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Rotating electroosmotic flow of an Eyring fluid

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Abstract

A perturbation analysis is presented in this paper for the electroosmotic (EO) flow of an Eyring fluid through a wide rectangular microchannel that rotates about an axis perpendicular to its own. Mildly shear-thinning rheology is assumed such that at the leading order the problem reduces to that of Newtonian EO flow in a rotating channel, while the shear thinning effect shows up in a higher-order problem. Using the relaxation time as the small ordering parameter, analytical solutions are deduced for the leading- as well as first-order problems in terms of the dimensionless Debye and rotation parameters. The velocity profiles of the Ekman–electric double layer (EDL) layer, which is the boundary layer that arises when the Ekman layer and the EDL are comparably thin, are also deduced for an Eyring fluid. It is shown that the present perturbation model can yield results that are close to the exact solutions even when the ordering parameter is as large as order unity. By this order of the relaxation time parameter, the enhancing effect on the rotating EO flow due to shear-thinning Eyring rheology can be significant.

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Acknowledgements

The authors are very thankful to the reviewers for their comments, which have helped to improve the manuscript to its present form. The work was financially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, through General Research Fund Project HKU 715510E and 17206615, and the University of Hong Kong through the Small Project Funding Scheme under Project Code 201309176109.

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China

    Cheng Qi & Chiu-On Ng

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  1. Cheng Qi
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Correspondence to Chiu-On Ng.

Appendix

Appendix

This Appendix contains the lengthy expressions for functions or parameters introduced in Sects. 4 and 5. In Eqs. (58) and (59), the four functions are given by

$$\begin{aligned} U_{\kappa }({\hat{z}})= & {} \frac{{\hat{\kappa }}^2\eta ^2C_1C_2}{3({\hat{\kappa }}^4+4\eta ^4)} {{\mathrm{sech}}}({\hat{\kappa }})\cosh ({\hat{\kappa }} {\hat{z}})\nonumber \\&+\,{{\mathrm{sech}}}^3({\hat{\kappa }}) \cosh ({\hat{\kappa }} {\hat{z}}) [U_1 \sinh ^2({\hat{\kappa }} {\hat{z}})+U_2\cosh ^2({\hat{\kappa }} z)]\nonumber \\&+\,{{\mathrm{sech}}}^2({\hat{\kappa }})\cosh (2 {\hat{\kappa }} {\hat{z}}) [U_3\sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})\nonumber \\&+\,U_6 \cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}})]\nonumber \\&+\,{{\mathrm{sech}}}^2({\hat{\kappa }})\sinh (2 {\hat{\kappa }} {\hat{z}})[U_4\sin (\eta {\hat{z}})\cosh (\eta {\hat{z}})\nonumber \\&+\,U_5\cos (\eta {\hat{z}}) \sinh (\eta {\hat{z}})]\nonumber \\&+\,{{\mathrm{sech}}}({\hat{\kappa }})\cosh ({\hat{\kappa }} {\hat{z}}) [U_7 \sin (2 \eta {\hat{z}})\sinh (2 \eta {\hat{z}})\nonumber \\&+\,U_{10}\cos (2 \eta {\hat{z}})\cosh (2\eta {\hat{z}})]\nonumber \\&+\,{{\mathrm{sech}}}({\hat{\kappa }})\sinh ({\hat{\kappa }} {\hat{z}}) [U_8\sin (2 \eta {\hat{z}})\cosh (2 \eta {\hat{z}})\nonumber \\&+\, U_9\cos (2\eta {\hat{z}}) \sinh (2 \eta {\hat{z}})]\nonumber \\&+\,U_{11}{{\mathrm{sech}}}({\hat{\kappa }}) \sin (2 \eta {\hat{z}}) \sinh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\,U_{12}{{\mathrm{sech}}}({\hat{\kappa }}) \cos (2\eta {\hat{z}}) \cosh ({\hat{\kappa }} {\hat{z}})\nonumber \\&+\,U_{13}{{\mathrm{sech}}}({\hat{\kappa }}) \sinh (2\eta {\hat{z}})\sinh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\,U_{14}{{\mathrm{sech}}}({\hat{\kappa }}) \cosh (2\eta {\hat{z}})\cosh ({\hat{\kappa }}{\hat{z}}), \end{aligned}$$
(A1)
$$\begin{aligned} U_{\eta }({\hat{z}})= & {} \sin (\eta {\hat{z}})\sin (2\eta {\hat{z}}) [U_{15}\sinh (2\eta {\hat{z}}) \sinh (\eta {\hat{z}})\nonumber \\&+\,U_{16}\cosh (2\eta {\hat{z}})\cosh (\eta {\hat{z}})] \nonumber \\&+\,\sin (\eta {\hat{z}})\cos (2\eta {\hat{z}}) [U_{17}\cosh (2\eta {\hat{z}})\sinh (\eta {\hat{z}})\nonumber \\&+\,U_{18}\sinh (2\eta {\hat{z}})\cosh (\eta {\hat{z}})]\nonumber \\&+\, \cos (\eta {\hat{z}})\sinh (\eta {\hat{z}}) [U_{19}\sin (2\eta {\hat{z}})\cosh (2\eta {\hat{z}})\nonumber \\&+\,U_{20}\cos (2\eta {\hat{z}}) \sinh (2\eta {\hat{z}})]\nonumber \\&+\,U_{21}\sin (\eta {\hat{z}}) \sin (2\eta {\hat{z}}) \cosh (\eta {\hat{z}})\nonumber \\&+\,U_{22}\sin (\eta {\hat{z}})\cos (2\eta {\hat{z}})\sinh (\eta {\hat{z}})\nonumber \\&+\,U_{23}\sin (\eta {\hat{z}})\sinh (\eta {\hat{z}}) \cosh (2\eta {\hat{z}})\nonumber \\&+\,U_{24}\cos (\eta {\hat{z}}) \sinh (\eta {\hat{z}}) \sinh (2\eta {\hat{z}}),\end{aligned}$$
(A2)
$$\begin{aligned} V_\kappa ({\hat{z}})= & {} \frac{{\hat{\kappa }}^2\eta ^2( C_1^2-C_2^2)}{6({\hat{\kappa }}^4+4\eta ^4)} {{\mathrm{sech}}}({\hat{\kappa }})\cosh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\,\frac{{\hat{\kappa }}^4{{\mathrm{sech}}}^2({\hat{\kappa }})}{6({\hat{\kappa }}^4+4\eta ^4)} [C_2\cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}})\nonumber \\&-\,C_1 \sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})] \nonumber \\&+\,V_1 {{\mathrm{sech}}}^3({\hat{\kappa }})\cosh ({\hat{\kappa }} {\hat{z}}) \sinh ^2({\hat{\kappa }} {\hat{z}})\nonumber \\&+\,V_2{{\mathrm{sech}}}^3({\hat{\kappa }})\cosh ^3({\hat{\kappa }} z)\nonumber \\&+\,V_3{{\mathrm{sech}}}^2({\hat{\kappa }}) \sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})\nonumber \\&+\,V_4 {{\mathrm{sech}}}^2({\hat{\kappa }}) \cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}})\nonumber \\&+\,{{\mathrm{sech}}}^2({\hat{\kappa }})\cosh (2 {\hat{\kappa }}{\hat{z}}) [V_5 \sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})\nonumber \\&+\,V_8 \cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}}) ] \nonumber \\&+\, {{\mathrm{sech}}}^2({\hat{\kappa }})\sinh (2 {\hat{\kappa }} {\hat{z}}) [V_6\sin (\eta {\hat{z}})\cosh (\eta {\hat{z}})\nonumber \\&+\, V_7\cos (\eta {\hat{z}}) \sinh (\eta {\hat{z}})]\nonumber \\&+\,V_9{{\mathrm{sech}}}({\hat{\kappa }}) \sin (2 \eta {\hat{z}}) \sinh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\,V_{10} {{\mathrm{sech}}}({\hat{\kappa }})\cos (2 \eta {\hat{z}}) \cosh ({\hat{\kappa }} {\hat{z}})\nonumber \\&+\,V_{11}{{\mathrm{sech}}}({\hat{\kappa }}) \sinh (2 \eta {\hat{z}}) \sinh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\,V_{12} {{\mathrm{sech}}}({\hat{\kappa }})\cosh (2\eta {\hat{z}})\cosh ({\hat{\kappa }}{\hat{z}})\nonumber \\&+\, {{\mathrm{sech}}}({\hat{\kappa }})\cosh ({\hat{\kappa }}{\hat{z}}) [V_{13}\sin (2\eta {\hat{z}}) \sinh (2\eta {\hat{z}})\nonumber \\&+\,V_{16} \cos (2\eta {\hat{z}})\cosh (2\eta {\hat{z}}) ]\nonumber \\&+\, {{\mathrm{sech}}}({\hat{\kappa }})\sinh ({\hat{\kappa }}{\hat{z}}) [V_{14}\sin (2 \eta {\hat{z}})\cosh (2\eta {\hat{z}})\nonumber \\&+\, V_{15}\cos (2\eta {\hat{z}}) \sinh (2\eta {\hat{z}})],\end{aligned}$$
(A3)
$$\begin{aligned} V_\eta ({\hat{z}})= & {} -\frac{5}{12} (C_1^2+C_2^2) [C_1\cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}})\nonumber \\&-\,C_2\sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})]\nonumber \\&+\,V_{17} \sin (\eta {\hat{z}}) \sin (2\eta {\hat{z}})\cosh (\eta {\hat{z}})\nonumber \\&+\,V_{18} \sin (\eta {\hat{z}})\cos (2\eta {\hat{z}}) \sinh (\eta {\hat{z}}) \nonumber \\&+\,V_{19}\cos (\eta {\hat{z}}) \sin (2\eta {\hat{z}}) \sinh (\eta {\hat{z}})\nonumber \\&+\,V_{20}\cos (\eta {\hat{z}}) \cos (2\eta {\hat{z}}) \cosh (\eta {\hat{z}})\nonumber \\&+\,V_{21}\sin (\eta {\hat{z}}) \sinh (\eta {\hat{z}})\cosh (2\eta {\hat{z}})\nonumber \\&+\,V_{22}\sin (\eta {\hat{z}})\cosh (\eta {\hat{z}})\sinh (2\eta {\hat{z}})\nonumber \\&+\,V_{23}\cos (\eta {\hat{z}}) \sinh (\eta {\hat{z}}) \sinh (2\eta {\hat{z}})\nonumber \\&+\,V_{24}\cos (\eta {\hat{z}}) \cosh (\eta {\hat{z}}) \cosh (2\eta {\hat{z}})\nonumber \\&+\,\sin (\eta {\hat{z}})\sin (2\eta {\hat{z}}) [V_{25}\sinh (\eta {\hat{z}}) \sinh (2\eta {\hat{z}})\nonumber \\&+\, V_{26}\cosh (\eta {\hat{z}})\cosh (2\eta {\hat{z}})]\nonumber \\&+\, \sin (\eta {\hat{z}})\cos (2\eta {\hat{z}}) [V_{27}\sinh (\eta {\hat{z}})\cosh (2\eta {\hat{z}})\nonumber \\&+\, V_{28}\cosh (\eta {\hat{z}})\sinh (2\eta {\hat{z}})]\nonumber \\&+\,\cos (\eta {\hat{z}})\sin (2\eta {\hat{z}}) [V_{29}\sinh (\eta {\hat{z}})\cosh (2\eta {\hat{z}})\nonumber \\&+\, V_{30}\cosh (\eta {\hat{z}}) \sinh (2\eta {\hat{z}})]\nonumber \\&+\,\cos (\eta {\hat{z}}) \cos (2\eta {\hat{z}}) [V_{31}\sinh (\eta {\hat{z}}) \sinh (2\eta {\hat{z}})\nonumber \\&+\, V_{32}\cosh (\eta {\hat{z}}) \cosh (2\eta {\hat{z}})], \end{aligned}$$
(A4)

where, with the introduction of \(\xi ={\hat{\kappa }}/\eta \) (for \(\eta \ne 0\)), the constants \(U_{1{-}24}\) and \(V_{1{-}32}\) are given as follows

$$\begin{aligned} U_1= & {} \frac{\xi ^8 \left( -27\xi ^8+56\xi ^4+16\right) }{2 \left( \xi ^4+4\right) ^3 \left( 81 \xi ^4+4\right) },\end{aligned}$$
(A5)
$$\begin{aligned} U_2= & {} \frac{\xi ^{12} \left( 9 \xi ^4-44\right) }{\left( \xi ^4+4\right) ^3 \left( 81 \xi ^4+4\right) },\end{aligned}$$
(A6)
$$\begin{aligned} U_3= & {} \frac{\xi ^6\left[ \left( 3 \xi ^4-10\right) C_1-\xi ^2 C_2\right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2}, \end{aligned}$$
(A7)
$$\begin{aligned} U_4= & {} \frac{\xi ^3\left[ 2 \xi ^2 \left( \xi ^4+1\right) \left( C_1+C_2\right) -\left( 3 \xi ^8-2 \xi ^4-4\right) \left( C_1-C_2\right) \right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2},\nonumber \\\end{aligned}$$
(A8)
$$\begin{aligned} U_5= & {} \frac{\xi ^3\left[ 2 \xi ^2 \left( \xi ^4+1\right) \left( C_1-C_2\right) +\left( 3 \xi ^8-2 \xi ^4-4\right) \left( C_1+C_2\right) \right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4 \right) ^2},\nonumber \\ \end{aligned}$$
(A9)
$$\begin{aligned} U_6= & {} -\frac{\xi ^6\left[ \xi ^2C_1+\left( 3\xi ^4-10\right) C_2\right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2},\end{aligned}$$
(A10)
$$\begin{aligned} U_7= & {} \frac{\xi ^2\left[ \left( \xi ^4+180\right) \left( C_1^2-C_2^2\right) +16\xi ^2C_1 C_2 \right] }{6 \left( \xi ^4+4\right) \left( \xi ^4+324\right) }, \end{aligned}$$
(A11)
$$\begin{aligned} U_8= & {} -\frac{\xi \left[ \left( \xi ^4\,{-}\,12\xi ^2+108\right) \left( C_1^2\,{-}\,C_2^2\right) +2 \left( \xi ^4+12\xi ^2+108\right) C_1 C_2\right] }{3 \left( \xi ^4+4\right) \left( \xi ^4+324\right) },\nonumber \\\end{aligned}$$
(A12)
$$\begin{aligned} U_9= & {} -\frac{\xi \left[ \left( \xi ^4+12\xi ^2+108\right) \left( C_1^2\,{-}\,C_2^2\right) \,{-}\,2 \left( \xi ^4\,{-}\,12 \xi ^2+108\right) C_1 C_2\right] }{3\left( \xi ^4+4\right) \left( \xi ^4+324\right) },\nonumber \\ \end{aligned}$$
(A13)
$$\begin{aligned} U_{10}= & {} \frac{\xi ^2\left[ 4 \xi ^2 \left( C_1^2-C_2^2\right) -\left( \xi ^4+180\right) C_1 C_2\right] }{3 \left( \xi ^4+4\right) \left( \xi ^4+324\right) },\end{aligned}$$
(A14)
$$\begin{aligned} U_{11}= & {} \left[ \frac{2\xi \left( \xi ^2+2\right) \left( \xi ^2+10\right) }{3 \left( \xi ^4+4\right) \left( \xi ^4+16\xi ^2+100\right) }\right] \left( C_1^2+C_2^2\right) ,\nonumber \\\end{aligned}$$
(A15)
$$\begin{aligned} U_{12}= & {} \left[ \frac{\xi ^2\left( \xi ^4+12\xi ^2+44\right) }{3\left( \xi ^4+4 \right) \left( \xi ^4+16\xi ^2+100\right) }\right] \left( C_1^2+C_2^2\right) ,\nonumber \\\end{aligned}$$
(A16)
$$\begin{aligned} U_{13}= & {} \left[ \frac{2\xi \left( \xi ^2-10\right) \left( \xi ^2-2\right) }{3\left( \xi ^4+4\right) \left( \xi ^4-16\xi ^2+100\right) }\right] \left( C_1^2+C_2^2\right) ,\nonumber \\\end{aligned}$$
(A17)
$$\begin{aligned} U_{14}= & {} -\left[ \frac{\xi ^2\left( \xi ^4-12\xi ^2+44\right) }{3\left( \xi ^4+4 \right) \left( \xi ^4-16\xi ^2+100\right) }\right] \left( C_1^2+C_2^2\right) ,\nonumber \\\end{aligned}$$
(A18)
$$\begin{aligned} U_{15}= & {} -C_2 \left( C_1^2+C_2^2\right) /40,\end{aligned}$$
(A19)
$$\begin{aligned} U_{16}= & {} \left( -4 C_1^3+3 C_1^2 C_2-4 C_1 C_2^2+3 C_2^3\right) /240,\end{aligned}$$
(A20)
$$\begin{aligned} U_{17}= & {} -C_1 \left( C_1^2+C_2^2\right) /40,\end{aligned}$$
(A21)
$$\begin{aligned} U_{18}= & {} \left( 3 C_1^3+4 C_1^2 C_2+3 C_1 C_2^2+4 C_2^3\right) /240,\end{aligned}$$
(A22)
$$\begin{aligned} U_{19}= & {} \left( 3 C_1^3-4 C_1^2 C_2+3 C_1 C_2^2-4 C_2^3\right) /240,\end{aligned}$$
(A23)
$$\begin{aligned} U_{20}= & {} -\left( 4 C_1^3+3 C_1^2 C_2+4 C_1 C_2^2+3 C_2^3\right) /240,\end{aligned}$$
(A24)
$$\begin{aligned} U_{21}= & {} \left( 4 C_1^3-3 C_1^2 C_2+4 C_1 C_2^2-3 C_2^3\right) /48,\end{aligned}$$
(A25)
$$\begin{aligned} U_{22}= & {} -\left( 3 C_1^3+4 C_1^2 C_2+3 C_1 C_2^2+4 C_2^3\right) /48,\end{aligned}$$
(A26)
$$\begin{aligned} U_{23}= & {} -\left( 3 C_1^3-4 C_1^2 C_2+3 C_1 C_2^2-4 C_2^3\right) /48,\end{aligned}$$
(A27)
$$\begin{aligned} U_{24}= & {} \left( 4 C_1^3+3 C_1^2 C_2+4 C_1 C_2^2+3 C_2^3\right) /48,\end{aligned}$$
(A28)
$$\begin{aligned} V_1= & {} -\frac{16 \xi ^{10} \left( 3 \xi ^4+2\right) }{\left( \xi ^4+4\right) ^3 \left( 81\xi ^4+4\right) },\end{aligned}$$
(A29)
$$\begin{aligned} V_2= & {} \frac{2\xi ^{10}\left( 19\xi ^4-4\right) }{\left( \xi ^4+4\right) ^3 \left( 81\xi ^4+4\right) },\end{aligned}$$
(A30)
$$\begin{aligned} V_3= & {} \frac{\xi ^4 \left[ \left( 3\xi ^4+4\right) C_1-4\xi ^2 C_2\right] }{12 \left( \xi ^4+4\right) ^2},\end{aligned}$$
(A31)
$$\begin{aligned} V_4= & {} -\frac{\xi ^4 \left[ 4\xi ^2C_1+\left( 3\xi ^4+4\right) C_2\right] }{12 \left( \xi ^4+4\right) ^2},\end{aligned}$$
(A32)
$$\begin{aligned} V_5= & {} -\frac{\xi ^4\left[ \left( -9 \xi ^4+8\right) C_1+\xi ^2 \left( \xi ^4+6 \right) C_2\right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2}, \end{aligned}$$
(A33)
$$\begin{aligned} V_6= & {} \frac{\xi ^3\left[ \left( \xi ^8+6\xi ^4-4\right) \left( C_1+C_2\right) -2\xi ^2\left( 3\xi ^4 -1\right) \left( C_1-C_2\right) \right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2},\nonumber \\\end{aligned}$$
(A34)
$$\begin{aligned} V_7= & {} \frac{\xi ^3\left[ \left( \xi ^8+6\xi ^4-4\right) \left( C_1-C_2\right) +2\xi ^2\left( 3\xi ^4 -1\right) \left( C_1+C_2\right) \right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2},\nonumber \\ \end{aligned}$$
(A35)
$$\begin{aligned} V_8= & {} -\frac{\xi ^4\left[ \xi ^2\left( \xi ^4+6\right) C_1+\left( 9 \xi ^4-8 \right) C_2\right] }{24 \left( \xi ^4-1\right) \left( \xi ^4+4\right) ^2},\end{aligned}$$
(A36)
$$\begin{aligned} V_9= & {} \frac{16\xi \left( \xi ^2+5\right) \left( C_1^2+C_2^2\right) }{3 \left( \xi ^4+4\right) \left( \xi ^4+16\xi ^2+100\right) },\end{aligned}$$
(A37)
$$\begin{aligned} V_{10}= & {} \frac{4\xi ^2\left( \xi ^2+2\right) \left( C_1^2+C_2^2\right) }{3 \left( \xi ^4+4\right) \left( \xi ^4+16\xi ^2+100\right) },\end{aligned}$$
(A38)
$$\begin{aligned} V_{11}= & {} \frac{16\xi \left( \xi ^2-5\right) \left( C_1^2+C_2^2\right) }{3\left( \xi ^4+4\right) \left( \xi ^4-16 \xi ^2 +100\right) },\end{aligned}$$
(A39)
$$\begin{aligned} V_{12}= & {} -\frac{4\xi ^2\left( \xi ^2-2\right) \left( C_1^2+C_2^2\right) }{3\left( \xi ^4+4\right) \left( \xi ^4-16 \xi ^2 +100\right) },\end{aligned}$$
(A40)
$$\begin{aligned} V_{13}= & {} -\frac{\xi ^2\left[ -4\xi ^2 (C_1^2-C_2^2)+\left( \xi ^4+180\right) C_1 C_2\right] }{3 \left( \xi ^4+4\right) \left( \xi ^4+324\right) }, \end{aligned}$$
(A41)
$$\begin{aligned} V_{14}= & {} -\frac{\xi \left[ \left( \xi ^4+12\xi ^2+108\right) (C_1^2-C_2^2)-2 \left( \xi ^4-12\xi ^2+108\right) C_1 C_2\right] }{3 \left( \xi ^4+4\right) \left( \xi ^4+324\right) },\nonumber \\\end{aligned}$$
(A42)
$$\begin{aligned} V_{15}= & {} \frac{\xi \left[ \left( \xi ^4-12\xi ^2+108\right) (C_1^2-C_2^2)+2 \left( \xi ^4+12\xi ^2+108\right) C_1 C_2\right] }{3 \left( \xi ^4+4\right) \left( \xi ^4+324\right) },\nonumber \\ \end{aligned}$$
(A43)
$$\begin{aligned} V_{16}= & {} -\frac{\xi ^2\left[ \left( \xi ^4+180\right) (C_1^2-C_2^2)+16 \xi ^2 C_1 C_2\right] }{6 \left( \xi ^4+4\right) \left( \xi ^4+324\right) },\end{aligned}$$
(A44)
$$\begin{aligned} V_{17}= & {} \left( 2 C_1^3-5 C_1^2 C_2+2 C_1 C_2^2-5 C_2^3\right) /24,\end{aligned}$$
(A45)
$$\begin{aligned} V_{18}= & {} \left( C_1^3+5 C_1^2 C_2+C_1 C_2^2+5 C_2^3\right) /24, \end{aligned}$$
(A46)
$$\begin{aligned} V_{19}= & {} -\left( 6 C_1^3+7 C_1^2 C_2+6 C_1 C_2^2+7 C_2^3\right) /48, \end{aligned}$$
(A47)
$$\begin{aligned} V_{20}= & {} \left( 7 C_1^3-6 C_1^2 C_2+7 C_1 C_2^2-6 C_2^3\right) /48, \end{aligned}$$
(A48)
$$\begin{aligned} V_{21}= & {} -\left( C_1^3-5 C_1^2 C_2+C_1 C_2^2-5 C_2^3\right) /24, \end{aligned}$$
(A49)
$$\begin{aligned} V_{22}= & {} \left( 6 C_1^3-7 C_1^2 C_2+6 C_1 C_2^2-7 C_2^3\right) /48, \end{aligned}$$
(A50)
$$\begin{aligned} V_{23}= & {} -\left( 2 C_1^3+5 C_1^2 C_2+2 C_1 C_2^2+5 C_2^3\right) /24, \end{aligned}$$
(A51)
$$\begin{aligned} V_{24}= & {} \left( 7 C_1^3+6 C_1^2 C_2+7 C_1 C_2^2+6 C_2^3\right) /48, \end{aligned}$$
(A52)
$$\begin{aligned} V_{25}= & {} C_1 \left( C_1^2+C_2^2\right) /60, \end{aligned}$$
(A53)
$$\begin{aligned} V_{26}= & {} -\left( 7 C_1^3-4 C_1^2 C_2+7 C_1 C_2^2-4 C_2^3\right) /240, \end{aligned}$$
(A54)
$$\begin{aligned} V_{27}= & {} C_2 \left( C_1^2+C_2^2\right) /15, \end{aligned}$$
(A55)
$$\begin{aligned} V_{28}= & {} \left( 4 C_1^3-13 C_1^2 C_2+4 C_1 C_2^2-13 C_2^3\right) /240, \end{aligned}$$
(A56)
$$\begin{aligned} V_{29}= & {} -\left( 4 C_1^3+13 C_1^2 C_2+4 C_1 C_2^2+13 C_2^3\right) /240, \end{aligned}$$
(A57)
$$\begin{aligned} V_{30}= & {} C_2 \left( C_1^2+C_2^2\right) /24, \end{aligned}$$
(A58)
$$\begin{aligned} V_{31}= & {} \left( 7 C_1^3+4 C_1^2 C_2+7 C_1 C_2^2+4 C_2^3\right) /240, \end{aligned}$$
(A59)
$$\begin{aligned} V_{32}= & {} -C_1 \left( C_1^2+C_2^2\right) /24, \end{aligned}$$
(A60)

in which \(C_1\) and \(C_2\) are given in Eqs. (44) and (45), respectively.

Setting \({\hat{z}}=1\) in the functions \(U_\kappa ({\hat{z}}), U_\eta ({\hat{z}}), V_\kappa ({\hat{z}})\), and \(V_\eta ({\hat{z}})\), we obtain the parameters \(M_{\kappa , \eta }\) and \(N_{\kappa , \eta }\) as follows

$$\begin{aligned} M_\kappa= & {} \frac{\xi ^2 C_1C_2}{3\left( \xi ^4+4\right) } +U_1\tanh ^2({\hat{\kappa }} )+U_2\nonumber \\&+\,U_3 \left[ 1+\tanh ^2({\hat{\kappa }})\right] \sin (\eta ) \sinh (\eta ) \nonumber \\&+\, 2\tanh ({\hat{\kappa }}) \left[ U_4\sin (\eta ) \cosh (\eta ) + U_5\cos (\eta ) \sinh (\eta )\right] \nonumber \\&+\,U_6\left[ 1+\tanh ^2({\hat{\kappa }})\right] \cos (\eta ) \cosh (\eta )\nonumber \\&+\, U_7 \sin (2 \eta ) \sinh (2 \eta )\nonumber \\&+\, \tanh ({\hat{\kappa }})[U_8\sin (2 \eta ) \cosh (2 \eta )\nonumber \\&+\, U_9\cos (2 \eta ) \sinh (2 \eta )] \nonumber \\&+\, U_{10}\cos (2 \eta ) \cosh (2 \eta )\nonumber \\&+\,U_{11}\tanh ({\hat{\kappa }} )\sin (2 \eta ) +U_{12} \cos (2 \eta )\nonumber \\&+\, U_{13}\tanh ({\hat{\kappa }})\sinh (2\eta )+U_{14}\cosh (2 \eta ),\end{aligned}$$
(A61)
$$\begin{aligned} M_\eta= & {} \sin (\eta ) \sin (2 \eta )[U_{15} \sinh (\eta ) \sinh (2 \eta )\nonumber \\&+\,U_{16} \cosh (\eta ) \cosh (2 \eta )]\nonumber \\&+\, \sin (\eta ) \cos (2 \eta )[U_{17} \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, U_{18} \cosh (\eta ) \sinh (2 \eta )]\nonumber \\&+\, \cos (\eta )[U_{19} \sin (2 \eta ) \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, U_{20} \cos (2 \eta ) \sinh (\eta ) \sinh (2 \eta )]\nonumber \\&+\, U_{21} \sin (\eta ) \sin (2 \eta ) \cosh (\eta )\nonumber \\&+\, U_{22} \sin (\eta ) \cos (2 \eta ) \sinh (\eta )\nonumber \\&+\, U_{23} \sin (\eta ) \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, U_{24} \cos (\eta ) \sinh (\eta ) \sinh (2 \eta ),\end{aligned}$$
(A62)
$$\begin{aligned} N_\kappa= & {} \frac{\xi ^2\left( C_1^2-C_2^2\right) }{6\left( \xi ^4+4\right) }\nonumber \\&+\,\frac{\xi ^4{{\mathrm{sech}}}^2({\hat{\kappa }})}{6(\xi ^4+4)} [C_2 \cos (\eta ) \cosh (\eta )-C_1 \sin (\eta ) \sinh (\eta )]\nonumber \\&+\,V_1\tanh ^2({\hat{\kappa }})+ V_2+{{\mathrm{sech}}}^2({\hat{\kappa }}) [V_3\sin (\eta )\sinh (\eta ) \nonumber \\&+\, V_4\cos (\eta )\cosh (\eta )]\nonumber \\&+\,V_5[1+\tanh ^2({\hat{\kappa }})]\sin (\eta ) \sinh (\eta )\nonumber \\&+\,2 V_6\tanh ({\hat{\kappa }}) \sin (\eta ) \cosh (\eta ) \nonumber \\&+\,2V_7\tanh ({\hat{\kappa }})\cos (\eta )\sinh (\eta ) \nonumber \\&+\, V_8[1+\tanh ^2({\hat{\kappa }})]\cos (\eta ) \cosh (\eta ) \nonumber \\&+\, V_9\tanh ({\hat{\kappa }})\sin (2\eta )+V_{10}\cos (2\eta )\nonumber \\&+\,V_{11}\tanh ({\hat{\kappa }})\sinh (2\eta ) +V_{12}\cosh (2\eta ) \nonumber \\&+\, V_{13}\sin (2 \eta ) \sinh (2 \eta )+ \tanh ({\hat{\kappa }}) [V_{14}\sin (2 \eta ) \cosh (2 \eta )\nonumber \\&+\, V_{15}\cos (2 \eta ) \sinh (2 \eta )] \nonumber \\&+\, V_{16}\cos (2 \eta ) \cosh (2 \eta ),\end{aligned}$$
(A63)
$$\begin{aligned} N_\eta= & {} \frac{5}{12} (C_1^2+C_2^2)[C_2 \sin (\eta ) \sinh (\eta )\nonumber \\&-\,C_1 \cos (\eta ) \cosh (\eta )]\nonumber \\&+\,\sin (\eta )[V_{17}\cosh (\eta ) \sin (2 \eta )\nonumber \\&+\, V_{18}\sinh (\eta ) \cos (2 \eta )]\nonumber \\&+\, \cos (\eta )[V_{19}\sinh (\eta )\sin (2 \eta )\nonumber \\&+\, V_{20}\cosh (\eta ) \cos (2 \eta )] \nonumber \\&+\,\sin (\eta ) [V_{21} \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, V_{22} \sinh (2 \eta ) \cosh (\eta )]\nonumber \\&+\,\cos (\eta ) [V_{23} \sinh (\eta ) \sinh (2 \eta )\nonumber \\&+\, V_{24} \cosh (\eta ) \cosh (2 \eta )]\nonumber \\&+\, \sin (\eta ) \sin (2 \eta ) [V_{25} \sinh (\eta ) \sinh (2 \eta )\nonumber \\&+\, V_{26} \cosh (\eta ) \cosh (2 \eta )]\nonumber \\&+\, \sin (\eta ) \cos (2 \eta ) [V_{27} \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, V_{28} \sinh (2 \eta ) \cosh (\eta )]\nonumber \\&+\, \sin (2 \eta ) \cos (\eta ) [V_{29} \sinh (\eta ) \cosh (2 \eta )\nonumber \\&+\, V_{30} \sinh (2 \eta ) \cosh (\eta )]\nonumber \\&+\, \cos (\eta ) \cos (2 \eta ) [V_{31} \sinh (\eta ) \sinh (2 \eta )\nonumber \\&+\, V_{32} \cosh (\eta ) \cosh (2 \eta )]. \end{aligned}$$
(A64)

In Eq. (65), the two coefficients \(K_{\kappa , \eta }\) are

$$\begin{aligned} K_\kappa= & {} -\frac{2\xi (C_1^2-C_2^2)}{3 (\xi ^4+4)} \tanh ({\hat{\kappa }} )\nonumber \\&+\,\frac{4 \xi ^9 (4-119 \xi ^4)}{3 (\xi ^4+4)^3 (81 \xi ^4+4)} {{\mathrm{sech}}}^2({\hat{\kappa }} ) \tanh ({\hat{\kappa }} )\nonumber \\&+\,\frac{20 \xi ^9}{3 (\xi ^4+4)^2 (81 \xi ^4+4)} {{\mathrm{sech}}}^2({\hat{\kappa }} )\tanh ({\hat{\kappa }} )\cosh (2 {\hat{\kappa }} )\nonumber \\&+\,\frac{\xi ^4 \text {sech}^2({\hat{\kappa }} )}{6 (\xi ^4+4)} [\sin (\eta ) \cosh (\eta )(C_1-C_2)\nonumber \\&-\cos (\eta ) \sinh (\eta )(C_1+C_2)]\nonumber \\&-2{{\mathrm{sech}}}^2({\hat{\kappa }} )[\sin (\eta ) \cosh (\eta ) (V_3+V_4)\nonumber \\&-\cos (\eta )\sinh (\eta ) (V_3-V_4)]\nonumber \\&+\,\frac{\text {sech}^2({\hat{\kappa }})}{2\xi ^2-2\xi +1} [K_1 \sin (\eta ) \cosh (\eta -2 {\hat{\kappa }} )\nonumber \\&+\,K_2 \cos (\eta ) \sinh (\eta -2 {\hat{\kappa }} )]\nonumber \\&+\,\frac{\text {sech}^2({\hat{\kappa }} )}{2 \xi ^2+2\xi +1} [-K_3 \sin (\eta ) \cosh (\eta +2 {\hat{\kappa }} )\nonumber \\&+\,K_4 \cos (\eta ) \sinh (\eta +2 {\hat{\kappa }} )]\nonumber \\&+\,\frac{4}{4+\xi ^2} [-K_5 \sin (2 \eta ) +K_6 \tanh ({\hat{\kappa }})\cos (2 \eta ) ]\nonumber \\&+\,\frac{4}{4-\xi ^2} [K_7 \sinh (2\eta ) +K_8\tanh ({\hat{\kappa }})\cosh (2 \eta ) ]\nonumber \\&-\frac{4\sin (2 \eta )}{64+\xi ^4} [K_9 \tanh ({\hat{\kappa }} )\sinh (2 \eta )+K_{10} \cosh (2 \eta )]\nonumber \\&+\,\frac{4 \cos (2 \eta )}{64+\xi ^4} [K_{11} \sinh (2 \eta )+K_{12}\tanh ({\hat{\kappa }} ) \cosh (2 \eta ) ],\nonumber \\\end{aligned}$$
(A65)
$$\begin{aligned} K_\eta= & {} \frac{5}{6} (C_1^2+C_2^2) [(C_1-C_2) \sin (\eta ) \cosh (\eta )\nonumber \\&+\,(C_1+C_2) \cos (\eta ) \sinh (\eta )]\nonumber \\&+\,\frac{K_{13}}{5} \sin (\eta ) \cosh (3 \eta )+\frac{K_{14}}{5} \sin (3 \eta ) \cosh (\eta )\nonumber \\&+\,\frac{K_{15}}{5} \cos (\eta ) \sinh (3 \eta )\nonumber \\&+\,\frac{K_{16}}{5} \cos (3 \eta ) \sinh (\eta )+K_{17} \sin (\eta ) \cosh (\eta )\nonumber \\&+\,K_{18} \cos (\eta ) \sinh (\eta )\nonumber \\&+\, \frac{K_{19}}{2} \sin (\eta ) \cosh (\eta )+\frac{K_{20}}{2} \cos (\eta ) \sinh (\eta )\nonumber \\&+\, \frac{K_{21}}{6} \sin (3 \eta ) \cosh (3 \eta )\nonumber \\&+\,\frac{K_{22}}{6} \cos (3 \eta ) \sinh (3 \eta )+ \frac{K_{23}}{10} \sin (\eta ) \cosh (3 \eta )\nonumber \\&+\,\frac{K_{24}}{10} \sin (3 \eta ) \cosh (\eta )\nonumber \\&+\,\frac{K_{25}}{10} \cos (\eta ) \sinh (3 \eta )+\frac{K_{26}}{10} \cos (3 \eta ) \sinh (\eta ),\nonumber \\ \end{aligned}$$
(A66)

in which \(K_{1{-}26}\) are given as follows

$$\begin{aligned} K_1= & {} (1-2\xi )(-V_5+V_6)+V_7-V_8,\end{aligned}$$
(A67)
$$\begin{aligned} K_2= & {} V_5-V_6+(1-2\xi )(V_7-V_8),\end{aligned}$$
(A68)
$$\begin{aligned} K_3= & {} (1+2\xi )(V_5+V_6)+V_7+V_8, \end{aligned}$$
(A69)
$$\begin{aligned} K_4= & {} V_5+V_6-(1+2\xi )(V_7+V_8), \end{aligned}$$
(A70)
$$\begin{aligned} K_5= & {} \xi V_9+2V_{10}, \end{aligned}$$
(A71)
$$\begin{aligned} K_6= & {} 2V_9-\xi V_{10}, \end{aligned}$$
(A72)
$$\begin{aligned} K_7= & {} \xi V_{11}-2V_{12}, \end{aligned}$$
(A73)
$$\begin{aligned} K_8= & {} -2V_{11}+\xi V_{12} ,\end{aligned}$$
(A74)
$$\begin{aligned} K_9= & {} \xi ^3V_{13}+(16-2\xi ^2)V_{14}\nonumber \\&+\,(16+2\xi ^2)V_{15}-8\xi V_{16}, \end{aligned}$$
(A75)
$$\begin{aligned} K_{10}= & {} (16-2\xi ^2)V_{13}+\xi ^3V_{14} -8\xi V_{15}\nonumber \\&+\,(16+2\xi ^2)V_{16}, \end{aligned}$$
(A76)
$$\begin{aligned} K_{11}= & {} (16+2\xi ^2)V_{13}-8\xi V_{14} -\xi ^3 V_{15}\nonumber \\&-(16-2\xi ^2)V_{16}, \end{aligned}$$
(A77)
$$\begin{aligned} K_{12}= & {} -8\xi V_{13}+(16+2\xi ^2)V_{14}\nonumber \\&+\,(-16+2\xi ^2)V_{15}-\xi ^3V_{16}, \end{aligned}$$
(A78)
$$\begin{aligned} K_{13}= & {} -3V_{21}-3V_{22}-V_{23}-V_{24}, \end{aligned}$$
(A79)
$$\begin{aligned} K_{14}= & {} 3V_{17}-V_{18}-V_{19}-3V_{20}, \end{aligned}$$
(A80)
$$\begin{aligned} K_{15}= & {} V_{21}+V_{22}-3V_{23}-3V_{24}, \end{aligned}$$
(A81)
$$\begin{aligned} K_{16}= & {} V_{17}+3V_{18}+3V_{19}-V_{20}, \end{aligned}$$
(A82)
$$\begin{aligned} K_{17}= & {} -V_{17}+V_{18}-V_{19}-V_{20}\nonumber \\&+\,V_{21}-V_{22}+V_{23}-V_{24}, \end{aligned}$$
(A83)
$$\begin{aligned} K_{18}= & {} -V_{17}-V_{18}+V_{19}-V_{20}-V_{21}\nonumber \\&+\,V_{22}+V_{23}-V_{24}, \end{aligned}$$
(A84)
$$\begin{aligned} K_{19}= & {} V_{25}-V_{26}-V_{27}+V_{28}+V_{29}\nonumber \\&-V_{30}+V_{31}-V_{32} ,\end{aligned}$$
(A85)
$$\begin{aligned} K_{20}= & {} V_{25}-V_{26}+V_{27}-V_{28}-V_{29}\nonumber \\&+\,V_{30}+V_{31}-V_{32}, \end{aligned}$$
(A86)
$$\begin{aligned} K_{21}= & {} V_{25}+V_{26}-V_{27}-V_{28}-V_{29}\nonumber \\&-V_{30}-V_{31}-V_{32}, \end{aligned}$$
(A87)
$$\begin{aligned} K_{22}= & {} V_{25}+V_{26}+V_{27}+V_{28}+V_{29}\nonumber \\&+\,V_{30}-V_{31}-V_{32}, \end{aligned}$$
(A88)
$$\begin{aligned} K_{23}= & {} -V_{25}-V_{26}+3V_{27}+3V_{28}-3V_{29}\nonumber \\&-3V_{30}-V_{31}-V_{32}, \end{aligned}$$
(A89)
$$\begin{aligned} K_{24}= & {} -3V_{25}+3V_{26}+V_{27}-V_{28}+V_{29}\nonumber \\&-V_{30}+3V_{31}-3V_{32}, \end{aligned}$$
(A90)
$$\begin{aligned} K_{25}= & {} -3V_{25}-3V_{26}-V_{27}-V_{28}+V_{29}\nonumber \\&+\,V_{30}-3V_{31}-3V_{32}, \end{aligned}$$
(A91)
$$\begin{aligned} K_{26}= & {} -V_{25}+V_{26}-3V_{27}+3V_{28}-3V_{29}\nonumber \\&+\,3V_{30}+V_{31}-V_{32}. \end{aligned}$$
(A92)

Under the assumptions of \({\hat{\kappa }}\gg 1\) and \(\eta \gg 1\), the four functions \(U_{\kappa , \eta }({\hat{z}})\) and \(V_{\kappa , \eta }({\hat{z}})\) in Eqs. (A1)–(A4) can be simplified and expressed in terms of the variable \(z^*\) as follows

$$\begin{aligned} U_\kappa ^*(z^*)= & {} -\frac{8 \xi ^3 \left( \xi ^5+10 \xi ^4+46 \xi ^3+84 \xi ^2+16 \xi -24\right) \mathrm{e}^{-(\xi +2) z^*}}{3 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^2+6 \xi +10\right) \left( \xi ^2+6 \xi +18\right) }\nonumber \\&+\,\frac{\xi ^4 \left( 6 \xi ^6+9 \xi ^5+9 \xi ^4+6 \xi ^3-4 \xi ^2-4 \xi -4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) }\nonumber \\&\times \,\mathrm{e}^{-(2 \xi +1) z^*} \sin \left( z^*\right) \nonumber \\&-\frac{\xi ^4 \left( 3 \xi ^5+5 \xi ^4+8 \xi ^3+8 \xi ^2+8 \xi +4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) }\nonumber \\&\times \, \mathrm{e}^{-(2 \xi +1) z^*} \cos \left( z^*\right) +\frac{\xi ^8 \left( 4-9 \xi ^4\right) \mathrm{e}^{-3\xi z^*}}{2 \left( \xi ^4+4\right) ^2 \left( 81 \xi ^4+4\right) }, \end{aligned}$$
(A93)
$$\begin{aligned} U_{\eta }^*(z^*)= & {} \frac{8 \xi ^3 \left[ (7 \xi -6) \sin \left( z^*\right) +(\xi -8) \cos \left( z^*\right) \right] }{15 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) } \mathrm{e}^{-3 z^*}, \end{aligned}$$
(A94)
$$\begin{aligned} V_\kappa ^*(z^*)= & {} -\frac{10 \xi ^{10} \mathrm{e}^{-3 \xi z^*}}{\left( \xi ^4+4\right) ^2 \left( 81 \xi ^4+4\right) }\nonumber \\&-\frac{64 \xi ^3 (\xi +2) \mathrm{e}^{-(\xi +2) z^*}}{3 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^2+6 \xi +10\right) }\nonumber \\&+\,\frac{\xi ^4 \left( \xi ^5+15 \xi ^4+20 \xi ^3+20 \xi ^2+16 \xi +4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) }\nonumber \\&\times \, \mathrm{e}^{-(2 \xi +1) z^*} \sin \left( z^*\right) \nonumber \\&+\,\frac{\xi ^4 \left( 2 \xi ^6+3 \xi ^5+3 \xi ^4-2 \xi ^3-4 \xi -4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) }\nonumber \\&\times \, \mathrm{e}^{-(2 \xi +1) z^*} \cos \left( z^*\right) \nonumber \\&+\,\frac{8 \xi ^3 \left[ 6 (\xi +2) \cos \left( 2z^*\right) -\left( \xi ^3+4 \xi ^2+8 \xi -12\right) \sin \left( 2 z^*\right) \right] }{3 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^2+6 \xi +18\right) }\nonumber \\&\times \, \mathrm{e}^{-(\xi +2) z^*}, \end{aligned}$$
(A95)
$$\begin{aligned} V_\eta ^*(z^*)= & {} \frac{8 \xi ^3 \left[ (7 \xi -6) \cos \left( z^*\right) -(\xi -8) \sin \left( z^*\right) \right] }{15 \left( \xi ^2+2 \xi +2\right) \left( \xi ^4+4\right) ^2} \mathrm{e}^{-3 z^*}. \end{aligned}$$
(A96)

Similarly, the six coefficients \(M_{\kappa ,\eta }^*, N_{\kappa , \eta }^*\), and \(K_{\kappa , \eta }^*\) corresponding to \(M_{\kappa , \eta }, N_{\kappa , \eta }\), and \(K_{\kappa , \eta }\) reduce to

$$\begin{aligned} M_\kappa ^*= & {} \frac{\xi ^8 \left( 4-9 \xi ^4\right) }{2 \left( \xi ^4+4\right) ^2 \left( 81 \xi ^4+4\right) }\nonumber \\&-\frac{\xi ^4 \left( 3 \xi ^5+5 \xi ^4+8 \xi ^3+8 \xi ^2+8 \xi +4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) }\nonumber \\&-\frac{8 \xi ^3 \left( \xi ^5+10 \xi ^4+46 \xi ^3+84 \xi ^2+16 \xi -24\right) }{3 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^2+6 \xi +10\right) \left( \xi ^2+6 \xi +18\right) }, \nonumber \\\end{aligned}$$
(A97)
$$\begin{aligned} M_\eta ^*= & {} \frac{8\xi ^3(\xi -8)}{15 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) }, \end{aligned}$$
(A98)
$$\begin{aligned} N_\kappa ^*= & {} -\frac{10 \xi ^{10}}{\left( \xi ^4+4\right) ^2 \left( 81 \xi ^4+4\right) }\nonumber \\&+\,\frac{\xi ^4 \left( 2 \xi ^6+3 \xi ^5+3 \xi ^4-2 \xi ^3-4 \xi -4\right) }{6 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^3+\xi ^2+\xi +1\right) } \nonumber \\&-\frac{16 \xi ^3 (\xi +2) \left( \xi ^2+6 \xi +42\right) }{3 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) \left( \xi ^2+6 \xi +10\right) \left( \xi ^2+6 \xi +18\right) }, \nonumber \\&\end{aligned}$$
(A99)
$$\begin{aligned} N_\eta ^*= & {} \frac{8 \xi ^3 (7 \xi -6)}{15 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) }, \end{aligned}$$
(A100)
$$\begin{aligned} K_\kappa ^*= & {} \left[ \frac{32 \left( \xi ^3+7 \xi ^2+16 \xi +42\right) }{\left( \xi ^2+6 \xi +10\right) \left( \xi ^2+6 \xi +18\right) }\right. \nonumber \\&\left. -\frac{\xi ^2 \left( \xi ^4+\xi ^3+\xi ^2+2 \xi +2\right) }{\xi ^3+\xi ^2+\xi +1}\right] \frac{2\xi ^3 }{3 \left( \xi ^2+2 \xi +2\right) \left( \xi ^4+4\right) ^2}\nonumber \\&+\, \frac{40 \xi ^9}{3 \left( \xi ^4+4\right) ^2 \left( 81 \xi ^4+4\right) }, \end{aligned}$$
(A101)
$$\begin{aligned} K_\eta ^*= & {} -\frac{32 \xi ^3 (2 \xi -1)}{15 \left( \xi ^4+4\right) ^2 \left( \xi ^2+2 \xi +2\right) }. \end{aligned}$$
(A102)

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Qi, C., Ng, CO. Rotating electroosmotic flow of an Eyring fluid. Acta Mech. Sin. 33, 295–315 (2017). https://doi.org/10.1007/s10409-016-0629-4

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