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The effects of stress jump conditions on the hydrodynamic permeability of filtration processes using effective medium approximation

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Abstract

This work concerns the analytical study of a uniform steady flow of an incompressible, viscous and electrically conducting fluid across an array of parallel porous cylindrical fibres subjected to a transverse uniform magnetic field with the assumption that the stress jump conditions for tangential stresses at the fluid–porous interface are applied. The effective medium approximation has been used for predicting the overall bed permeability (OBP) of fibrous filtration beds. The mathematically governed equations are formulated as Stokes’ equation in the fluid region, while Brinkman’s equation is in the porous regions. These equations have been solved using the stream function method, and the corresponding flow field expressions within each region are obtained. The derivation of the drag force acting on the surface of the porous circular fibre has been presented. The effect of various physical parameters on the velocity profiles is graphically illustrated and discussed. Interestingly, the OBP is considerably increased by implementing the jump boundary conditions, and this has significant applications in filtration systems design and may have physical implications in biological systems. This model is in agreement with the published results of the existing models.

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References

  1. H C Brinkman, Flow Turbulence Combust. 1(1), 27 (1949)

    Article  Google Scholar 

  2. J Happel, AIChE J. 4(2), 197 (1958)

    Article  Google Scholar 

  3. S Kuwabara, J. Phys. Soc. Jpn. 14(4), 527 (1959)

    Article  ADS  Google Scholar 

  4. S Deo, Sadhana 29(4), 381 (2004)

    Article  Google Scholar 

  5. S Deo, P K Yadav and A Tiwari, Appl. Math. Model. 34(5), 1329 (2010)

    Article  MathSciNet  Google Scholar 

  6. Y Li and C W Park, Chem. Eng. Sci. 54(5), 633 (1999)

    Article  Google Scholar 

  7. B Albusairi and J T Hsu, Chem. Eng. J. 100(1–3), 79 (2004)

    Article  Google Scholar 

  8. G P Raja Sekhar, Appl. Anal. 89(6), 833 (2010)

    Article  MathSciNet  Google Scholar 

  9. J A Ochoa-Tapia and S Whitaker, Int. J. Heat Mass Transf. 38(14), 2635 (1995)

    Article  Google Scholar 

  10. A V Kuznetsov, Appl. Sci. Res. 56, 53 (1996)

    Article  ADS  Google Scholar 

  11. A Bhattacharyya and G P Raja Sekhar, Chem. Eng. Sci. 59, 4481 (2004)

    Article  Google Scholar 

  12. A Bhattacharyya and G P Raja Sekhar, Z. Angew. Math. Phys. 56, 475 (2005)

    Article  MathSciNet  Google Scholar 

  13. M K Partha, P V Murthy and G P Raja Sekhar, J. Eng. Mech. 131, 1291 (2005)

    Article  Google Scholar 

  14. M K Partha, P V Murthy and G P Raja Sekhar, J. Porous Media 9, 745 (2006)

    Article  Google Scholar 

  15. P K Yadav, A Tiwari, S Deo, A Filippov and S Vasin, Acta Mech. 15(1), 193 (2010)

    Article  Google Scholar 

  16. S Deo, A Filippov, A Tiwari and S Vasin, Adv. Colloid Interface Sci. 164(1–2), 21 (2011)

    Article  Google Scholar 

  17. S Vasin, E Sherysheva and A Filippov, Colloid J. 73(2), 167 (2011)

    Article  Google Scholar 

  18. M Elbehairy, Y Gamiel, S T Assar and M Kamel El Sayed, J. Eng. Res. 6(3), 76 (2022)

    Google Scholar 

  19. E I Saad, Meccanica 48(7), 1747 (2013)

    Article  MathSciNet  Google Scholar 

  20. M K Prasad and T Bucha, Pramana – J. Phys. 94, 24 (2020)

    Article  ADS  Google Scholar 

  21. A Jarray, Z Mehrez and A El Cafsi, Pramana – J. Phys. 94, 156 (2020)

    Article  ADS  Google Scholar 

  22. F Ali et al, Pramana – J. Phys. 93, 47 (2019)

    Google Scholar 

  23. B G Srivastava and S Deo, Appl. Math. Comput. 219(17), 8959 (2013)

    MathSciNet  Google Scholar 

  24. A Tiwari, S Deo and A N Filippov, Colloid J. 74(4), 515 (2012)

    Article  Google Scholar 

  25. P K Yadav, S Jaiswal, T Asim and R Mishra, Eur. Phys. J. Plus 133(7), 1 (2018)

    Article  Google Scholar 

  26. M Saleem, M N Tufail and Q A Chaudhry, Pramana – J Phys. 95, 28 (2021)

    Article  ADS  Google Scholar 

  27. K P Kashyap, O Ojjela and S K Das, Pramana – J Phys. 92, 73 (2019)

    Article  ADS  Google Scholar 

  28. N T Eldabe, G M Moatimid and A Sayed, Pramana – J Phys. 95, 76 (2021)

    Article  ADS  Google Scholar 

  29. S Deo, D K Maurya and A N Filippov, Colloid J. 82(6), 649 (2020)

    Article  Google Scholar 

  30. H P Mazumdar, U N Ganguly and S K Venkatesan, Indian J. Pure Appl. Math. 27(5), 519 (1996)

    Google Scholar 

  31. A R Rao and K S Deshikachar, J. Indian Inst. Sci. 68(7 &8), 247 (1988)

    Google Scholar 

  32. S Kenjí, Int. J. Heat Fluid Flow 29(3), 752 (2008)

    Article  Google Scholar 

  33. G Varshney, V Katiyar and S Kumar, Int. J. Eng. Sci. Technol. 2(2), 967 (2008)

    Google Scholar 

  34. P K Yadav, S Jaiswal, J Y Puchakatla and M K Yadav, Eur. Phys. J. Plus 136(1), 1 (2021)

    Article  Google Scholar 

  35. Y Gamiel, M Elbehairy and M Kamel El-Sayed, Alex. Eng. J. 61(11), 8667 (2022)

    Article  Google Scholar 

  36. J Happel and H Brenner, Low Reynolds number hydrodynamics (Martinus Nijoff, The Hague, Netherlands, 1983)

    Book  MATH  Google Scholar 

  37. B Albusairi and J T Hsu, Chem. Eng. J. 100, 79 (2004)

    Article  Google Scholar 

  38. A N Filippov, S I Vasin and V M Starov, Colloids Surf. A Physicochem. Eng. Asp. 282–283, 272 (2006)

    Article  Google Scholar 

  39. P K Yadav, A Tiwari, S Deo, A Filippov and S Vasin, Acta Mech. 215, 209 (2010), https://doi.org/10.1007/s00707-010-0331-8

    Article  Google Scholar 

  40. J Prakash and G P Raja Sekhar, Chem. Eng. Commun. 198, 85 (2011), https://doi.org/10.1080/00986445.2010.493125

    Article  Google Scholar 

  41. H Tan and K M Pillai, Comput. Fluids 38(6), (2009)

    Article  MathSciNet  Google Scholar 

  42. R H Davis and H A Stone, Chem. Eng. Sci. 48(23), 3993 (1993)

    Article  Google Scholar 

  43. G Neale, N Epstein and W Nader, Chem. Eng. Sci. 28(10), 1865 (1973)

    Article  Google Scholar 

  44. Y Li and C W Park, Adv. Colloid Interface Sci. 87(1), 1 (2000)

    Article  Google Scholar 

  45. W Wang and A S Sangani, Phys. Fluid 9(6), 1529 (1997)

    Article  ADS  Google Scholar 

  46. G H Neale and W K Nader, AIChE 19(1), 112 (1973)

    Article  Google Scholar 

  47. G H Neale and W K Nader, AIChE 20(3), 530 (1974)

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to express their sincere thanks to Prof. S T Assar, Engineering Physics and Mathematics Department, Faculty of Engineering, Tanta University, for her support and helpful suggestions during the work. The authors also appreciate the reviewers for their valuable comments and suggestions, which led to much improvement in the presentation of the paper.

Author information

Authors and Affiliations

  1. Engineering Physics and Mathematics Department, Faculty of Engineering, Tanta University, Tanta, 31734, Egypt

    Yasser Gamiel

  2. Basic Science Department, Kafrelsheikh Higher Institute of Engineering and Technology, Kafrelsheikh, 33511, Egypt

    M Kamel El-Sayed & Marwa Elbehairy

Authors
  1. Yasser Gamiel
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  2. M Kamel El-Sayed
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  3. Marwa Elbehairy
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Corresponding author

Correspondence to Marwa Elbehairy.

Appendix

Appendix

As previously mentioned, expressing the coefficients indicated in the solution is cumbersome, hence we present specific coefficients employed in the drag as given in eq. (21)

$$\begin{aligned} \xi= & {} (-S^2+\eta _i\chi _i), \quad \Gamma =(2 \eta _i \chi _i+S^2),\\ \kappa= & {} (-S^2-\eta _i\chi _i), \quad L =(M^2+S^2- \eta _i \chi _i),\\ \omega= & {} (M^2-S^2- \eta _i \chi _i),\quad \Lambda =(\Gamma M^2+S^4+S^2 \eta _i \chi _i),\\ \Pi= & {} (\Gamma M^2-S^4-S^2 \eta _i \chi _i), \end{aligned}$$
$$\begin{aligned} C= & {} 2 R ((-(((M^2-R^2) \lambda -\eta _e \chi _e)\\{} & {} \times (\omega I_1(S)+S I_0(S) \eta _i \chi _i) (K I_1+I K_1)S^2)\\{} & {} -(\lambda R^2+\eta _e \chi _e) \lambda ^2(S (M L K_1\\{} & {} -\chi _i \eta _i K (M^2+S^2)) I_0(S)+(K \Lambda -2 M \omega K_1)\\{} & {} \times I_1(S)) M I_0(M \lambda )-((L M I_1\\{} & {} +I (M^2+S^2) \eta _i \chi _i) S I_0(S)+(-i \Lambda -2 M \omega I_1)\\{} & {} \times I_1(S))(\lambda R^2+\eta _e \chi _e)\\{} & {} \times \lambda ^2 M K_0(M \lambda )+(S (M L K_1-\chi _i \\{} & {} \times \eta _i K (M^2+S^2)) I_0(S)+(K \Lambda -2 M \omega K_1) \\{} & {} \times I_1(S)) \lambda I_1(M \lambda )(-((M^2-R^2) \lambda \\{} & {} -\eta _e \chi _e))+((L M I_1+I(M^2+S^2) \eta _i \chi _i) \\{} & {} \times S I_0(S)+(-i \Lambda -2 M \omega I_1) I_1(S))\lambda K_1(M \lambda )\\{} & {} \times ((M^2-R^2) \lambda -\eta _e \chi _e)) K_1(R \lambda ) \\{} & {} +R \lambda K_0(R \lambda )((S (M L K_1-\chi _i \eta _i K (M^2+S^2)) \\{} & {} \times I_0(S)+(K \Lambda -2 M \omega K_1) I_1(S)) (-\lambda )\\{} & {} \times I_1(M \lambda ) (M^2 \lambda -\chi _e \eta _e)+((L M I_1+I (M^2+S^2) \\{} & {} \times \eta _i\chi _i)~ S I_0(S)+(-I \Lambda -2 M \omega I_1)~ I_1(S))\\{} & {} \times \lambda K_1(M \lambda ) (M^2 \lambda -\chi _e \eta _e)-(-((\omega I_1(S)\\{} & {} +S I_0(S) \eta _i \chi _i)(K I_1+I K_1) S^2)+M~ (S (M L K_1 \\{} & {} -\chi _i \eta _iK (M^2+S^2)) I_0(S)+(K \Lambda -2 M \omega K_1)~ \\{} & {} \times I_1(S))\lambda ^2 I_0(M \lambda )+M ((L M I_1+I (M^2+S^2) \eta _i \\{} & {} \times \chi _i) S I_0(S)+(-i \Lambda -2 M \omega I_1) I_1(S))\lambda ^2 K_0(M \lambda )) \\{} & {} \times \chi _e \eta _e))\biggr { /} ((R^2 S^2 K_0(M \lambda )\eta _e (2 \omega I_1(S) \\{} & {} +2 S I_0(S) \eta _i \chi _i)\chi _e \lambda ^2-((L M^2 (M^2+R^2) \\{} & {} \times \lambda ^3-\eta _e\chi _e (M^2+2 R^2) L \\{} & {} \times \lambda ^2-(M+R)(M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M-R) \lambda +(M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i)\eta _e \chi _e) M K_1-\chi _i \\{} & {} \times (M^2 (M^2+R^2) (M^2+S^2) \lambda ^3\\{} & {} -\eta _e \chi _e (M^2+S^2) (M^2+2 R^2)\lambda ^2\\{} & {} +(-M^6+(R^2+S^2)M^4-R^2 S^2 M^2)\lambda \\{} & {} +M^2 (M^2-S^2) \eta _e \chi _e) K \eta _i)\\{} & {} \times I_0(S) S+I_1(S) ((2 M^2-2 S^2\\{} & {} -2 \chi _i \eta _i) M ((M^4+R^2 M^2)\lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2)\lambda ^2\\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2)\lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) K_1 \\{} & {} -K (M^2 (M^2+R^2) \Lambda \lambda ^3 \\{} & {} -\Lambda \chi _e (M^2+2 R^2) \eta _e \lambda ^2\\{} & {} -(M+R) \Pi (M-R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e))) I_1(M \lambda ) \\{} & {} +(R^2 S^2 I_0(M \lambda )\eta _e (2 \omega I_1(S)\\{} & {} +2 S I_0(S) \eta _i \chi _i) \chi _e \lambda ^2\\{} & {} +(M I_1 (L M^2 (M^2+R^2) \lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2) L \lambda ^2\\{} & {} -(M+R) (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \\{} & {} \times (M-R) \lambda +(M^4+\kappa M^2 \\{} & {} +2 S^2\eta _i \chi _i) \eta _e \chi _e) \\{} & {} +I (M^2 (M^2+R^2)(M^2+S^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+S^2) (M^2+2 R^2) \lambda ^2\\{} & {} +(-M^6+(R^2+S^2) M^4-R^2 S^2 M^2) \lambda \\{} & {} +M^2 (M^2-S^2) \eta _e \chi _e) \eta _i \chi _i) \\{} & {} \times S I_0(S)+(-I (M^2 (M^2+R^2) \Lambda \lambda ^3\\{} & {} -\Lambda \chi _e (M^2+2 R^2) \eta _e \lambda ^2 \\{} & {} -(M+R) \Pi (M-R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e)+M ((M^4+R^2 M^2)\lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2) \lambda ^2\\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) I_1 (-(2 M^2-2 S^2\\{} & {} -2 \chi _i \eta _i))) I_1(S)) K_1(M \lambda ) \\{} & {} -(-((2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i) \\{} & {} \times (K I_1+I K_1) R^2 S^2)+((((M^2+R^2) \Lambda \lambda ^2 \\{} & {} -(M+R) \Pi (M-R)) K M\\ {}{} & {} -(-2 M^4+2 (R^2+S^2) M^2\\{} & {} -2 R^2 S^2+2 (M^4+R^2 M^2) \lambda ^2) \omega K_1) \\{} & {} \times I_1(S)+S I_0(S) ((M^2 (M^2+R^2) L \lambda ^2 \\{} & {} -(M+R)(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R)) \\{} & {} \times K_1-\chi _i K (-M^4+(R^2+S^2) M^2-R^2 S^2 \\{} & {} +(M^2+R^2) (M^2+S^2) \lambda ^2) M \eta _i)) I_0(M \lambda ) \\{} & {} +(((M^2 (M^2+R^2) L \lambda ^2-(M+R) (M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M-R)) I_1+I M(-M^4+(R^2+S^2) M^2\\{} & {} -R^2 S^2+(M^2+R^2) (M^2+S^2) \lambda ^2) \\{} & {} \times \eta _i \chi _i)S I_0(S)+(-M I ((M^2+R^2) \Lambda \lambda ^2\\{} & {} -(M+R) \Pi (M-R))\\ {}{} & {} -(-2 M^4+2 (R^2+S^2) M^2 \\{} & {} -2 R^2 S^2+2 (M^4+R^2 M^2) \lambda ^2) \\{} & {} \times \omega I_1) I_1(S)) K_0(M \lambda )) \lambda \chi _e \eta _e) K_0(R \lambda )\\{} & {} -R ((K I_1+I K_1) (\omega I_1(S)\\{} & {} +S I_0(S) \eta _i \chi _i) (2 (M^2-R^2) \lambda -2 \eta _e \chi _e)S^2\\{} & {} +((((M^2+R^2) \Lambda \lambda ^3 +\Lambda \eta _e \chi _e \lambda ^2 \\{} & {} -(M+R) \Pi (M-R) \lambda +\Pi \eta _e \chi _e)\\{} & {} \times K M-(2 M^2-2 S^2-2 \chi _i \eta _i) ((M^4+R^2 M^2) \lambda ^3\\{} & {} +M^2 \eta _e \chi _e \lambda ^2 +(-M^4+(R^2+S^2) M^2-R^2 S^2)~ \lambda \\{} & {} +(M^2-S^2) ~\eta _e \chi _e) ~K_1) \\{} & {} \times I_1(S)+ S I_0(S)\lambda ^3 ((L M^2 (M^2+R^2) \\{} & {} +L M^2 \eta _e \chi _e \lambda ^2 \\{} & {} -(M+R) (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \\{} & {} \times (M-R) \lambda +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times \eta _e \chi _e) K_1-((M^2+R^2) (M^2+S^2)\lambda ^3 \\{} & {} +(M^2+S^2) \eta _e \chi _e \lambda ^2 \\{} & {} +(-M^4\!+\!(R^2+S^2) M^2\!-\!R^2 S^2) \lambda \\{} & {} +(M^2\!-\!S^2) \eta _e\chi _e) \chi _i K M \eta _i)) \\{} & {} \times I_0(M \lambda )+((I_1 (L M^2 (M^2+R^2) \lambda ^3 \\{} & {} +L M^2 \eta _e \chi _e \lambda ^2-(M+R) \\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times (M-R) \lambda +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\eta _e \chi _e)\\{} & {} +I M ((M^2+R^2) (M^2+S^2) \lambda ^3\\{} & {} +(M^2+S^2) \eta _e \chi _e \lambda ^2 \\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) \eta _i \chi _i) S I_0(S)\\{} & {} +(-M I ((M^2+R^2) \Lambda \lambda ^3 +\Lambda \eta _e \chi _e \lambda ^2 \\{} & {} -(M+R) \Pi (M-R) \lambda +\Pi \eta _e \chi _e) \\{} & {} +((M^4+R^2 M^2) \lambda ^3 +M^2 \eta _e \chi _e \lambda ^2 \\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) I_1 \\{} & {} \times (-(2 M^2-2 S^2-2 \chi _i \eta _i))) \\{} & {} \times I_1(S)) K_0(M \lambda )-(2 (M^2-R^2) \lambda \\{} & {} -2 \eta _e \chi _e) \lambda (I_0(M \lambda ) (-(\omega I_1(S)\\{} & {} +S I_0(S) \eta _i \chi _i)) S^2 \\{} & {} +(L M I_1+I (M^2+S ^2) \eta _i \chi _i) \\{} & {} \times I_0(S) S+(-I \Lambda -2 M \omega I_1) \\{} & {} \times I_1(S)) K_1(M \lambda )+(K_0(M \lambda )\\{} & {} \times (\omega I_1(S)+S I_0(S) \eta _i \chi _i) S^2 \\{} & {} +(M L K_1-\chi _i \eta _i K (M^2+S^2)) I_0(S)S\\{} & {} +(K \Lambda -2 M \omega K_1) I_1(S)) \\{} & {} \times \lambda I_1(M \lambda ) (2 (M^2-R^2) \lambda \\{} & {} -2 \eta _e \chi _e)) K_1(R \lambda ), \end{aligned}$$
$$\begin{aligned} D= & {} 2 R \lambda ((-M ((K I_1+I K_1) (\lambda M^2\\{} & {} -R^2 \lambda -\chi _e \eta _e) (\omega I_1(S) \\{} & {} +S I_0(S) \eta _i \chi _i) S^2 \\{} & {} +(S ((M^4+\kappa M^2+2 S^2 \eta _i \chi _i)K_1\\{} & {} -M K \chi _i \eta _i (M^2-S^2)) I_0(S) \\{} & {} +(\Pi K M-(2 M-2 S) (M+S) \omega K_1) \\{} & {} \times I_1(S))I_0(M \lambda ) (\lambda R^2 \\{} & {} +\eta _e \chi _e)+(((M^4+\kappa M^2 +2 S^2 \eta _i \chi _i) I_1\\{} & {} +I M (M^2-S^2) \eta _i \chi _i) S I_0(S) \\{} & {} +(-M I \Pi +(M+S) (-(2 M-2 S)) \omega I_1) \\{} & {} \times I_1(S)) K_0(M \lambda ) (\lambda R^2 \\{} & {} +\eta _e \chi _e)) \lambda +(S ((M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) K_1-M K \chi _i \eta _i \\{} & {} \times (M^2-S^2)) I_0(S)\!+\!(\Pi K M\!-\!(2 M-2 S) (M\!+\!S) \\{} & {} \times \omega K_1) I_1(S)) I_1(M \lambda )\\{} & {} \times (-(\lambda M^2-R^2 \lambda -\chi _e \eta _e))\\{} & {} +(((M^4+\kappa M^2+2 S^2 \eta _i \chi _i) I_1\\{} & {} +I M (M^2-S^2) \eta _i \chi _i) S I_0(S) \\{} & {} +(-M I \Pi +(M+S) (-(2 M-2 S)) \omega I_1) I_1(S)) \\{} & {} \times K_1(M \lambda ) (\lambda M^2-R^2 \lambda -\chi _e \eta _e))K_1(R \lambda ) \\{} & {} +R \lambda K_0(R \lambda ) ((((M^4+\kappa M^2 \\{} & {} +2S^2 \eta _i \chi _i) I_1+I M (M^2-S^2) \eta _i \chi _i) \\{} & {} \times S I_0(S)\!+\!(-M I \Pi \!+\!(M+S) (-(2 M\!-\!2 S)) \omega I_1) \\{} & {} \times I_1(S)) K_1(M \lambda ) (M^2 \lambda -\chi _e \eta _e) \\{} & {} -\lambda \chi _e M (-((\omega I_1(S) \\{} & {} +S I_0(S) \eta _i \chi _i) (K I_1+I K_1) S^2) \\{} & {} +(S ((M^4+\kappa M^2+2 S^2 \eta _i \chi _i) K_1\\{} & {} -M K \chi _i \eta _i (M^2-S^2)) \\{} & {} \times I_0(S)+(\Pi K M-(2 M-2 S) (M+S) \omega K_1)\\{} & {} \times I_1(S)) I_0(M \lambda )+(((M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) I_1+I M (M^2-S^2) \\{} & {} \times \eta _i \chi _i) S I_0(S)\!+\!(-M I \Pi \!+\!(M+S) (-(2 M\!-\!2 S))\\{} & {} \times \omega I_1) I_1(S)) K_0(M \lambda )) \eta _e \\{} & {} +(S ((M^4+\kappa M^2+2 S^2 \eta _i \chi _i) K_1 \\{} & {} -M K \chi _i \eta _i (M^2-S^2)) I_0(S)\\{} & {} +(\Pi K M-(2 M-2 S) (M+S) \omega K_1)\\{} & {} \times I_1(S)) I_1(M \lambda )(-(M^2 \lambda -\chi _e \eta _e))))\biggr { /}\\{} & {} (((R^2 S^2 K_0(M \lambda ) \eta _e (2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)\\{} & {} \times \chi _e \lambda ^2-(((\lambda ^3-\lambda ) M^6\\{} & {} +((R^2+S^2-\chi _i \eta _i) \lambda ^3 \\{} & {} -\eta _e \chi _e \lambda ^2+(R^2+S^2+\eta _i \chi _i)\lambda \\{} & {} +\eta _e \chi _e) M^4 +(R^2 (S^2-\chi _i \eta _i) \lambda ^3\\{} & {} -(2 R^2+S^2-\chi _i \eta _i) \chi _e \eta _e \lambda ^2 \\{} & {} +((-R^2-2 \chi _i \eta _i) S^2-R^2 \chi _i \eta _i) \lambda \\{} & {} -\eta _e \chi _e \xi ) M^2-2 R^2 \eta _e \chi _e (S^2-\chi _i \eta _i) \lambda ^2 \\{} & {} +2 R^2 S^2 \lambda \eta _i \chi _i \\{} & {} +2 S^2 \eta _e \eta _i \chi _e \chi _i) M K_1 -\chi _i ((\lambda ^3-\lambda ) M^6 \\{} & {} +((R^2+S^2) \lambda ^3 -\eta _e \chi _e \lambda ^2+(R^2+S^2) \lambda \\{} & {} +\eta _e \chi _e) M^4+(R^2 S^2 \lambda ^3 \\{} & {} -(2 R^2+S^2) \chi _e \eta _e \lambda ^2 \\{} & {} -R^2 S^2 \lambda -S^2 \eta _e \chi _e) M^2 \\{} & {} -2 R^2 S^2 \eta _e \lambda ^2 \chi _e) \\{} & {} \times K \eta _i) I_0(S) S+((2 M^2-2 S^2-2 \chi _i \eta _i)\\{} & {} \times M ((\lambda ^3-\lambda ) M^4+(R^2 \lambda ^3 -\eta _e \chi _e \lambda ^2\\{} & {} +(R^2+S^2) \lambda +\eta _e \chi _e) M^2-R^2 S^2 \lambda \\{} & {} -2 R^2 \eta _e \lambda ^2 \chi _e-S^2 \eta _e \chi _e) K_1\\{} & {} -K ((\lambda ^2-1) \Gamma \lambda M^6\\{} & {} +((S^4+(R^2+\eta _i \chi _i) S^2 \\{} & {} +2 R^2 \eta _i \chi _i) \lambda ^3 \\{} & {} -\eta _e \chi _e \Gamma \lambda ^2 +(S^4+(R^2+\eta _i \chi _i) S^2+2 R^2 \eta _i \chi _i)\lambda \\{} & {} +\Gamma \eta _e \chi _e) M^4\\{} & {} +\left( R^2 S^2 \xi \lambda ^3-(S^4+2 \left( R^2 +\frac{\eta _i \chi _i}{2}\right) S^2\right. \\{} & {} \left. +4 R^2 \eta _i \chi _i) \chi _e \eta _e \lambda ^2 \right. \\{} & {} \left. -R^2 S^2 \xi \lambda -S^2 \eta _e \chi _e \xi \right) M^2 \\{} & {} -2 R^2 S^2 \eta _e \lambda ^2 \chi _e \xi )) I_1(S)) I_1(M \lambda )\\{} & {} +(R^2 S^2 I_0(M \lambda ) \eta _e (2 \omega I_1(S) \\{} & {} +2 S I_0(S) \eta _i \chi _i) \chi _e \lambda ^2\\{} & {} +(M I_1 ((\lambda ^3-\lambda ) M^6 \\{} & {} +((R^2+S^2-\chi _i \eta _i) \lambda ^3 \\{} & {} -\eta _e \chi _e \lambda ^2+(R^2+S^2+\eta _i \chi _i) \lambda \\{} & {} +\eta _e \chi _e) M^4 +(R^2 (S^2-\chi _i \eta _i) \lambda ^3 \\{} & {} \times (2 R^2+S^2-\chi _i \eta _i) \chi _e \eta _e \lambda ^2 \\{} & {} +((-R^2-2 \chi _i \eta _i) S^2-R^2 \chi _i \eta _i) \lambda \\{} & {} -\eta _e \chi _e \xi ) M^2-2 R^2 \eta _e \chi _e \\{} & {} \times (S^2-\chi _i \eta _i) \lambda ^2+2 R^2 S^2 \lambda \\{} & {} \times \eta _i \chi _i+2 S^2 \eta _e \eta _i \chi _e \chi _i) \\{} & {} +I \eta _i ((\lambda ^3-\lambda ) M^6+((R^2+S^2) \lambda ^3 \\{} & {} -\eta _e \chi _e \lambda ^2+(R^2+S^2) \lambda +\eta _e \chi _e) M^4 \\{} & {} +(R^2 S^2 \lambda ^3-(2 R^2+S^2) \chi _e \eta _e \lambda ^2 \\{} & {} -R^2 S^2 \lambda -S^2 \eta _e \chi _e) M^2 \\{} & {} -2 R^2 S^2 \eta _e \lambda ^2 \chi _e) \chi _i) S I_0(S) \\{} & {} +(-I ((\lambda ^2-1) \Gamma \lambda M^6+((S^4+(R^2+\eta _i \chi _i)\\{} & {} \times S^2+2 R^2 \eta _i \chi _i) \lambda ^3 -\eta _e \chi _e \Gamma \lambda ^2 \\{} & {} +(S^4+(R^2+\eta _i \chi _i) S^2+2 R^2 \eta _i \chi _i) \lambda +\Gamma \eta _e \chi _e) \\{} & {} \times M^4+\left( R^2 S^2 \xi \lambda ^3-(S^4+2 \left( R^2 +\frac{\eta _i \chi _i}{2}\right) S^2\right. \\{} & {} \left. +4 R^2 \eta _i \chi _i) \chi _e \eta _e \lambda ^2 -R^2 S^2 \xi \lambda -S^2 \eta _e \chi _e \xi \right) M^2 \\{} & {} -2 R^2 S^2 \eta _e \lambda ^2 \chi _e \xi ) +M I_1 (-(2 M^2-2 S^2-2 \chi _i \eta _i)) \\{} & {} \times ((\lambda ^3-\lambda ) M^4+(R^2 \lambda ^3 \\{} & {} -\eta _e \chi _e \lambda ^2+(R^2+S^2) \lambda \\{} & {} +\eta _e \chi _e) M^2-R^2 S^2 \lambda -2 R^2 \eta _e \lambda ^2 \chi _e\\{} & {} -S^2 \eta _e \chi _e)) I_1(S))K_1(M \lambda )\\{} & {} -(-((2 \omega I_1(S)+2 S I_0(S) \\{} & {} \times \eta _i \chi _i) (K I_1+I K_1) R^2 S^2) \\{} & {} +((((\lambda ^2-1) \Gamma M^4+(\lambda ^2+1) \\{} & {} \times (S^4+(R^2+\eta _i \chi _i) S^2+2 R^2 \eta _i \chi _i)M^2\\{} & {} +(\lambda ^2-1) R^2 S^2 \xi ) K M \\{} & {} -(2 (\lambda ^2\!-\!1) M^4\!+\!2 (\lambda ^2 R^2\!+\!R^2\!+\!S^2) M^2\\{} & {} -2 R^2 S^2)\omega K_1) I_1(S)+S I_0(S) (((\lambda ^2-1) M^6 \\{} & {} +(R^2+S^2+\lambda ^2 (R^2+S^2-\chi _i \eta _i) \\{} & {} +\eta _i \chi _i) M^4+(-\chi _i \eta _i R^2 \\{} & {} +\lambda ^2 (S^2-\chi _i \eta _i) R^2\\{} & {} +S^2 (-R^2-2 \chi _i \eta _i)) M^2 \\{} & {} +2 R^2 S^2 \eta _i \chi _i) K_1-\chi _i K ((\lambda ^2-1) M^4 \\{} & {} +(\lambda ^2+1) (R^2+S^2) M^2+(\lambda ^2-1) R^2 S^2)\\{} & {} \times M \eta _i)) I_0(M \lambda )+((I_1 ((\lambda ^2-1) M^6 \\{} & {} +(R^2+S^2+\lambda ^2 (R^2+S^2-\chi _i \eta _i) \\{} & {} +\eta _i \chi _i) M^4+(-\chi _i \eta _i R^2 \\{} & {} +\lambda ^2 (S^2-\chi _i \eta _i) R^2 \\{} & {} +S^2 (-R^2-2 \chi _i \eta _i)) M^2 \\{} & {} +2 R^2 S^2 \eta _i \chi _i)+((\lambda ^2-1) M^4 \\{} & {} +(\lambda ^2+1) (R^2+S^2) M^2+(\lambda ^2-1) R^2 S^2)\\{} & {} \times I M \eta _i \chi _i) S I_0(S)+(-M I ((\lambda ^2-1) \\{} & {} \times \Gamma M^4+(\lambda ^2+1) (S^4+(R^2+\eta _i \chi _i) S^2 \\{} & {} +2 R^2 \eta _i \chi _i) M^2+(\lambda ^2-1) R^2 S^2 \xi ) \\{} & {} -(2 (\lambda ^2-1) M^4+2 (\lambda ^2 R^2+R^2+S^2) M^2\\{} & {} -2 R^2 S^2) \omega I_1) I_1(S)) K_0(M \lambda )) \lambda \chi _e \eta _e) K_0(R \lambda ) \\{} & {} -R K_1(R \lambda ) ((K I_1+I K_1) (2 \lambda M^2-2 R^2 \lambda \\{} & {} -2 \chi _e \eta _e) (\omega I_1(S) \\{} & {} +S I_0(S) \eta _i \chi _i) S^2+((((\lambda ^2-1) \Gamma \lambda M^4\\{} & {} +(\lambda ^2+1) (\lambda (S^4+(R^2+\eta _i \chi _i)S^2\\{} & {} +2 R^2 \eta _i \chi _i)+\Gamma \eta _e \chi _e)M^2\\{} & {} +(\lambda ^2-1) S^2 \xi (\lambda R^2+\eta _e \chi _e))K M\\{} & {} -(2 M^2-2 S^2-2 \chi _i \eta _i) ((\lambda ^3-\lambda ) M^4 \\{} & {} +(R^2 \lambda ^3+\eta _e \chi _e \lambda ^2+(R^2+S^2)\lambda \\{} & {} +\eta _e \chi _e) M^2-S^2 (\lambda R^2+\eta _e \chi _e)) K_1)I_1(S)\\{} & {} +S I_0(S) (((\lambda ^3-\lambda ) M^6 \\{} & {} +((R^2+S^2-\chi _i \eta _i) \lambda ^3+\eta _e \chi _e \lambda ^2 \\{} & {} +(R^2+S^2+\eta _i \chi _i) \lambda +\eta _e \chi _e) M^4 \\{} & {} +(R^2 (S^2-\chi _i \eta _i) \lambda ^3+\eta _e (S^2-\chi _i \eta _i) \chi _e \lambda ^2 \\{} & {} +((-R^2-2 \chi _i \eta _i) S^2-R^2 \chi _i \eta _i) \lambda \\{} & {} -\eta _e \chi _e \xi ) M^2+2 S^2(\lambda R^2+\eta _e \chi _e) \eta _i \chi _i)K_1\\{} & {} -((\lambda ^3-\lambda ) M^4+(\lambda ^2+1)\\{} & {} \times ((R^2+S^2) \lambda +\eta _e \chi _e) M^2+(\lambda ^2-1) \\{} & {} \times S^2 (\lambda R^2+\eta _e \chi _e)) \chi _i K M \\{} & {} \times \eta _i)) I_0(M \lambda )+((I_1 ((\lambda ^3-\lambda )M^6\\{} & {} +((R^2+S^2-\chi _i \eta _i) \lambda ^3\\{} & {} +\eta _e \chi _e \lambda ^2+(R^2+S^2 \\{} & {} +\eta _i \chi _i) \lambda +\eta _e \chi _e) M^4+(R^2 \\{} & {} \times (S^2-\chi _i \eta _i) \lambda ^3+\eta _e (S^2-\chi _i \eta _i) \chi _e \lambda ^2 \\{} & {} +((-R^2-2 \chi _i \eta _i) S^2-R^2 \chi _i \eta _i) \lambda \\{} & {} -\eta _e \chi _e \xi ) M^2 \\{} & {} +2 S^2 (\lambda R^2+\eta _e \chi _e) \eta _i \chi _i)\\{} & {} +I M ((\lambda ^3-\lambda ) M^4+(\lambda ^2+1) \\{} & {} \times ((R^2+S^2) \lambda +\eta _e \chi _e) M^2\\{} & {} +(\lambda ^2-1) S^2 (\lambda R^2+\eta _e \chi _e))\\{} & {} \times \eta _i \chi _i) S I_0(S)+(-M I ((\lambda ^2-1) \Gamma \lambda M^4\\{} & {} +(\lambda ^2+1) (\lambda (S^4+(R^2+\eta _i \chi _i) S^2 \\{} & {} +2 R^2 \eta _i \chi _i)+\Gamma \eta _e \chi _e) M^2\\{} & {} +(\lambda ^2-1) S^2 \xi (\lambda R^2+\eta _e \chi _e))\\{} & {} +I_1 ((\lambda ^3-\lambda ) M^4+(R^2 \lambda ^3+\eta _e \chi _e \lambda ^2 \\{} & {} +(R^2+S^2) \lambda +\eta _e \chi _e) M^2-S^2 (\lambda R^2\\{} & {} +\eta _e \chi _e)) (-(2 M^2-2 S^2-2 \chi _i \eta _i))) I_1(S))\\{} & {} \times K_0(M \lambda )-(2 \lambda M^2-2 R^2 \lambda \\{} & {} -2 \chi _e \eta _e) \lambda (I_0(M \lambda )\\{} & {} \times (-(\omega I_1(S)+S I_0(S) \eta _i \chi _i)) S^2+(L M I_1 \\{} & {} +I (M^2+S^2) \eta _i \chi _i) I_0(S) S \\{} & {} +(-i \Lambda -2 M \omega I_1) I_1(S)) K_1(M \lambda ) \\{} & {} +(K_0(M \lambda ) (\omega I_1(S) +S I_0(S) \eta _i \chi _i) S^2 \\{} & {} +(M L K_1-\chi _i \eta _i K (M^2+S^2)) \\{} & {} \times I_0(S) S+(K \Lambda -2 M \omega K_1) I_1(S)) \\{} & {} \times \lambda I_1(M \lambda ) (2 \lambda M^2-2 R^2 \lambda \\{} & {} -2 \chi _e \eta _e))) M, \end{aligned}$$
$$\begin{aligned} E= & {} -2 R ((-((2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)\\{} & {} \times (\lambda R^2+\eta _e \chi _e) \lambda ^2\\{} & {} \times M K_0(M \lambda ) S^2)+\lambda K_1(M \lambda ) \\{} & {} \times (\omega I_1(S)+S I_0(S) \eta _i \chi _i) (2 (M^2-R^2) \lambda \\{} & {} -2 \eta _e \chi _e) S^2+(((S^4+\eta _i \chi _i S^2\\{} & {} +\lambda ^2 \Lambda +M^2 (-S^2-2 \chi _i \eta _i)) K M\\{} & {} -(2\lambda ^2 M^2-2 M^2+2 S^2)\\{} & {} \times \omega K_1) I_1(S)+S I_0(S) ((-M^4+L\lambda ^2 M^2+\xi M^2 \\{} & {} -2 S^2 \chi _i \eta _i) K_1\!-\!\chi _i K(-M^2\!+\!S^2\!+\!(M^2\!+\!S^2) \lambda ^2)\\{} & {} \times M \eta _i)) ((M^2-R^2) \lambda -\eta _e \chi _e)) \\{} & {} \times K_1(R \lambda ) -R \lambda ((M S^2 K_0(M \lambda ) \\{} & {} \times (2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i) \lambda ^2\\{} & {} +((S^4\!+\!\eta _i \chi _i S^2\!+\!\lambda ^2 \Lambda \!+\!M^2 (-S^2\!-\!2 \chi _i \eta _i)) K M\\{} & {} -(2 \lambda ^2 M^2-2 M^2+2 S^2)\\{} & {} \times \omega K_1) I_1(S)+S I_0(S)((-M^4+L \lambda ^2 M^2+\xi M^2 \\{} & {} -2 S^2 \chi _i \eta _i) K_1\!-\!\chi _iK (-M^2\!+\!S^2\!+\!(M^2\!+\!S^2) \lambda ^2)\\{} & {} \times M \eta _i)) \chi _e \eta _e-(2 \omega I_1(S)\\{} & {} +2 S I_0(S) \eta _i \chi _i) \lambda (M^2 \lambda \\{} & {} -\chi _e \eta _e) S^2 K_1(M \lambda )) K_0(R \lambda )) \biggr { /}\\{} & {} (((K I_1+I K_1) R^2 \lambda \eta _e (2 \omega I_1(S)+2 S I_0(S) \\{} & {} \times \eta _i \chi _i) \chi _e S^2 -\lambda ((((M^2+R^2) \Lambda \lambda ^2-(M+R)\\{} & {} \times \Pi (M-R)) K M-(-2 M^4+2 (R^2+S^2) M^2\\{} & {} -2 R^2 S^2+2 (M^4+R^2 M^2) \lambda ^2) \omega K_1) I_1(S)+S I_0(S)\\{} & {} \times ((M^2 (M^2+R^2) L \lambda ^2-(M+R)\\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times (M-R)) K_1-\chi _i K (-M^4+(R^2+S^2) M^2\\{} & {} -R^2 S^2+(M^2+R^2)(M^2+S^2) \lambda ^2) M \eta _i)) \\{} & {} \times \chi _e \eta _e I_0(M \lambda )+(((2 M^2-2 S^2-2 \chi _i \eta _i)\\{} & {} \times M ((M^4+R^2 M^2) \lambda ^3-\eta _e \chi _e (M^2+2 R^2)\lambda ^2\\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) K_1-K (M^2 (M^2+R^2) \\{} & {} \times \Lambda \lambda ^3-\Lambda \chi _e (M^2+2 R^2)\eta _e \lambda ^2 \\{} & {} -(M+R) \Pi (M-R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e)) I_1(S)-((L M^2 (M^2+R^2)\lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2) L \lambda ^2-(M+R)\\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R) \lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \eta _e \chi _e)M K_1\\{} & {} -\chi _i (M^2 (M^2+R^2) (M^2+S^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+S^2) (M^2+2 R^2)\lambda ^2\\{} & {} +(-M^6+(R^2+S^2) M^4-R^2 S^2 M^2) \lambda \\{} & {} +M^2 (M^2-S^2)\eta _e \chi _e) K \eta _i) I_0(S) S) I_1(M \lambda )\\{} & {} -\lambda \chi _e \eta _e (R^2 \lambda I_1(M \lambda )\\{} & {} \times (-(2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)) S^2\\{} & {} +((M^2 (M^2+R^2) L \lambda ^2-(M+R) (M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M\!-\!R)) I_1\!+\!I M(-M^4\!+\!(R^2\!+\!S^2) M^2 \\{} & {} -R^2 S^2+(M^2+R^2) (M^2+S^2) \lambda ^2) \eta _i \chi _i) I_0(S) S \\{} & {} +(-M I ((M^2+R^2) \Lambda \lambda ^2-(M+R) \Pi \\{} & {} \times (M-R))-(-2 M^4+2 (R^2+S^2) M^2\\{} & {} -2 R^2 S^2 +2 (M^4+R^2 M^2) \lambda ^2) \omega I_1) I_1(S))\\{} & {} \times K_0(M \lambda )+(R^2 S^2 I_0(M \lambda ) \eta _e \\{} & {} \times (2 \omega I_1(S) +2 S I_0(S) \eta _i \chi _i)\chi _e \lambda ^2\\{} & {} +(M I_1 (L M^2 (M^2+R^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+2 R^2) L \lambda ^2-(M+R)\\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R) \lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \eta _e \chi _e) \\{} & {} +I (M^2 (M^2+R^2) (M^2+S^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+S^2) (M^2+2R^2) \lambda ^2 \\{} & {} +(-M^6+(R^2+S^2) M^4-R^2 S^2 M^2) \lambda \\{} & {} +M^2(M^2-S^2) \eta _e \chi _e) \eta _i \chi _i) S I_0(S) \\{} & {} +(-I (M^2 (M^2+R^2) \Lambda \lambda ^3-\Lambda \chi _e\\{} & {} \times (M^2+2 R^2) \eta _e\lambda ^2 -(M+R) \Pi (M-R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e)+M ((M^4+R^2 M^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+2 R^2) \lambda ^2\\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2)\\{} & {} \times \lambda +(M^2-S^2) \eta _e \chi _e) I_1 \\{} & {} \times (-(2 M^2-2 S^2-2 \chi _i \eta _i))) I_1(S)) K_1(M \lambda )) \\{} & {} \times K_0(R \lambda )-R (((((M^2+R^2) \Lambda \lambda ^3 \\{} & {} +\Lambda \eta _e \chi _e \lambda ^2 -(M+R) \Pi (M-R) \lambda \\{} & {} +\Pi \eta _e \chi _e) K M-(2 M^2-2 S^2-2 \chi _i \eta _i)\\{} & {} \times ((M^4+R^2 M^2) \lambda ^3+M^2 \eta _e \chi _e \lambda ^2 \\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2)\lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) K_1) I_1(S)+S I_0(S)\\{} & {} \times ((L M^2 (M^2+R^2) \lambda ^3+L M^2 \eta _e \chi _e \lambda ^2\\{} & {} -(M+R) (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R)\lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \\{} & {} \times \eta _e \chi _e) K_1-((M^2+R^2) (M^2+S^2) \lambda ^3\\{} & {} +(M^2+S^2) \eta _e \chi _e \lambda ^2+(-M^4+(R^2+S^2) M^2\\{} & {} -R^2 S^2) \lambda +(M^2-S^2) \eta _e \chi _e)\\{} & {} \times \chi _i K M \eta _i)) I_0(M \lambda ) \\{} & {} +(\lambda I_1(M \lambda ) (\omega I_1(S)+S I_0(S) \eta _i \chi _i)\\{} & {} \times (2 (M^2-R^2) \lambda -2 \eta _e \chi _e) S^2\\{} & {} +(I_1 (L M^2 (M^2+R^2) \lambda ^3+L M^2\\{} & {} \times \eta _e \chi _e \lambda ^2-(M+R) (M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M-R) \lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \eta _e \chi _e)\\{} & {} +I M ((M^2+R^2) (M^2+S^2) \lambda ^3 \\{} & {} +(M^2+S^2) \eta _e \chi _e \lambda ^2 \\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e)\eta _i \chi _i) I_0(S) S\\{} & {} +(-M I ((M^2+R^2) \Lambda \lambda ^3 +\Lambda \eta _e \chi _e \lambda ^2 \\{} & {} -(M+R) \Pi (M-R) \lambda +\Pi \eta _e \chi _e)\\{} & {} +((M^4+R^2 M^2) \lambda ^3 \\{} & {} +M^2 \eta _e \chi _e \lambda ^2 +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) I_1 (-(2 M^2-2 S^2-2 \chi _i \eta _i)))\\{} & {} \times I_1(S)) K_0(M \lambda )+(I_0(M \lambda )\\{} & {} \times (-(\omega I_1(S) +S I_0(S) \eta _i \chi _i))S^2\\{} & {} +(L M I_1+I (M^2+S^2) \eta _i \chi _i) I_0(S)S\\{} & {} +(-i\Lambda -2 M \omega I_1) I_1(S)) \\{} & {} \times \lambda K_1(M \lambda ) (-(2 (M^2-R^2) \lambda \\{} & {} -2 \eta _e \chi _e))+((K I_1+I K_1)\\{} & {} \times (\omega I_1(S)+S I_0(S) \eta _i \chi _i)S^2\\{} & {} +(S (M L K_1-\chi _i \eta _i K (M^2+S^2)) I_0(S)\\{} & {} +(K \Lambda -2 M \omega K_1) I_1(S)) \lambda \\{} & {} \times I_1(M \lambda )) (2 (M^2-R^2) \lambda -2 \eta _e \chi _e))K_1(R \lambda )) M, \end{aligned}$$
$$\begin{aligned} G= & {} 2 R ((M \lambda ^2 I_0(M \lambda ) (\lambda R^2\\{} & {} +\eta _e \chi _e) (2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)S^2\\{} & {} +\lambda I_1(M \lambda ) (\omega I_1(S)+S I_0(S) \eta _i \chi _i) (2 (M^2-R^2) \lambda \\{} & {} -2 \eta _e \chi _e) S^2+(((-M^4+L \lambda ^2 M^2 \\{} & {} +\xi M^2-2 S^2 \chi _i \eta _i) I_1+I M \\{} & {} \times (-M^2+S^2+(M^2+S^2) \lambda ^2) \eta _i \chi _i)\\{} & {} \times S I_0(S)+(-(2 \lambda ^2 M^2-2 M^2+2 S^2)\\{} & {} \times \omega I_1-(S^4+\eta _i \chi _i S^2+\lambda ^2 \Lambda \\{} & {} +M^2 (-S^2-2 \chi _i \eta _i)) M I) I_1(S))\\{} & {} \times ((M^2-R^2) \lambda -\eta _e \chi _e)) K_1(R \lambda )\\{} & {} -R \lambda ((M S^2 I_0(M \lambda )\\{} & {} \times (-(2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i))\lambda ^2\\{} & {} +((-M^4+L \lambda ^2 M^2+\xi M^2\\{} & {} -2 S^2 \chi _i \eta _i) I_1+I M (-M^2+S^2+(M^2+S^2)\\{} & {} \times \lambda ^2)\eta _i \chi _i) S I_0(S)+(-(2 \lambda ^2 M^2-2M^2\\{} & {} +2 S^2) \omega I_1-(S^4+\eta _i \chi _i S^2+\lambda ^2\Lambda \\{} & {} +M^2 (-S^2-2 \chi _i \eta _i)) M I) I_1(S)) \chi _e \eta _e\\{} & {} -(2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)\lambda (M^2 \lambda -\chi _e \eta _e)\\{} & {} \times S^2 I_1(M \lambda )) K_0(R \lambda ))\biggr { /} \\{} & {} (((K I_1+I K_1) R^2 \lambda \eta _e (2 \omega I_1(S)+2 S I_0(S)\\{} & {} \times \eta _i \chi _i) \chi _e S^2-\lambda \chi _e(R^2 \lambda K_1(M \lambda )\\{} & {} \times (-(2 \omega I_1(S)+2 S I_0(S) \eta _i \chi _i)) S^2\\{} & {} +I_0(S) ((M^2 (M^2+R^2)L \lambda ^2\\{} & {} -(M+R) (M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M-R)) K_1-\chi _i K \\{} & {} \times (-M^4+(R^2+S^2) M^2-R^2 S^2+(M^2+R^2)\\{} & {} \times (M^2+S^2) \lambda ^2) M \eta _i) S+(((M^2+R^2) \Lambda \lambda ^2\\{} & {} -(M+R) \Pi (M-R)) K M\\{} & {} -(-2 M^4+2 (R^2+S^2)M^2\\{} & {} -2 R^2 S^2+2 (M^4+R^2 M^2) \lambda ^2) \omega K_1)\\{} & {} \times I_1(S)) \eta _e I_0(M \lambda )+(R^2 S^2 K_0(M \lambda )\\{} & {} \times \eta _e (2 \omega I_1(S)+2 S I_0(S) \\{} & {} \times \eta _i \chi _i) \chi _e \lambda ^2-((L M^2 (M^2+R^2)\lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2)L \lambda ^2\\{} & {} -(M\!+\!R) (M^4\!+\!\kappa M^2+2 S^2 \eta _i \chi _i) (M-R)\lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times \eta _e \chi _e) M K_1-\chi _i (M^2 (M^2+R^2)\\{} & {} \times (M^2+S^2) \lambda ^3-\eta _e \chi _e (M^2+S^2)(M^2+2 R^2) \lambda ^2\\{} & {} +(-M^6+(R^2+S^2) M^4-R^2 S^2 M^2)\lambda \\{} & {} +M^2 (M^2-S^2) \eta _e \chi _e)K \eta _i) I_0(S)S\\{} & {} +I_1(S)((2 M^2-2 S^2-2 \chi _i \eta _i)\\{} & {} \times M ((M^4+R^2 M^2) \lambda ^3-\eta _e \chi _e (M^2+2 R^2) \lambda ^2\\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) K_1-K (M^2 (M^2+R^2) \Lambda \lambda ^3\\{} & {} -\Lambda \chi _e (M^2+2 R^2) \eta _e\lambda ^2\\{} & {} -(M+R) \Pi (M-R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e))) I_1(M \lambda ) \\{} & {} -\lambda (((M^2 (M^2+R^2) L \lambda ^2-(M+R) \\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R)) I_1 \\{} & {} +I M (-M^4\!+\!(R^2\!+\!S^2) M^2\!-\!R^2 S^2\!+\!(M^2+R^2)\\{} & {} \times (M^2+S^2) \lambda ^2) \eta _i \chi _i)S I_0(S)\\{} & {} +(-M I ((M^2+R^2) \Lambda \lambda ^2-(M+R) \Pi \\{} & {} \times (M\!-\!R))\!-\!(-2 M^4\!+\!2 (R^2\!+\!S^2) M^2\!-\!2 R^2 S^2\!+\!2\\{} & {} \times (M^4+R^2 M^2) \lambda ^2) \omega I_1) I_1(S)) \\{} & {} \times \chi _e \eta _e K_0(M \lambda )+((M I_1 (L M^2 (M^2+R^2) \lambda ^3 \\{} & {} -\eta _e \chi _e (M^2+2 R^2) L \lambda ^2 \\{} & {} -(M+R) ~(M^4+\kappa ~ M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times (M-R) \lambda +~(M^4+\kappa M^2+2 S^2 \\{} & {} \times \eta _i \chi _i) \eta _e \chi _e)+I ~(M^2 \\{} & {} \times (M^2+R^2) (M^2+S^2) \lambda ^3-\eta _e \\{} & {} \times \chi _e (M^2+S^2) (M^2+2 R^2) \lambda ^2\\{} & {} +(-M^6+(R^2+S^2) M^4-R^2 S^2 M^2)\lambda \\{} & {} +M^2 (M^2-S^2) \eta _e \chi _e) \eta _i \chi _i)\\{} & {} \times I_0(S)S+(-I (M^2(M^2+R^2) \Lambda \lambda ^3\\{} & {} -\Lambda \chi _e (M^2\!+\!2 R^2) \eta _e \lambda ^2-(M\!+\!R) \Pi (M\!-\!R) M^2 \lambda \\{} & {} +M^2 \Pi \eta _e \chi _e)+ M ((M^4+R^2 M^2) \lambda ^3\\{} & {} -\eta _e \chi _e (M^2+2 R^2) \lambda ^2 \\{} & {} +(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) I_1 (-(2 M^2-2 S^2-2 \chi _i \eta _i)))\\{} & {} \times I_1(S)) K_1(M \lambda )) K_0(R \lambda )-R ((\lambda K_1(M \lambda )~\\{} & {} \times (\omega I_1(S)+S I_0(S) \eta _i \chi _i)\\{} & {} \times (2 (M^2-R^2) \lambda -2 \eta _e \chi _e)S^2\\{} & {} +I_0(S) ((L M^2 (M^2+R^2) \lambda ^3 \\{} & {} +L M^2 \eta _e \chi _e \lambda ^2-(M+R) (M^4+\kappa M^2 \\{} & {} +2 S^2 \eta _i \chi _i) (M-R) \lambda +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i)\\{} & {} \times \eta _e \chi _e) K_1\!-\!((M^2\!+\!R^2) (M^2\!+\!S^2) \lambda ^3\!+\!(M^2+S^2)\\{} & {} \times \eta _e \chi _e \lambda ^2+(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) \chi _i K M \eta _i) S\\{} & {} +(((M^2+R^2) \Lambda \lambda ^3+\Lambda \eta _e \chi _e \lambda ^2 \\{} & {} -(M+R)~ \Pi ~ (M-R) \lambda +\Pi \eta _e \chi _e) ~K M\\{} & {} -(2 M^2-2 S^2-2 \chi _i \eta _i)~ ((M^4+R^2 M^2) ~\lambda ^3\\{} & {} +M^2 \eta _e \chi _e \lambda ^2+(-M^4+(R^2\!+\!S^2)~ M^2\!-\!R^2 S^2) ~\lambda \\{} & {} +(M^2-S^2)\eta _e \chi _e) ~K_1)~ I_1(S)) I_0(M \lambda )\\{} & {} -(2 (M^2\!-\!R^2) \lambda \!-\!2 \eta _e \chi _e) (((L M I_1+I (M^2\!+\!S^2) \\{} & {} \times \eta _i \chi _i) S I_0(S)+(-I \Lambda -2 M\\{} & {} \times \omega I_1) I_1(S)) \lambda K_1(M \lambda )\\{} & {} -(\omega I_1(S)+S I_0(S) \eta _i \chi _i)\\{} & {} \times (K I_1+I K_1) S^2)+((I_1 (L M^2 (M^2+R^2)\lambda ^3\\{} & {} +L M^2 \eta _e \chi _e \lambda ^2-(M+R)\\{} & {} \times (M^4+\kappa M^2+2 S^2 \eta _i \chi _i) (M-R)\lambda \\{} & {} +(M^4+\kappa M^2+2 S^2 \eta _i \chi _i) \eta _e \chi _e)\\{} & {} +I M ((M^2+R^2) (M^2+S^2)\lambda ^3+(M^2+S^2) \eta _e \\{} & {} \times \chi _e \lambda ^2+(-M^4+(R^2+S^2)M^2\\{} & {} -R^2 S^2)~ \lambda +(M^2-S^2) \eta _e \chi _e)~ \eta _i \chi _i)\\{} & {} \times S ~I_0(S)+(-M I ((M^2+R^2)~ \Lambda \lambda ^3\\{} & {} +\Lambda \eta _e\chi _e \lambda ^2-(M+R)~ \Pi ~ (M-R)\lambda \\{} & {} +\Pi \eta _e \chi _e)+((M^4+R^2 M^2) \lambda ^3 \\{} & {} +M^2 \eta _e \chi _e \lambda ^2+(-M^4+(R^2+S^2) M^2-R^2 S^2) \lambda \\{} & {} +(M^2-S^2) \eta _e \chi _e) I_1 (-(2 M^2-2 S^2-2 \chi _i \eta _i)))\\{} & {} \times I_1(S)) K_0(M \lambda )+(K_0(M \lambda ) (\omega I_1(S)\\{} & {} + S I_0(S) \eta _i \chi _i) S^2 \\{} & {} +(M L K_1-\chi _i \eta _i K (M^2+S^2)) I_0(S) S \\{} & {} +(K \Lambda -2 M \omega K_1) I_1(S))\\{} & {} \times \lambda I_1(M \lambda ) (2 (M^2-R^2)\lambda \\{} & {} -2 \eta _e \chi _e)) K_1(R \lambda )) M. \end{aligned}$$

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Gamiel, Y., El-Sayed, M.K. & Elbehairy, M. The effects of stress jump conditions on the hydrodynamic permeability of filtration processes using effective medium approximation. Pramana - J Phys 97, 83 (2023). https://doi.org/10.1007/s12043-023-02554-9

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  • DOI: https://doi.org/10.1007/s12043-023-02554-9

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