MHD peristaltic flow of nanofluid in a vertical channel with multiple slip features: an application to chyme movement

Abstract

The mathematical modelling of biological fluids is of utmost importance due to its applications in various fields of medicine. The peristaltic mechanism plays a crucial role in understanding numerous biological flows. The current paper emphasizes on the MHD peristalsis of Jeffrey nanofluid flowing through a vertical channel when subjected to the combined heat/mass transportation. The equations for the current flow scenario are developed with relevant assumptions for which the perturbation technique is followed to simulate the solution. The expressions of velocity, temperature and concentration are obtained, and the solutions of skin-friction coefficient, Nusselt number and Sherwood number at the wall are acquired. Further, the influence of relevant parameters on various physical quantities for both non-Newtonian Jeffery and viscous fluid is graphically analyzed. The outcomes are deliberated in detail Further, it is renowned that the current study has many biomechanical applications such as the movement of chyme motion in the gastrointestinal tract and during the surgery to take control of the flow of blood by adjusting the magnetic field intensity.

Appendix

$$A_{1} = {\text{Mn}}^{2} \left( {1 + \lambda_{1} } \right)$$

$$A_{2} = \left( {\frac{{{\text{d}}p_{0} }}{{{\text{d}}x}} - {\text{G}} + {\text{Br}}} \right)\left( {1 + \lambda_{1} } \right)$$

$$A_{3} = \frac{{A_{2} }}{{A_{1} \left( {{\text{cos}}h\sqrt {A_{1} } h + \alpha_{1} {\text{sin}}h\sqrt {A_{1} } h} \right)}}$$

$$A_{4} = \frac{{A_{1} A_{3}^{2} }}{2} + \frac{{Mn^{2} A_{3}^{2} }}{2}$$

$$A_{5} = 2{\text{Mn}}^{2} A_{2} A_{3}$$

$$A_{6} = \frac{1}{2}\left( {{\text{Mn}}^{2} \left( {A_{3}^{2} + 2A_{2}^{2} } \right) - A_{3}^{2} A_{1} } \right)$$

$$A_{7} = {\text{G}} + \frac{{{\text{Nt}}}}{{{\text{Nb}}}}{\text{Br}}$$

$$A_{8} = \left( {\alpha_{2} {\text{G}} - \alpha_{3} \frac{{{\text{Nt}}}}{{{\text{Nb}}}}{\text{Br}}} \right)\left( {\frac{{A_{4} }}{{2\sqrt {A_{1} } }}{\text{sin}}h2\sqrt {A_{1} } h + } \right.\left. {\frac{{A_{5} }}{{\sqrt {A_{1} } }}{\text{sin}}h\sqrt {A_{1} } h - A_{6} h} \right)$$

$$A_{{9}} = \frac{{A_{4} }}{{4A_{1}^{2} }}{\text{cos}}h2\sqrt {A_{1} } h + \frac{{A_{5} }}{{A_{1} }}{\text{cos}}h\sqrt {A_{1} } h + \frac{{A_{8} }}{{A_{1} A_{7} }}$$

$$\begin{aligned} A_{{10}} & = - 3A_{1} ^{4} A_{3} A_{9} h - 6\sqrt {A_{1} } A_{2} A_{5} \;h\;{\text{cos}}h\left( {\sqrt {A_{1} } h} \right) + 6A_{2} A_{5} \;{\text{sin}}h\left( {\sqrt {A_{1} } h} \right) + \\ & \quad 3A_{1} ^{{7/2}} A_{3} A_{9} \sin h\left( {\sqrt {A_{1} } h} \right) + 6\alpha _{1} A_{1} A_{2} A_{5} A_{9} h\;{\text{sin}}h\left( {\sqrt {A_{1} } h} \right) + \\ & \quad 6A_{1} ^{{3/2}} A_{3} A_{5} h\sin h^{2} \left( {\sqrt {A_{1} } h} \right) + A_{1} ^{{3/2}} A_{3} A_{4} \sin h^{3} \left( {\sqrt {A_{1} } h} \right) \\ & \quad - \frac{1}{2}\sqrt {A_{1} } A_{2} A_{4} \sin h\left( {2\sqrt {A_{1} } h} \right) \\ \end{aligned}$$

$$\begin{aligned} A_{{11}} & = A_{1} ^{{5/2}} A_{3} A_{4} A_{9} h\sin h^{2} \left( {\sqrt {A_{1} } h} \right) + 3{\text{ }}A_{1} ^{{5/2}} A_{3} A_{6} A_{9} h\sin h^{2} \left( {\sqrt {A_{1} } h} \right) \\ & \quad + A_{1} ^{2} A_{3} A_{4} A_{9} \cos h^{3} \left( {\sqrt {A_{1} } h} \right)\sin h\left( {\sqrt {A_{1} } h} \right) \\ & \quad + A_{1} ^{{5/2}} A_{3} A_{4} A_{9} h\sin h^{4} \left( {\sqrt {A_{1} } h} \right) \\ \end{aligned}$$

$$\begin{aligned} A_{{12}} & = \cos h\left( {\sqrt {A_{1} } h} \right)\left( {6\alpha _{1} A_{1} ^{{3/2}} A_{2} A_{5} A_{9} h^{2} } \right. + \frac{1}{2}\alpha _{1} A_{1} ^{2} A_{3} A_{9} \left( { - 5A_{4} + 6A_{1} A_{6} h^{2} } \right. + {\text{ }} \\ {\text{ }} & \quad \left. {\left. {3A_{4} \cos h\left( {2\sqrt {A_{1} } h} \right)\sin h\left( {\sqrt {A_{1} } h} \right)} \right)} \right) \\ \end{aligned}$$

$$A_{{13}} = A_{1} ^{{3/2}} A_{3} A_{4} \cosh ^{2} \left( {\sqrt {A_{1} } h} \right)\left( {1 + 3\alpha _{1} A_{1} A_{9} h\sinh ^{2} \left( {\sqrt {A_{1} } h} \right)} \right)$$