Permeability conditions for the physiological viscous nanofluid: endoscopic analysis for uniform and non-uniform tubes

Abstract

The analysis of endoscopic effect in uniform and non-uniform tubes for the peristaltic flow of a viscous nanofluid with permeable condition is discussed. We have used copper as a nanoparticle and blood as its base fluid. The present problem is modeled and exact solutions for non-dimensional differential equations are found under low Reynolds number and long wavelength approximation. The possessions of all physical parameters on peristaltic flow and heat transfer characteristics are witnessed from graphical depictions. It is found that temperature profile decreases when we increase nanoparticle concentration into our base fluid for both uniform and non-uniform tubes. It is also found that velocity profile increases near the endoscopic tube for increasing values of Darcy number.

Appendix

$$\begin{aligned} a_{1} &= \frac{{k_{f} }}{{k_{\text{nf}} }},\, \, a_{2} = \frac{{\mu_{f} }}{{\mu_{\text{nf}} }},\, a_{3} = \frac{{(\rho \beta )_{\text{nf}} }}{{(\rho \beta )_{f} }},\, a_{4} = Ba_{1} a_{2} a_{3} Gr ,\, a_{5} = - a_{2} a_{3} C_{1} Gr,\\ a_{6} &= - C_{2} a_{2} a_{3} Gr ,\,a_{7} = \frac{{a_{6} }}{4} - 2\frac{{a_{5} }}{8},\, a_{8} = \frac{{a_{4} }}{64},\, a_{9} = \frac{{a_{5} }}{4} \\ C_{1} & = - \frac{{ - 4 - a_{1} Br_{1}^{2} + a_{1} Br_{2}^{2} }}{{4({\text{Log}}[r_{1} ] - {\text{Log}}[r_{2} ])}},\quad C_{2} = \frac{{ - a_{1} Br_{2}^{2} {\text{Log}}[r_{1} ] + 4{\text{Log}}[r_{2} ] + a_{1} Br_{1}^{2} {\text{Log}}[r_{2} ]}}{{4({\text{Log}}[r_{1} ] - {\text{Log}}[r_{2} ])}} \\ A_{1} &= ((r_{1} - r_{2} )(4\sqrt {Da} (45 + 5(6a_{7} + a_{9} )r_{1}^{2} + 36a_{8} r_{1}^{4} + r_{1} ( - 15a_{7} + 5a_{9} - 9a_{8} r_{1}^{2} )r_{2} \\ &\quad -\, (105a_{7} + 40a_{9} + 9a_{8} r_{1}^{2} )r_{2}^{2} - 9a_{8} r_{1} r_{2}^{3} - 189a_{8} r_{2}^{4} ) + 45r_{2} (a_{2} a_{3} DaGr ( - 4C_{2} \\ &\quad +\, a_{1} Br_{2}^{2} ) - 4(r_{1}^{2} - r_{2}^{2} )(a_{7} + a_{8} (r_{1}^{2} + r_{2}^{2} )))\xi ) + 120a_{9} r_{1}^{3} r_{2} \xi {\text{Log}}[r_{1} ]^{2} \\ & \quad + \,{\text{Log}}[r_{1} ](60\sqrt {Da} (2a_{9} r_{1}^{3} - 3a_{9} r_{1}^{2} r_{2} + 3(2a_{7} + a_{9} )r_{2}^{3} + 12a_{8} r_{2}^{5} ) + r_{2} (4r_{1} (45 + 10(3a_{7} \\ & \quad -\, 4a_{9} )r_{1}^{2} + 36a_{8} r_{1}^{4} ) + 180( - 1 + a_{2} a_{3} C_{2} DaGr + a_{9} r_{1}^{2} )r_{2} - 5(24a_{7} + 4a_{9} \\ & \quad +\, 9a_{1} a_{2} a_{3} BDaGr )r_{2}^{3} - 144a_{8} r_{2}^{5} )\xi + 60r_{2} (6a_{9} \sqrt {Da} r_{2}^{2} + 3a_{2} a_{3} C_{1} DaGr r_{2} \xi \\ & \quad -\, 2a_{9} (r_{1}^{3} + r_{2}^{3} )\xi ){\text{Log}}[r_{2} ]) + r_{2} {\text{Log}}[r_{2} ]( - 120\sqrt {Da} r_{2} (3a_{9} r_{1} + 3a_{7} r_{2} \\ & \quad -\, 2a_{9} r_{2} + 6a_{8} r_{2}^{3} ) + ( - 4r_{1} (45 + 45a_{2} a_{3} C_{1} DaGr + 5(6a_{7} + a_{9} )r_{1}^{2} + 36a_{8} r_{1}^{4} ) \\ & \quad +\, 180(1 + a_{2} a_{3} (C_{1} - C_{2} )DaGr )r_{2} + 180a_{9} r_{1} r_{2}^{2} + 5(24a_{7} - 32a_{9} + 9a_{1} a_{2} a_{3} BDaGr ) \\ &\quad r_{2}^{3} + 144a_{8} r_{2}^{5} )\xi + 60r_{2} ( - 3a_{2} a_{3} C_{1} DaGr \xi + 2a_{9} r_{2} ( - 3\sqrt {Da} + r_{2} \xi )){\text{Log}}[r2]))/ \\ & \quad (180(\sqrt {Da} + r_{2} \xi ({\text{Log}}[r_{1} ] - {\text{Log}}[r_{2} ]))), \\ A_{2} &= - \frac{1}{{(12(\sqrt {Da} + r_{2} \xi ({\text{Log}}[r_{1} ] - {\text{Log}}[r_{2} ])))}}(a_{2} ((r_{1} - r_{2} )(\sqrt {Da} (2r_{1}^{2} - r_{1} r_{2} \\ &\quad -\, 7r_{2}^{2} ) + 3r_{2} (4Da - r_{1}^{2} + r_{2}^{2} )\xi ) + 2r_{2} ( - 3\sqrt {Da} r_{2}^{2} + ( - r_{1}^{3} + 6Dar_{2} + r_{2}^{3} )\xi )( - {\text{Log}}[r_{1} ] + {\text{Log}}[r_{2} ])). \end{aligned}$$