon \,\cosh \left( \lambda \,\varepsilon \right) \\ +{\mathrm{e}^{-\lambda \,\varepsilon }}{l}^{ 3}{\lambda }^{3}\varepsilon \,\sinh \left( \lambda \right) +3\,{\mathrm{e}^{- \lambda \,\varepsilon }}{l}^{3}\lambda \,\varepsilon \,\sinh \left( \lambda \right) -3\,{\mathrm{e}^{-\lambda \,\varepsilon }}{l}^{3}{\lambda }^{2} \varepsilon \,\cosh \left( \lambda \right) -3\,{\mathrm{e}^{-\lambda }}{l}^{3} \sinh \left( \lambda \,\varepsilon \right) +3\,{\mathrm{e}^{-\lambda \, \varepsilon }}{l}^{3}\sinh \left( \lambda \right) \end{array}} \end{aligned}$$

and the coefficients M1, M2 are given as follows

$$\begin{aligned} M1= & \, -\zeta \lambda ^2\left[ \left( -18\,{\lambda }^{4}{l}^{6}+6\, \zeta \,{\lambda }^{3}-60 \,{\lambda }^{2}{l}^{6}+15\,{\lambda }^{2}{l}^{3}+4\, \zeta \,{\lambda }^{5}+18\,{\lambda }^{4}{l}^{5}+72\,{\lambda }^{2}{l} ^{5}+4\, \zeta \,{\lambda }^{5}{l}^{6}-9\, \zeta \,{\lambda }^{5}{l}^{5}+9\, \zeta \,{ \lambda }^{3}{l}^{5}+45\, \zeta \,{\lambda }^{3}{l}^{3}- 9\, \zeta \,{\lambda }^{5}l\right. \right. \\& \left. +\,10\, \zeta \,{\lambda }^{5}{l}^{3}-60\, \zeta \,{\lambda }^{3}{l}^{ 6} \right) +\left( 2\,{\lambda }^{3}-18\, \zeta \,{\lambda }^{4}-12\, \zeta \,{\lambda }^{2}-8\, \zeta \,{ \lambda }^{6}-10\,{\lambda }^{3}{l}^{3}+24\,{\lambda }^{5}{l}^{6}+3\,{ \lambda }^{5}{l}^{3}-27\,{\lambda }^{5}{l}^{5}+120\,\lambda \,{l}^{6}-30 \,\lambda \,{l}^{3}\right. \\& +\,116\,{\lambda }^{3}{l}^{6}-144\,\lambda \,{l}^{5}-144 \,{\lambda }^{3}{l}^{5}-90\, \zeta \,{\lambda }^{4}{l}^{ 3}+120\, \zeta \,{\lambda }^{2}{l}^{6}+72\, \zeta \,{\lambda }^{4}{l}^{6}+18\, \zeta \,{ \lambda }^{6}{l}^{5}+18\, \zeta \,{\lambda }^{6}l-20\, \zeta \,{\lambda }^{6}{l}^{3}-8\, \zeta \,{\lambda }^{6}{l}^{6}+18\, \zeta \,{\lambda } ^{4}{l}^{5}\\& -\,\left. 18\, \zeta \,{\lambda }^{2}{l}^{5}-90\, \zeta \,{\lambda }^{2}{l}^{3}+18\, \zeta \,{\lambda }^{4}l \right) \tanh (\lambda ) +\left( -12\, \zeta \,{\lambda }^{5}{l}^{6}-27\, \zeta \,{\lambda }^{5}{l}^{5}-9\, \zeta \,{\lambda }^ {3}{l}^{5}-18\, \zeta \,{\lambda }^{5}l+80\, \zeta \,{\lambda }^{3}{l}^{3}+45\, \zeta \,{ \lambda }^{5}{l}^{3}-76\, \zeta \,{\lambda }^{3}{l}^{6}\right. \\& -\, 2\,{\lambda }^{2}+15\,{l}^{3}-2\,{\lambda }^{4}-60\,{l}^{6}+72\,{l}^{5}- 44\,{\lambda }^{4}{l}^{6}+14\, \zeta \,{\lambda }^{3}-98 \,{\lambda }^{2}{l}^{6}+10\,{\lambda }^{2}{l}^{3}+12\, \zeta \,{\lambda }^{5}+63\,{\lambda }^{4}{l}^{5}+126\,{\lambda }^{2}{l }^{5}+45\, \zeta \,\lambda \,{l}^{3}\\& -\,9\, \zeta \,{\lambda }^{3}l+9\, \zeta \,\lambda \,{l}^{5} +6\, \zeta \,\lambda -6\,{\lambda }^{6}{l}^{6}+9\,{ \lambda }^{6}{l}^{5}-3\,{\lambda }^{6}{l}^{3}+4\, \zeta \,{\lambda }^{7}-8\,{\lambda }^{4}{l}^{3}-9\, \zeta \,{ \lambda }^{7}{l}^{5}-9\, \zeta \,{\lambda }^{7}l+10\, \zeta \,{\lambda }^{7}{l}^{3}\\& +\,\left. \left. 4\, \zeta \,{\lambda }^{7}{l}^{6}-60\, \zeta \,\lambda \, {l}^{6} \right) \tanh ^2(\lambda ) \right] \\ M2= & \, \left( -4\,{\zeta }^{2}{\lambda }^{7}-6\,{\zeta }^{2}{\lambda }^{5}-87\, \zeta \,{\lambda }^{4}{l}^{5}-18\, \zeta \,{\lambda }^{6}{l}^{5}-90\,{\zeta }^{2}{\lambda }^{3}{l}^{5}- 90\, \zeta \,{\lambda }^{2}{l}^{5}+4\,{\zeta }^{2}{ \lambda }^{7}{l}^{5}-54\,{\zeta }^{2}{\lambda }^{5}{l}^{5} \right) +\left( -8\,{\zeta }^{2}{\lambda }^{8}{l}^{5}+62\,{\zeta }^{2}{\lambda }^{6}{l}^ {5}\right. \\& +\,198\,{\zeta }^{2}{\lambda }^{4}{l}^{5}+24\, \zeta \,{\lambda }^{7}{l}^{5}+234\, \zeta \,{\lambda }^{3}{l}^ {5}+145\, \zeta \,{\lambda }^{5}{l}^{5}+180\,{\zeta }^{ 2}{l}^{5}{\lambda }^{2}+180\, \zeta \,\lambda \,{l}^{5}+ 18\,{\zeta }^{2}{\lambda }^{6}+12\,{\zeta }^{2}{\lambda }^{4}-2\, \zeta \,{\lambda }^{5}-30\,{\lambda }^{2}{l}^{5}\\& -\,\left. 9\,{ \lambda }^{4}{l}^{5}+8\,{\zeta }^{2}{\lambda }^{8} \right) \tanh (\lambda )+ \left( 4\,{\zeta }^{2}{\lambda }^{9}{l}^{5}-6\, \zeta \,{ \lambda }^{8}{l}^{5}-137\, \zeta \,{\lambda }^{4}{l}^{5} -52\, \zeta \,{\lambda }^{6}{l}^{5}-90\,{\zeta }^{2}{l} ^{5}\lambda -144\,{\zeta }^{2}{\lambda }^{3}{l}^{5}-86\,{\zeta }^{2}{ \lambda }^{5}{l}^{5}-8\,{\zeta }^{2}{\lambda }^{7}{l}^{5}\right. \\& \left. -\,147\, \zeta \,{\lambda }^{2}{l}^{5}-4\,{\zeta }^{2}{\lambda }^{9}-14 \,{\zeta }^{2}{\lambda }^{5}-12\,{\zeta }^{2}{\lambda }^{7}+2\, \zeta \,{\lambda }^{6}-6\,{\zeta }^{2}{\lambda }^{3}-90\, \zeta \,{l}^{5}+30\,\lambda \,{l}^{5}+3\,{\lambda }^{5} {l}^{5}+19\,{\lambda }^{3}{l}^{5}+2\, \zeta \,{\lambda } ^{4} \right) \tanh ^2(\lambda ) \end{aligned}$$