Free Convective Boundary Layer Flow of Radiating and Reacting MHD Fluid Past a Cosinusoidally Fluctuating Heated Plate

Abstract

The work is focused on unsteady magneto-hydrodynamics free convective flow past a moving vertical porous plate in the presence of thermal radiation and first order chemical reaction with viscous dissipation. The temperature of the plate is assumed spanwise cosinusoidally fluctuating with time by taking into account of heat generation. The second order perturbation technique is employed to investigate the non-linear partial differential equations governing the fluid flow which are non-dimensionalized by introducing the similarity transformations. For the numerical computation, flex PDE Solver is used to study the effects of magnetic intensity, radiation, porous permeability, Eckert number, Schmidt number and heat generation/absorption parameters on velocity and temperature profiles. Also, the skin friction coefficients, the rate of heat transfer and the rate of mass transfer at the surface of the plate are computed numerically. The results show that within the boundary layer, the velocity and temperature are found to decrease with the increasing values of Prandtl number and radiation parameter however the trend is reverse with respect to porous permeability and heat generation/ absorption parameters.

Appendix

$$\begin{aligned} N= & {} M^{2}+\frac{1}{K}\,,\,\alpha _1 =N+\pi ^{2}-i\omega \\ \alpha _2= & {} Re^{2}R-Pr Q\\ \alpha _3= & {} Re^{2}R+\pi ^{2}-i\omega Pr-QPr\\ \alpha _4= & {} 4\pi ^{2}+N-2i\omega \\ \alpha _5= & {} 4\pi ^{2}+Re^{2}R N-2i \omega Pr-Q Pr\\ m_1= & {} 0.5\left[ {Re Sc+\sqrt{Re^{2}Sc^{2}+4\gamma Sc}} \right] \\ m_2= & {} 0.5\left[ {Re Pr+\sqrt{Re^{2}Pr^{2}+4 \alpha _2 }} \right] \\ m_3= & {} 0.5\left[ {Re+\sqrt{Re^{2}+4N}} \right] \\ m_4= & {} 0.5\left[ {Re Pr+\sqrt{Re^{2} Pr^{2}+4\alpha _3 }} \right] \\ m_5= & {} 0.5\left[ {Re+\sqrt{Re^{2}+4\alpha _1 }} \right] \\ m_6= & {} 0.5\left[ {Re Pr+\sqrt{Re^{2}Pr^{2}+4\alpha _5 }} \right] \\ m_7= & {} 0.5\left[ {Re+\sqrt{Re^{2}+4\alpha _4 }} \right] \\ A_1= & {} \frac{Re^{2}Gr}{m_2^2 -m_2 Re-N}\\ A_2= & {} \frac{Re^{2}Gm}{m_1^2 -m_1 Re-N}\\ A_3= & {} A_1 +A_2\\ A_4= & {} \frac{Pr\,m_3 A_3 }{m_3^2 -m_3 Re Pr-\alpha _2 }\\ A_5= & {} \frac{Pr\,m_2 A_1 }{m_2^2 -m_2 Re Pr-\alpha _2 }\\ A_6= & {} \frac{Pr\,m_1 A_2 }{m_1^2 -m_1 Re Pr-\alpha _2 }\\ A_7= & {} -A_4 +A_5 +A_6\\ A_8= & {} \frac{Re^{2}\,Gr A_7 }{m_2^2 -m_2 Re-N}\\ \end{aligned}$$

$$\begin{aligned} A_9= & {} \frac{Re^{2}\,Gr A_4 }{m_3^2 -m_3 Re-N}\\ A_{10}= & {} \frac{Re^{2}\,Gr A_5 }{m_2^2 -m_2 Re-N}\\ A_{11}= & {} \frac{Re^{2}\,Gr A_6 }{m_1^2 -m_1 Re-N}\\ A_{12}= & {} A_8 +A_9 -A_{10} -A_{11}\\ A_{13}= & {} m_3 A_{12} , A_{14} =m_2 A_8\\ A_{15}= & {} m_3 A_9 , A_{16} =m_2 A_{10}\\ A_{17}= & {} m_1 A_{11} , A_{18} =-2Pr m_3 A_3 A_{13}\\ A_{19}= & {} -2Pr m_3 A_3 A_{14} , A_{20} =-2Pr m_3 A_3 A_{15}\\ A_{21}= & {} -2Pr m_3 A_3 A_{16} , A_{22} =-2Pr m_3 A_3 A_{17}\\ A_{23}= & {} -2Pr m_2 A_1 A_{13} , A_{24} =-2Pr m_2 A_1 A_{14}\\ A_{25}= & {} -2Pr m_2 A_1 A_{15} , A_{26} =-2Pr m_2 A_1 A_{16}\\ A_{27}= & {} -2Pr m_2 A_1 A_{17} , A_{28} =-2Pr m_1 A_2 A_{13}\\ A_{29}= & {} -2Pr m_1 A_2 A_{14} , A_{30} =-2Pr m_1 A_2 A_{15}\\ A_{31}= & {} -2Pr\,m_1 A_2 A_{16} , A_{32} =-2Pr m_1 A_2 A_{17}\\ A_{33}= & {} \frac{A_{18} }{4m_3^2 -2m_3 Re Pr-\alpha _2 }\\ A_{34}= & {} \frac{A_{19} }{\left( {m_3 +m_2 } \right) ^{2}-\left( {m_3 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{35}= & {} \frac{A_{20} }{4m_3^2 -2m_3 Re Pr-\alpha _2 }\\ A_{36}= & {} \frac{A_{21} }{\left( {m_3 +m_2 } \right) ^{2}-\left( {m_3 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{37}= & {} \frac{A_{22} }{\left( {m_3 +m_1 } \right) ^{2}-\left( {m_3 +m_1 } \right) Re Pr-\alpha _2 }\\ A_{38}= & {} \frac{A_{23} }{\left( {m_3 +m_2 } \right) ^{2}-\left( {m_3 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{39}= & {} \frac{A_{24} }{4m_2^2 -2m_2 Re Pr-\alpha _2 }\\ A_{40}= & {} \frac{A_{25} }{\left( {m_3 +m_2 } \right) ^{2}-\left( {m_3 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{41}= & {} \frac{A_{26} }{4m_2^2 -2m_2 Re Pr-\alpha _2 }\\ A_{42}= & {} \frac{A_{27} }{\left( {m_1 +m_2 } \right) ^{2}-\left( {m_1 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{43}= & {} \frac{A_{28} }{\left( {m_1 +m_3 } \right) ^{2}-\left( {m_1 +m_3 } \right) Re Pr-\alpha _2 }\\ A_{44}= & {} \frac{A_{29} }{\left( {m_1 +m_2 } \right) ^{2}-\left( {m_1 +m_2 } \right) Re Pr-\alpha _2 }\\ \end{aligned}$$

$$\begin{aligned} A_{45}= & {} \frac{A_{30} }{\left( {m_1 +m_3 } \right) ^{2}-\left( {m_1 +m_3 } \right) Re Pr-\alpha _2 }\\ A_{46}= & {} \frac{A_{31} }{\left( {m_1 +m_2 } \right) ^{2}-\left( {m_1 +m_2 } \right) Re Pr-\alpha _2 }\\ A_{47}= & {} \frac{A_{32} }{4m_1^2 -2m_2 Re Pr-\alpha _2 }\\ A_{48}= & {} \left( {{\begin{array}{l} {A_{33} -A_{34} -A_{35} +A_{36} } \\ {+A_{37} -A_{38} +A_{39} +A_{40} } \\ {-A_{41} -A_{42} -A_{43} +A_{44} } \\ {+A_{45} -A_{46} -A_{47} } \\ \end{array} }} \right) \\ A_{49}= & {} \frac{Re^{2}Gr A_{48} }{m_2^2 -m_2 Re-N}\\ A_{50}= & {} \frac{Re^{2}Gr A_{33} }{4m_3^2 -2m_3 Re-N}\\ A_{51}= & {} \frac{Re^{2}Gr A_{34} }{\left( {m_2 +m_3 } \right) ^{2}-\left( {m_2 +m_3 } \right) Re-N}\\ A_{52}= & {} \frac{Re^{2}Gr A_{35} }{4m_3^2 -2m_3 Re-N}\\ A_{53}= & {} \frac{Re^{2}Gr A_{36} }{\left( {m_2 +m_3 } \right) ^{2}-\left( {m_2 +m_3 } \right) Re-N}\\ A_{54}= & {} \frac{Re^{2}Gr A_{37} }{\left( {m_1 +m_3 } \right) ^{2}-\left( {m_1 +m_3 } \right) Re-N}\\ A_{55}= & {} \frac{Re^{2}Gr A_{38} }{\left( {m_2 +m_3 } \right) ^{2}-\left( {m_2 +m_3 } \right) Re-N}\\ A_{56}= & {} \frac{Re^{2}Gr A_{39} }{4m_2^2 -2m_2 Re-N}\\ A_{57}= & {} \frac{Re^{2}Gr A_{40} }{\left( {m_2 +m_3 } \right) ^{2}-\left( {m_2 +m_3 } \right) Re-N}\\ A_{58}= & {} \frac{Re^{2}Gr A_{41} }{4m_2^2 -2m_2 Re-N}\\ A_{59}= & {} \frac{Re^{2}Gr A_{42} }{\left( {m_2 +m_1 } \right) ^{2}-\left( {m_2 +m_1 } \right) Re-N}\\ A_{60}= & {} \frac{Re^{2}Gr A_{43} }{\left( {m_1 +m_3 } \right) ^{2}-\left( {m_1 +m_3 } \right) Re-N}\\ A_{61}= & {} \frac{Re^{2}Gr A_{44} }{\left( {m_2 +m_1 } \right) ^{2}-\left( {m_2 +m_1 } \right) Re-N}\\ A_{62}= & {} \frac{Re^{2}Gr A_{45} }{\left( {m_1 +m_3 } \right) ^{2}-\left( {m_1 +m_3 } \right) Re-N}\\ A_{63}= & {} \frac{Re^{2}Gr A_{46} }{\left( {m_2 +m_1 } \right) ^{2}-\left( {m_2 +m_1 } \right) Re-N}\\ \end{aligned}$$

$$\begin{aligned} A_{64}= & {} \frac{Re^{2}Gr A_{47} }{4m_1^2 -2\,m_1 \,Re-N}\\ A_{65}= & {} \left( {{\begin{array}{l} {A_{49} -A_{50} +A_{51} +A_{52} } \\ {-A_{53} -A_{54} +A_{55} -A_{56} } \\ {-A_{57} +A_{58} +A_{59} -A_{61} } \\ {-A_{61} -A_{62} +A_{63} +A_{64} } \\ \end{array} }} \right) \\ A_{66}= & {} \frac{Re^{2}Gr A_{47} }{m_4^2 -Re-\alpha _1 }\\ A_{67}= & {} 2Pr\,m_3 \,m_5 \,A_3 A_6\\ A_{68}= & {} 2Pr m_3 \,m_4 \,A_3 A_{66}\\ A_{69}= & {} 2Pr\,m_2 \,m_5 \,A_1 A_6\\ A_{70}= & {} 2Pr\,m_2 \,m_4 \,A_1 A_{66}\\ A_{71}= & {} 2Pr\,m_1 \,m_5 \,A_2 A_6\\ A_{72}= & {} 2Pr\,m_1 \,m_4 \,A_2 A_{66}\\ A_{73}= & {} \frac{A_{67} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re\,Pr-\alpha _3 }\\ A_{74}= & {} \frac{A_{68} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Re\,Pr-\alpha _3 }\\ A_{75}= & {} \frac{A_{69} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Re\,Pr-\alpha _3 }\\ A_{76}= & {} \frac{A_{70} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Re\,Pr-\alpha _3 }\\ A_{77}= & {} \frac{A_{71} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Re\,Pr-\alpha _3 }\\ A_{78}= & {} \frac{A_{72} }{\left( {m_1 +m_4 } \right) ^{2}-\left( {m_1 +m_4 } \right) Re\,Pr-\alpha _3 }\\ A_{79}= & {} \left( {{\begin{array}{l} {A_{73} -A_{74} -A_{75} +A_{76} } \\ {-A_{77} +A_{78} } \\ \end{array} }} \right) \\ A_{80}= & {} \frac{Re^{2}Gr A_{79} }{m_4^2 -m_4 Re-\alpha _1 }\\ A_{81}= & {} \frac{Re^{2}Gr A_{73} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{82}= & {} \frac{Re^{2}Gr A_{74} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{83}= & {} \frac{Re^{2}Gr A_{75} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{84}= & {} \frac{Re^{2}Gr A_{76} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Re-\alpha _1 }\\ A_{85}= & {} \frac{Re^{2}Gr A_{77} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Re-\alpha _1 }\\ \end{aligned}$$

$$\begin{aligned} A_{86}= & {} \frac{Re^{2}Gr A_{78} }{\left( {m_1 +m_4 } \right) ^{2}-\left( {m_1 +m_4 } \right) Re-\alpha _1 }\\ A_{87}= & {} \left( {\begin{array}{l} A_{80} -A_{81} +A_{82} +A_{83} \\ -A_{84} +A_{78} \\ \end{array}} \right) \\ A_{88}= & {} 2Pr\,m_3 \,m_5 \,A_3 A_{87}\\ A_{89}= & {} 2Pr m_3 \,m_5 \,A_3 A_{80}\\ A_{90}= & {} 2Pr\,m_3 \,(m_3 +m_5 )\,A_3 A_{81}\\ A_{91}= & {} 2Pr\,m_3 \,(m_3 +m_4 )\,A_3 A_{82}\\ A_{92}= & {} 2Pr\,m_3 \,(m_2 +m_5 )\,A_3 A_{83}\\ A_{93}= & {} 2Pr\,m_3 \,(m_2 +m_4 )\,A_3 A_{84}\\ A_{94}= & {} 2Pr m_3 \,(m_1 +m_5 )\,A_3 A_{85}\\ A_{95}= & {} 2Pr\,m_3 \,(m_1 +m_4 )\,A_3 A_{86}\\ A_{96}= & {} 2Pr m_2 \,m_5 \,A_1 A_{87}\\ A_{97}= & {} 2Pr\,m_2 \,m_5 \,A_1 A_{80}\\ A_{98}= & {} 2Pr\,m_2 \,(m_3 +m_5 )\,A_1 A_{81}\\ A_{99}= & {} 2Pr\,m_2 \,(m_3 +m_4 )\,A_1 A_{82}\\ A_{100}= & {} 2Pr m_2 \,(m_2 +m_5 )\,A_1 A_{83}\\ A_{101}= & {} 2Pr\,m_2 \,(m_2 +m_4 )\,A_1 A_{84}\\ A_{102}= & {} 2Pr\,m_2 \,(m_1 +m_5 )\,A_1 A_{85}\\ A_{103}= & {} 2 Pr m_2 \,(m_1 +m_4 )\,A_1 A_{86}\\ A_{104}= & {} 2Pr \,m_1 \,m_5 \,A_2 A_{87}\\ A_{105}= & {} 2Pr \,m_1 \,m_5 \,A_2 A_{80}\\ A_{106}= & {} 2Pr \,m_1 \,(m_3 +m_5 )\,A_2 A_{81}\\ A_{107}= & {} 2Pr \,m_1 \,(m_3 +m_4 )\,A_2 A_{82}\\ A_{108}= & {} 2Pr \,m_1 \,(m_2 +m_5 )\,A_2 A_{83}\\ A_{109}= & {} 2Pr \,m_1 \,(m_2 +m_4 )\,A_2 A_{84}\\ A_{110}= & {} 2Pr m_1 \,(m_1 +m_5 )\,A_2 A_{85}\\ A_{111}= & {} 2Pr \,m_1 \,(m_1 +m_4 )\,A_2 A_{86}\\ A_{112}= & {} 2Pr \,m_5 \,A_6 A_{13} , A_{113} =2Pr \,m_4 \,A_{66} A_{13}\\ A_{114}= & {} 2Pr \,m_5 \,A_6 A_{14} , A_{115} =2Pr \,m_4 \,A_{66} A_{14}\\ A_{116}= & {} 2Pr \,m_5 \,A_6 A_{15} , A_{117} =2Pr \,m_4 \,A_{66} A_{15}\\ A_{118}= & {} 2Pr \,m_5 \,A_{16} A_6 , A_{119} =2Pr \,m_4 \,A_{16} A_{66}\\ A_{120}= & {} 2Pr \,m_5 \,A_{17} A_6 , A_{121} =2Pr m_4 \,A_{17} A_{66}\\ A_{122}= & {} \frac{A_{88} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{123}= & {} \frac{A_{89} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{124}= & {} \frac{A_{90} }{\left( {2m_3 +m_5 } \right) ^{2}-\left( {2m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ \end{aligned}$$

$$\begin{aligned} A_{125}= & {} \frac{A_{91} }{\left( {2m_3 +m_4 } \right) ^{2}-\left( {2m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{126}= & {} \frac{A_{93} }{\left( {m_2 +m_3 +m_4 } \right) ^{2}-\left( {m_2 +m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{127}= & {} \frac{A_{94} }{\left( {m_1 +m_3 +m_5 } \right) ^{2}-\left( {m_1 +m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{128}= & {} \frac{A_{95} }{\left( {m_1 +m_3 +m_4 } \right) ^{2}-\left( {m_1 +m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{129}= & {} \frac{A_{96} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{130}= & {} \frac{A_{97} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{131}= & {} \frac{A_{98} }{\left( {m_2 +m_3 +m_5 } \right) ^{2}-\left( {m_2 +m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{132}= & {} \frac{A_{99} }{\left( {m_2 +m_3 +m_4 } \right) ^{2}-\left( {m_2 +m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{133}= & {} \frac{A_{100} }{\left( {2m_2 +m_5 } \right) ^{2}-\left( {2m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{134}= & {} \frac{A_{101} }{\left( {2m_2 +m_4 } \right) ^{2}-\left( {2m_2 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{135}= & {} \frac{A_{102} }{\left( {m_1 +m_2 +m_5 } \right) ^{2}-\left( {m_1 +m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{136}= & {} \frac{A_{103} }{\left( {m_1 +m_2 +m_4 } \right) ^{2}-\left( {m_1 +m_2 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{137}= & {} \frac{A_{104} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{138}= & {} \frac{A_{105} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{139}= & {} \frac{A_{106} }{\left( {m_1 +m_3 +m_5 } \right) ^{2}-\left( {m_1 +m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{140}= & {} \frac{A_{107} }{\left( {m_1 +m_3 +m_4 } \right) ^{2}-\left( {m_1 +m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{141}= & {} \frac{A_{108} }{\left( {m_1 +m_2 +m_5 } \right) ^{2}-\left( {m_1 +m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{142}= & {} \frac{A_{109} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{143}= & {} \frac{A_{110} }{\left( {2m_1 +m_5 } \right) ^{2}-\left( {2m_1 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{144}= & {} \frac{A_{111} }{\left( {2m_1 +m_5 } \right) ^{2}-\left( {2m_1 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{145}= & {} \frac{A_{112} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ \end{aligned}$$

$$\begin{aligned} A_{146}= & {} \frac{A_{113} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{147}= & {} \frac{A_{114} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{148}= & {} \frac{A_{115} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{149}= & {} \frac{A_{116} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{150}= & {} \frac{A_{117} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{151}= & {} \frac{A_{118} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{152}= & {} \frac{A_{119} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{153}= & {} \frac{A_{120} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Pr Re-\alpha _3 }\\ A_{154}= & {} \frac{A_{121} }{\left( {m_1 +m_4 } \right) ^{2}-\left( {m_1 +m_4 } \right) Pr Re-\alpha _3 }\\ A_{155}= & {} \left( {\begin{array}{l} -A_{122} -A_{123} +A_{124} -A_{125} \\ +A_{126} -A_{127} +A_{128} -A_{129} \\ +A_{130} -A_{131} +A_{132} +A_{133} \\ -A_{134} +A_{135} +A_{136} -A_{137} \\ +A_{138} -A_{139} +A_{140} +A_{141} \\ -A_{142} +A_{143} -A_{144} +A_{145} \\ -A_{146} -A_{147} +A_{148} -A_{149} \\ +A_{150} +A_{151} -A_{152} +A_{153} \\ -A_{154} \\ \end{array}} \right) \\ A_{156}= & {} \frac{Re^{2}Gr A_{155} }{m_4^2 -m_4 Re-\alpha _1 }\\ A_{157}= & {} \frac{Re^{2}Gr A_{122} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{158}= & {} \frac{Re^{2}Gr A_{123} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{159}= & {} \frac{Re^{2}Gr A_{124} }{\left( {2m_3 +m_4 } \right) ^{2}-\left( {2m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{160}= & {} \frac{Re^{2}Gr A_{125} }{\left( {2m_3 +m_4 } \right) ^{2}-\left( {2m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{161}= & {} \frac{Re^{2}Gr A_{126} }{\left( {m_2 +m_3 +m_4 } \right) ^{2}-\left( {m_2 +m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{162}= & {} \frac{Re^{2}Gr A_{127} }{\left( {m_1 +m_3 +m_5 } \right) ^{2}-\left( {m_1 +m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{163}= & {} \frac{Re^{2}Gr A_{128} }{\left( {m_1 +m_3 +m_4 } \right) ^{2}-\left( {m_1 +m_3 +m_4 } \right) Re-\alpha _1 }\\ \end{aligned}$$

$$\begin{aligned} A_{164}= & {} \frac{Re^{2}Gr A_{129} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{165}= & {} \frac{Re^{2}Gr A_{130} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{166}= & {} \frac{Re^{2}Gr A_{131} }{\left( {m_2 +m_3 +m_5 } \right) ^{2}-\left( {m_2 +m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{167}= & {} \frac{Re^{2}Gr A_{132} }{\left( {m_2 +m_3 +m_4 } \right) ^{2}-\left( {m_2 +m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{168}= & {} \frac{Re^{2}Gr A_{133} }{\left( {2m_2 +m_5 } \right) ^{2}-\left( {2m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{169}= & {} \frac{Re^{2}Gr A_{134} }{\left( {2m_2 +m_4 } \right) ^{2}-\left( {2m_2 +m_4 } \right) Re-\alpha _1 }\\ A_{170}= & {} \frac{Re^{2}Gr A_{135} }{\left( {m_1 +m_2 +m_5 } \right) ^{2}-\left( {m_1 +m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{171}= & {} \frac{Re^{2}Gr A_{136} }{\left( {m_1 +m_2 +m_4 } \right) ^{2}-\left( {m_1 +m_2 +m_4 } \right) Re-\alpha _1 }\\ A_{172}= & {} \frac{Re^{2}Gr A_{137} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Re-\alpha _1 }\\ A_{173}= & {} \frac{Re^{2}Gr A_{138} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Re-\alpha _1 }\\ A_{174}= & {} \frac{Re^{2}Gr A_{139} }{\left( {m_1 +m_3 +m_5 } \right) ^{2}-\left( {m_1 +m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{175}= & {} \frac{Re^{2}Gr A_{140} }{\left( {m_1 +m_3 +m_4 } \right) ^{2}-\left( {m_1 +m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{176}= & {} \frac{Re^{2}Gr A_{141} }{\left( {m_1 +m_2 +m_5 } \right) ^{2}-\left( {m_1 +m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{177}= & {} \frac{Re^{2}Gr A_{142} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Re-\alpha _1 }\\ A_{178}= & {} \frac{Re^{2}Gr A_{143} }{\left( {2m_1 +m_5 } \right) ^{2}-\left( {2m_1 +m_5 } \right) Re-\alpha _1 }\\ A_{179}= & {} \frac{Re^{2}Gr A_{144} }{\left( {2m_1 +m_5 } \right) ^{2}-\left( {2m_1 +m_5 } \right) Re-\alpha _1 }\\ A_{180}= & {} \frac{Re^{2}Gr A_{145} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{181}= & {} \frac{Re^{2}Gr A_{146} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{182}= & {} \frac{Re^{2}Gr A_{147} }{\left( {m_5 +m_2 } \right) ^{2}-\left( {m_5 +m_2 } \right) Re-\alpha _1 }\\ A_{183}= & {} \frac{Re^{2}Gr A_{148} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Re-\alpha _1 }\\ \end{aligned}$$

$$\begin{aligned} A_{184}= & {} \frac{Re^{2}Gr A_{149} }{\left( {m_3 +m_5 } \right) ^{2}-\left( {m_3 +m_5 } \right) Re-\alpha _1 }\\ A_{185}= & {} \frac{Re^{2}Gr A_{150} }{\left( {m_3 +m_4 } \right) ^{2}-\left( {m_3 +m_4 } \right) Re-\alpha _1 }\\ A_{186}= & {} \frac{Re^{2}Gr A_{151} }{\left( {m_2 +m_5 } \right) ^{2}-\left( {m_2 +m_5 } \right) Re-\alpha _1 }\\ A_{187}= & {} \frac{Re^{2}Gr A_{152} }{\left( {m_2 +m_4 } \right) ^{2}-\left( {m_2 +m_4 } \right) Re-\alpha _1 }\\ A_{188}= & {} \frac{Re^{2}Gr A_{153} }{\left( {m_1 +m_5 } \right) ^{2}-\left( {m_1 +m_5 } \right) Re-\alpha _1 }\\ A_{189}= & {} \frac{Re^{2}Gr A_{154} }{\left( {m_1 +m_4 } \right) ^{2}-\left( {m_1 +m_4 } \right) Re-\alpha _1 }\\ A_{190}= & {} \left( {\begin{array}{l} A_{156} -A_{157} +A_{158} -A_{159} \\ +A_{160} -A_{161} +A_{162} -A_{163} \\ +A_{164} -A_{165} +A_{166} -A_{167} \\ -A_{168} +A_{169} -A_{170} -A_{171} \\ +A_{172} -A_{173} +A_{174} -A_{175} \\ -A_{176} +A_{177} -A_{178} +A_{179} \\ -A_{180} +A_{181} +A_{182} -A_{183} \\ +A_{184} -A_{185} -A_{186} +A_{187} \\ -A_{188} +A_{189} \\ \end{array}} \right) \\ A_{191}= & {} Pr m_5^2 A_6^2 , A_{192} =Pr m_4^2 A_{66}^2\\ A_{193}= & {} 2Pr m_4 m_5 A_6 A_{66}\\ A_{194}= & {} \frac{A_{157} }{4m_5^2 -2Pr Re m_5 -\alpha _5 }\\ A_{195}= & {} \frac{A_{158} }{4m_4^2 -2Pr Re m_4 -\alpha _5 }\\ A_{196}= & {} \frac{A_{159} }{\left( {m_5 +m_4 } \right) ^{2}-\left( {m_5 +m_4 } \right) Re-\alpha _5 }\\ A_{197}= & {} A_{194} +A_{195} -A_{196}\\ A_{198}= & {} Re^{2}Gr A_{163} , A_{199} =Re^{2}Gr A_{160}\\ A_{200}= & {} Re^{2}\,Gr\,A_{161} , A_{201} =Re^{2}\,Gr\,A_{162}\\ A_{202}= & {} \frac{A_{198} }{m_6^2 -\,Re\,m_6 -\alpha _4 }\\ A_{203}= & {} \frac{A_{199} }{4m_5^2 -\,2Re\,m_5 -\alpha _5 }\\ A_{204}= & {} \frac{A_{200} }{4m_4^2 -\,2Re\,m_4 -\alpha _5 }\\ A_{205}= & {} \frac{A_{201} }{\left( {m_5 +m_4 } \right) ^{2}-\left( {m_5 +m_4 } \right) Re-\alpha _5 }\\ A_{206}= & {} A_{202} -A_{203} -A_{204} +A_{205}\\ A_{207}= & {} 2\,Pr \,m_5^2 \,A_6 \,A_{87}\\ \end{aligned}$$

$$\begin{aligned} A_{208}= & {} 2\,Pr m_5^2 \,A_6 \,A_{80}\\ A_{209}= & {} 2\,Pr \,m_5 \,\left( {m_3 +m_5 } \right) A_6 \,A_{81}\\ A_{210}= & {} 2Pr m_5 \,\left( {m_3 +m_4 } \right) A_6 \,A_{82}\\ A_{211}= & {} 2Pr m_5 \left( {m_2 +m_5 } \right) A_6 \,A_{83}\\ A_{212}= & {} 2Pr m_5 \left( {m_2 +m_4 } \right) A_6 \,A_{84}\\ A_{213}= & {} 2Pr m_5 \left( {m_1 +m_5 } \right) A_6 \,A_{85}\\ A_{214}= & {} 2Pr m_5 \left( {m_1 +m_4 } \right) A_6 A_{86}\\ A_{215}= & {} 2Pr m_4 m_5 \,A_{66} A_{80}\\ A_{216}= & {} 2Pr m_4 \left( {m_3 +m_5 } \right) A_{66} A_{81}\\ A_{217}= & {} 2Pr m_4 \left( {m_3 +m_4 } \right) A_{66} A_{82}\\ A_{218}= & {} 2Pr\,m_4 \,\left( {m_2 +m_5 } \right) A_{66} \,A_{83}\\ A_{219}= & {} 2Pr m_4 \left( {m_2 +m_4 } \right) A_{66} \,A_{84}\\ A_{220}= & {} 2Pr m_4 \left( {m_1 +m_{5.} } \right) A_{66} \,A_{85}\\ A_{221}= & {} 2Pr m_4 \left( {m_1 +m_{4.} } \right) A_{66} \,A_{86}\\ A_{222}= & {} A_3 \,A_{172} , A_{223} =A_3 \,A_{168}\\ A_{224}= & {} A_3 \,A_{169} , A_{225} =A_3 \,A_{170}\\ A_{226}= & {} A_3 \,A_{171} , A_{227} =A_1 \,A_{172}\\ A_{228}= & {} A_1 \,A_{168} , A_{229} =A_1 \,A_{169}\\ A_{230}= & {} A_1 \,A_{170} , A_{231} =A_1 \,A_{171}\\ A_{232}= & {} A_2 \,A_{172} , A_{233} =A_2 \,A_{168}\\ A_{234}= & {} A_2 \,A_{169} , A_{235} =A_2 \,A_{170}\\ A_{236}= & {} A_2 \,A_{171}\\ A_{237}= & {} \frac{A_{207} }{4m_5^2 -\,2Re Pr \,m_5 -\alpha _5 }\\ A_{238}= & {} \frac{A_{208} }{4m_5^2 -\,2Re Pr \,m_5 -\alpha _5 }\\ A_{239}= & {} \frac{A_{209} }{\left( {m_3 +2m_5 } \right) ^{2}-\left( {m_3 +2m_5 } \right) Pr \,Re-\alpha _5 }\\ A_{240}= & {} \frac{A_{210} }{\left( {m_3 +m_4 +m_5 } \right) ^{2}-\left( {m_3 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{241}= & {} \frac{A_{211} }{\left( {2m_5 +m_2 } \right) ^{2}-\left( {2m_5 +m_2 } \right) Pr Re-\alpha _5 }\\ A_{242}= & {} \frac{A_{212} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) PrRe-\alpha _5 }\\ A_{243}= & {} \frac{A_{213} }{\left( {m_1 +2m_5 } \right) ^{2}-\left( {m_1 +2m_5 } \right) Pr \,Re-\alpha _5 }\\ A_{244}= & {} \frac{A_{214} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{245}= & {} \frac{A_{215} }{\left( {m_4 +m_5 } \right) ^{2}-\left( {m_4 +m_5 } \right) Pr \,Re-\alpha _5 }\\ \end{aligned}$$

$$\begin{aligned} A_{246}= & {} \frac{A_{216} }{\left( {m_3 +m_4 +m_5 } \right) ^{2}-\left( {m_3 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{247}= & {} \frac{A_{217} }{\left( {m_3 +2m_4 } \right) ^{2}-\left( {m_3 +2m_4 } \right) Pr \,Re-\alpha _5 }\\ A_{248}= & {} \frac{A_{218} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) PrRe-\alpha _5 }\\ A_{249}= & {} \frac{A_{219} }{\left( {m_2 +2m_4 } \right) ^{2}-\left( {m_2 +2m_4 } \right) Pr Re-\alpha _5 }\\ A_{250}= & {} \frac{A_{220} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{251}= & {} \frac{A_{221} }{\left( {m_1 +2m_4 } \right) ^{2}-\left( {m_1 +2m_4 } \right) Pr Re-\alpha _5 }\\ A_{252}= & {} \frac{A_{222} }{\left( {m_3 +m_7 } \right) ^{2}-\left( {m_3 +m_7 } \right) Pr Re-\alpha _5 }\\ A_{253}= & {} \frac{A_{223} }{\left( {m_3 +m_6 } \right) ^{2}-\left( {m_3 +m_6 } \right) Pr Re-\alpha _5 }\\ A_{254}= & {} \frac{A_{224} }{\left( {m_3 +2m_5 } \right) ^{2}-\left( {m_3 +2m_5 } \right) Pr\,Re-\alpha _5 }\\ A_{255}= & {} \frac{A_{225} }{\left( {m_3 +2m_4 } \right) ^{2}-\left( {m_3 +2m_4 } \right) Pr Re-\alpha _5 }\\ A_{256}= & {} \frac{A_{226} }{\left( {m_3 +m_4 +m_5 } \right) ^{2}-\left( {m_3 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{257}= & {} \frac{A_{227} }{\left( {m_2 +m_7 } \right) ^{2}-\left( {m_2 +m_7 } \right) Pr \,Re-\alpha _5 }\\ A_{258}= & {} \frac{A_{228} }{\left( {m_2 +m_6 } \right) ^{2}-\left( {m_2 +m_6 } \right) Pr \,Re-\alpha _5 }\\ A_{259}= & {} \frac{A_{229} }{\left( {m_2 +2m_5 } \right) ^{2}-\left( {m_2 +2m_5 } \right) Pr \,Re-\alpha _5 }\\ A_{260}= & {} \frac{A_{230} }{\left( {m_2 +2m_4 } \right) ^{2}-\left( {m_2 +2m_4 } \right) Pr Re-\alpha _5 }\\ A_{261}= & {} \frac{A_{231} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ A_{262}= & {} \frac{A_{232} }{\left( {m_1 +m_7 } \right) ^{2}-\left( {m_1 +m_7 } \right) Pr \,Re-\alpha _5 }\\ A_{263}= & {} \frac{A_{233} }{\left( {m_1 +2m_4 } \right) ^{2}-\left( {m_1 +2m_4 } \right) Pr \,Re-\alpha _5 }\\ A_{264}= & {} \frac{A_{234} }{\left( {m_1 +2m_5 } \right) ^{2}-\left( {m_1 +2m_5 } \right) Pr Re-\alpha _5 }\\ A_{265}= & {} \frac{A_{235} }{\left( {m_1 +2m_4 } \right) ^{2}-\left( {m_1 +2m_4 } \right) Pr \,Re-\alpha _5 }\\ A_{266}= & {} \frac{A_{236} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) Pr Re-\alpha _5 }\\ \end{aligned}$$

$$\begin{aligned} A_{267}= & {} \left( {\begin{array}{l} A_{237} -A_{238} -A_{239} -A_{240} \\ -A_{241} +A_{242} -A_{243} +A_{244} \\ -A_{245} -A_{246} +A_{247} +A_{248} \\ -A_{249} +A_{250} -A_{251} -A_{252} \\ +A_{253} -A_{254} -A_{255} +A_{256} \\ -A_{257} -A_{258} +A_{259} +A_{260} \\ -A_{261} +A_{262} -A_{263} +A_{264} \\ +A_{265} -A_{266} \\ \end{array}} \right) \\ A_{268}= & {} \frac{Re^{2}Gr A_{267} }{m_6^2 -\,Re\,m_6 -\alpha _4 }\\ A_{269}= & {} \frac{Re^{2}Gr A_{237} }{4m_5^2 -\,2Re\,m_5 -\alpha _4 }\\ A_{270}= & {} \frac{Re^{2}Gr A_{238} }{4m_5^2 -\,2Re\,m_5 -\alpha _4 }\\ A_{271}= & {} \frac{Re^{2}Gr A_{239} }{\left( {m_3 +2m_5 } \right) ^{2}-\left( {m_3 +2m_5 } \right) \,Re-\alpha _4 }\\ A_{272}= & {} \frac{Re^{2}Gr A_{240} }{\left( {m_3 +m_4 +m_5 } \right) ^{2}-\left( {m_3 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{273}= & {} \frac{Re^{2}Gr A_{241} }{\left( {2m_5 +m_2 } \right) ^{2}-\left( {2m_5 +m_2 } \right) \,Re-\alpha _4 }\\ A_{274}= & {} \frac{Re^{2}Gr A_{242} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{275}= & {} \frac{Re^{2}Gr A_{243} }{\left( {m_1 +2m_5 } \right) ^{2}-\left( {m_1 +2m_5 } \right) \,Re-\alpha _4 }\\ A_{276}= & {} \frac{Re^{2}Gr A_{244} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{277}= & {} \frac{Re^{2}Gr A_{245} }{\left( {m_4 +m_5 } \right) ^{2}-\left( {m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{278}= & {} \frac{Re^{2}Gr A_{246} }{\left( {m_3 +m_4 +m_5 } \right) ^{2}-\left( {m_3 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{279}= & {} \frac{Re^{2}Gr A_{247} }{\left( {m_3 +2m_4 } \right) ^{2}-\left( {m_3 +2m_4 } \right) \,Re-\alpha _4 }\\ A_{280}= & {} \frac{Re^{2}Gr A_{248} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{281}= & {} \frac{Re^{2}Gr A_{249} }{\left( {m_2 +2m_4 } \right) ^{2}-\left( {m_2 +2m_4 } \right) \,Re-\alpha _4 }\\ A_{282}= & {} \frac{Re^{2}Gr A_{250} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{283}= & {} \frac{Re^{2}Gr A_{251} }{\left( {m_1 +2m_4 } \right) ^{2}-\left( {m_1 +2m_4 } \right) \,Re-\alpha _4 }\\ \end{aligned}$$

$$\begin{aligned} A_{284}= & {} \frac{Re^{2}Gr A_{252} }{\left( {m_3 +m_7 } \right) ^{2}-\left( {m_3 +m_7 } \right) \,Re-\alpha _4 }\\ A_{285}= & {} \frac{Re^{2}Gr A_{253} }{\left( {m_3 +m_6 } \right) ^{2}-\left( {m_3 +m_6 } \right) \,Re-\alpha _4 }\\ A_{286}= & {} \frac{Re^{2}Gr A_{254} }{\left( {m_3 +2m_5 } \right) ^{2}-\left( {m_3 +2m_5 } \right) Re-\alpha _4 }\\ A_{287}= & {} \frac{Re^{2}Gr A_{255} }{\left( {m_3 +2m_4 } \right) ^{2}-\left( {m_3 +2m_4 } \right) Re-\alpha _4 }\\ A_{288}= & {} \frac{Re^{2}Gr A_{256} }{\left( {m_3 +2m_4 } \right) ^{2}-\left( {m_3 +2m_4 } \right) \,Re-\alpha _4 }\\ A_{289}= & {} \frac{Re^{2}Gr A_{257} }{\left( {m_2 +m_7 } \right) ^{2}-\left( {m_2 +m_7 } \right) \,Re-\alpha _4 }\\ A_{290}= & {} \frac{Re^{2}Gr A_{258} }{\left( {m_2 +m_6 } \right) ^{2}-\left( {m_2 +m_6 } \right) \,Re-\alpha _4 }\\ A_{291}= & {} \frac{Re^{2}Gr A_{259} }{\left( {m_2 +2m_5 } \right) ^{2}-\left( {m_2 +2m_5 } \right) \,Re-\alpha _4 }\\ A_{292}= & {} \frac{Re^{2}Gr A_{260} }{\left( {m_2 +2m_4 } \right) ^{2}-\left( {m_2 +2m_4 } \right) \,Re-\alpha _4 }\\ A_{293}= & {} \frac{Re^{2}Gr A_{261} }{\left( {m_2 +m_4 +m_5 } \right) ^{2}-\left( {m_2 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{294}= & {} \frac{Re^{2}Gr A_{262} }{\left( {m_1 +m_7 } \right) ^{2}-\left( {m_1 +m_7 } \right) \,Re-\alpha _4 }\\ A_{295}= & {} \frac{Re^{2}Gr A_{263} }{\left( {m_1 +m_6 } \right) ^{2}-\left( {m_1 +m_6 } \right) \,Re-\alpha _4 }\\ A_{296}= & {} \frac{Re^{2}Gr A_{264} }{\left( {m_1 +2m_5 } \right) ^{2}-\left( {m_1 +2m_5 } \right) \,Re-\alpha _4 }\\ A_{297}= & {} \frac{Re^{2}Gr A_{265} }{\left( {m_1 +2m_4 } \right) ^{2}-\left( {m_1 +2m_4 } \right) \,Re-\alpha _4 }\\ A_{298}= & {} \frac{Re^{2}Gr A_{266} }{\left( {m_1 +m_4 +m_5 } \right) ^{2}-\left( {m_1 +m_4 +m_5 } \right) \,Re-\alpha _4 }\\ A_{299}= & {} \left( {\begin{array}{l} A_{268} -A_{269} +A_{270} +A_{271} \\ +A_{272} +A_{273} -A_{274} +A_{275} \\ -A_{276} +A_{277} +A_{278} -A_{279} \\ -A_{280} +A_{281} -A_{282} +A_{283} \\ +A_{284} -A_{285} +A_{286} +A_{287} \\ -A_{288} +A_{289} +A_{290} -A_{291} \\ -A_{292} +A_{293} -A_{294} +A_{295} \\ -A_{296} -A_{297} +A_{298} \\ \end{array}} \right) \end{aligned}$$