Heat and mass transfer analysis on non-Newtonian fluid motion driven by peristaltic pumping in an asymmetric curved channel

Abstract

The foremost aim of the present study is to present a novel exploration of the peristaltic flow of Johnson–Segalman fluid in asymmetric curved channel in the presence of heat and mass transfer. The exceedingly non-linear governing equations are relatively simplified by adopting the assumptions of long wavelength and small Reynolds number approximations. The analytical expressions for the stream function, pressure gradient, temperature and concentration of the fluid have obtained by using the perturbation technique. Further, the solution acquired by the perturbation method is validated with the existing results made available in the literature. Lastly, the physical features of pertinent flow variables and geometric parameters have been discussed by plotting the graphs and stream functions.

Appendix

$$\begin{aligned} k_{1}= & {} \frac{1}{k}, A_{1} =\frac{(k_{1} h_{2} +1)^{2}-(k_{1} h_{1} +1)^{2}}{(k_{1} h_{1} +1)\left( {k_{1} h_{2} +1} \right) }, \\ A_{2}= & {} 2k_{1} (k_{1} h_{1} +1)(k_{1} h_{2} +1)\log \left( {\frac{k_{1}h_{2} +1}{k_{1} h_{1} +1}} \right) , \\ A_{3}= & {} \frac{(k_{1} h_{2} +1)^{2}-(k_{1} h_{1} +1)^{2}}{(k_{1} h_{1} +1)} \\&-2(k_{1} h_{1} +1)\log \left( {\frac{k_{1} h_{2} +1}{k_{1} h_{1} +1}} \right) ,\\ A_{4}= & {} \left( {(k_{1} h_{2} +1)^{2}-(k_{1} h_{1} +1)^{2}} \right) k_{1} \left( {k_{1} h_{1} +1} \right) \\&\left( {1+2\log \left( {k_{1} h_{1} +1} \right) } \right) \\&-2k_{1} (k_{1} h_{1} +1)\left( {(k_{1} h_{2} +1)^{2}\log \left( {k_{1} h_{2} +1} \right) } \right) \\&-(k_{1} h_{1} +1)^{2}\log \left( {k_{1} h_{1} +1} \right) \\ B_{1}= & {} (k_{1} h_{1} +1)^{2}-(k_{1} h_{2} +1)^{2} \\&-k_{1} \left( {k_{1} h_{1} +1} \right) (q_{0} +h_{1} -h_{2} ),\\ A= & {} \frac{2k_{1} A_{4} (h_{1} -h_{2} )-2A_{2} B_{1} }{k_{1} (A_{1} A_{4} -A_{2} A_{3} )},\\ B= & {} \frac{F_{0} }{2}-A\log (k_{1} h_{2} +1)-C\log (k_{1} h_{2} +1)(k_{1} h_{2} +1)^{2} \\&-D(k_{1} h_{2} +1)^{2}-\frac{(k_{1} h_{2} +1)}{k_{1} }, \\ C= & {} \frac{2B_{1} -k_{1} AA_{3} }{A_{4} }, \end{aligned}$$

$$\begin{aligned} D= & {} \frac{F_{0} +h_{1} -h_{2} -A\log \left( {\frac{k_{1} h_{2} +1}{k_{1} h_{1} +1}} \right) -C\left( {\log (k_{1} h_{2} +1)(k_{1} h_{2} +1)^{2}-\log (k_{1} h_{1} +1)(k_{1} h_{1} +1)^{2}} \right) }{(k_{1} h_{2} +1)^{2}-(k_{1} h_{1} +1)^{2}} \end{aligned}$$

$$\begin{aligned} J_{1}= & {} \frac{(1-\xi ^{2})\eta _{1} }{\eta _{1} +\mu },R_{1} =(1-\xi ^{2}), \\ l_{01}= & {} -2\left( {2+4k_{1}^{3} C^{2}+\left( {C-2D} \right) \left( {-k_{1} +4k_{1}^{2} } \right) } \right) , \\ l_{02}= & {} k_{1}^{2} \left( {C-4k_{1}^{2} C^{2}-2D} \right) , \\ l_{03}= & {} 4k_{1} A\left( {2+k_{1} \left( {A+2h_{1} } \right) } \right) , \end{aligned}$$

$$\begin{aligned} l_{1}= & {} \frac{Br}{4}\left\{ \frac{l_{01} h_{1} +l_{02} h_{1}^{2}-l_{03} }{(1+k_{1} h_{1} )^{2}} \right. \\&\quad +8A(-1+8k_{1} C)\log (k_{1} h_{1} +1)^{2} \\&\quad \left. + \frac{8[1-k_{1} (C-2D)-2k_{1}^{2} (4D+C(-2+h_{1} ))+k_{1}^{3} C(4C-(-8+h_{1} )h_{1} )]}{k_{1} }\right\} \\ l_{2}= & {} \frac{Br}{4}\left\{ \frac{l_{01} h_{2} +l_{02} h_{2}^{2}-l_{03} }{(1+k_{1} h_{2} )^{2}}+8A(-1+8k_{1} C)\log (k_{1} h_{2} +1)^{2}\right. \\&\quad \left. + \frac{8[1-k_{1} (C-2D)-2k_{1}^{2} (4D+C(-2+h_{2} ))+k_{1}^{3} C(4C-(-8+h_{2} )h_{2} )]}{k_{1} }\right\} \end{aligned}$$

$$\begin{aligned} C_{1}= & {} \frac{k_{1} \left( {1+l_{1} -l_{2} } \right) }{\log \left( {k_{1} h_{2} +1} \right) -\log \left( {k_{1} h_{1} +1} \right) },\\ C_{2}= & {} -l_{1} -\frac{C_{1} \log \left( {k_{1} h_{1} +1} \right) }{k_{1}},\\ j_{11}= & {} \frac{-192A^{3}J_{1} k_{1}^{4} }{576},j_{12} =\frac{192A^{2}CJ_{1} k_{1}^{4} }{64},\\ j_{01}= & {} k_{1} \left( {k_{1} h_{1} +1} \right) +\frac{j_{11} }{\left( {k_{1} h_{1} +1} \right) ^{4}}+\frac{j_{12} }{\left( {k_{1} h_{1} +1} \right) ^{2}}, \\ j_{02}= & {} k_{1}^{2} -\frac{2k_{1} j_{12} }{\left( {k_{1} h_{1} +1} \right) ^{3}}-\frac{4k_{1} j_{11} }{\left( {k_{1} h_{1} +1} \right) ^{5}}, \\ j_{03}= & {} k_{1} \left( {k_{1} h_{2} +1} \right) +\frac{j_{11} }{\left( {k_{1} h_{2} +1} \right) ^{4}}+\frac{j_{12} }{\left( {k_{1} h_{2} +1} \right) ^{2}}, \\ j_{04}= & {} k_{1}^{2} -\frac{2k_{1} j_{12} }{\left( {k_{1} h_{2} +1} \right) ^{3}}-\frac{4k_{1} j_{11} }{\left( {k_{1} h_{2} +1} \right) ^{5}},\\ A_{11}= & {} \log (k_{1} h_{2} +1)-\log (k_{1} h_{1} +1);\\ A_{12}= & {} (k_{1} h_{2} +1)^{2}-(k_{1} h_{1} +1)^{2}, \\ A_{13}= & {} (k_{1} h_{2} +1)^{2}\log (k_{1} h_{2} +1) \\&-(k_{1} h_{1} +1)^{2}\log (k_{1} h_{1} +1),\\ A_{14}= & {} q_{1} +j_{01} -j_{03}, \\ A_{15}= & {} (k_{1} h_{2} +1)^{2}\left\{ {1+2\log (k_{1} h_{2} +1)} \right\} \\&-(k_{1} h_{1} +1)^{2}\left\{ {1+\log (k_{1} h_{1} +1)} \right\} , \\ A_{16}= & {} j_{02} (k_{1} h_{1} +1)-j_{04} (k_{1} h_{2} +1),\\ A_{17}= & {} \frac{-j_{02} A_{11} (k_{1} h_{1} +1)}{k_{1} }-A_{14},\\ A_{18}= & {} 2A_{11} (k_{1} h_{1} +1)^{2}-A_{12}, \\ A_{19}= & {} A_{11} (k_{1} h_{1} +1)^{2}\left( {1+2\log (k_{1} h_{1} +1)} \right) -A_{13}, \\ c_{14}= & {} \frac{2k_{1} A_{12} A_{17} -A_{16} A_{18} }{2k_{1} A_{12} A_{19} -A_{15} A_{18} }, \\ c_{13}= & {} \frac{A_{17} -c_{14} A_{19} }{A_{18}}, \\ c_{12}= & {} \frac{A_{14} -c_{13} A_{12} -c_{14} A_{13} }{A_{11}}, \\ c_{11}= & {} \frac{F_{1} }{2}-c_{12} \log (k_{1} h_{2} +1) \\&-c_{14} \log (k_{1} h_{2} +1)(k_{1} h_{2} +1)^{2}-c_{13} (k_{1} h_{2} +1)^{2}-j_{03}, \\ j_{21}= & {} -8k_{1}^{4} (Ac_{14} +Cc_{12} -8AC^{3}R_{1} k_{1}^{4} ), \\ j_{22}= & {} -4k_{1}^{3} C(k_{1}^{2} -1),j_{23} =8k_{1}^{4} C(c_{14} -2C^{3}R_{1} k_{1}^{4} ), \\ j_{24}= & {} 4Ak_{1}^{3} (k_{1}^{2} -1),\\ j_{25}= & {} 8k_{1}^{4} (Ac_{12} +16A^{2}C^{2}J_{1} k_{1}^{4} -12A^{2}C^{2}R_{1} k_{1}^{4} ), \\ j_{26}= & {} 32A^{3}Ck_{1}^{8} (-5J_{1} +R_{1} ), \\ j_{27}= & {} 16A^{4}k_{1}^{8} (2J_{1} -R_{1}),\\ K_{1}= & {} -\frac{Br}{k_{1}^{2} }\left( \frac{j_{21} }{2}\left( {k_{1} h_{1} +1} \right) ^{2}+j_{22} \left( {k_{1} h_{1} +1} \right) ^{2} \right. \\&\quad +\frac{j_{23} }{4}\left( {k_{1} h_{1} +1} \right) ^{2}+\frac{j_{24} }{\left( {k_{1} h_{1} +1} \right) } \\&\quad +\frac{j_{25} }{4\left( {k_{1} h_{1} +1} \right) ^{2}}+\frac{j_{26}}{16\left( {k_{1} h_{1} +1} \right) ^{4}} \\&\quad \left. +\frac{j_{27} }{36\left( {k_{1} h_{1} +1} \right) ^{6}} \right) , \\ K_{2}= & {} replace\,h_{1\,\,} by\,\,h_{2} \,in\,\,K_{1}\\ b_{12}= & {} \frac{k_{1} \left( {K_{1} -K_{2} } \right) }{\log \left( {k_{1} h_{2} +1} \right) -\log \left( {k_{1} h_{1} +1} \right) }, \\ b_{11}= & {} -K_{1} -\frac{b_{12} \log \left( {k_{1} h_{1} +1} \right) }{k_{1} }. \end{aligned}$$