Alterations in peristaltic pumping of Jeffery nanoliquids with electric and magnetic fields

Abstract

The combined effects of electric and magnetic fields on peristaltic flow of Jeffery nanoliquids are analytically investigated. Double-diffusive convection in the asymmetric microchannel is also carried out. The walls of the microchannel are propagating with a finite phase difference in a sinusoidal manner. Rosseland diffusion flux model is employed to examine the thermal radiation effect. The zeta potential on the walls is considered very low to apply Hückel–Debye approximations. The coupled non-linear governing equations are simplified by using dimensional analysis and lubrication theory. The closed form solutions for potential function, nanoparticle fraction field, solute concentration field, temperature field, stream function, and axial velocity are derived under the appropriate boundary conditions. It is noteworthy that the pumping characteristics strongly depend on the magnetic fields, electric fields, electric double layer thickness, Jeffery parameter, thermal radiation and Grashof number. Furthermore, trapping phenomenon is analyzed under the effects of Hartmann number, Jeffrey parameter, Grashof number and Helmholtz–Smoluchowski velocity. The novelty of the present work is the amalgamation of biomimetics (peristaltic propulsion), electro-magneto-hydrodynamics and nanofluid dynamics to produce a smart pump system model for smart drug delivery systems.

Appendix

$$M = \left( {1 + \lambda_{1} } \right)\,Ha,$$

$$a_{1} = \frac{Nt + Nb}{{Nb\,(h_{2} - h_{1} )}},$$

$$a_{2} = \frac{{a_{1} Nb\,\Pr }}{{\Pr Rn - N_{CT} N_{TC} \Pr + 1}},$$

$$a_{3} = \frac{{Gr_{C} (1 + \lambda_{1} )(1 + N_{CT} )}}{{h_{1} - h_{2} }} - \frac{{Gr_{F} (1 + \lambda_{1} )(Nb + Nt)}}{{Nb\,(h_{1} - h_{2} )}},$$

$$a_{4} = \frac{{a_{2} (\cosh (a_{2} h_{1} ) + \sinh (a_{2} h_{1} ))(\cosh (a_{2} h_{2} ) + \sinh (a_{2} h_{2} ))(1 + \lambda_{1} )(Gr_{T} Nb + Gr_{F} Nt - Gr_{C} N_{CT} Nb)}}{{Nb\,(\cosh (a_{2} h_{1} ) - \cosh (a_{2} h_{2} ) + \sinh (a_{2} h_{1} ) - \sinh (a_{2} h_{2} ))}},$$

$$\begin{aligned} a_{5} = \frac{{Uhs\,\kappa^{3} \sinh (h_{1} \kappa )(1 + \lambda_{1} )}}{{\sinh (\kappa (h_{1} - h_{2} ))}},\quad a_{6} = \frac{{Uhs\,\kappa^{3} \cosh (h_{1} \kappa )(1 + \lambda_{1} )}}{{\sinh (\kappa (h_{1} - h_{2} ))}}, \hfill \\ \hfill \\ \end{aligned}$$

$$\begin{aligned} a_{7} = & \frac{{a_{3} (h_{1}^{2} - h_{2}^{2} )}}{{2M^{2} }} - \frac{{a_{4} \left( {\cosh (a_{2} h_{1} ) - \cosh (a_{2} h_{2} ) - \sinh (a_{2} h_{1} ) + \sinh (a_{2} h_{2} )} \right)}}{{a_{2}^{4} - M^{2} a_{2}^{2} }} \\ & \quad + \;\frac{{a_{5} \left( {\cosh (h_{1} \kappa ) - \cosh (h_{2} \kappa ) + \sinh (h_{1} \kappa ) - \sinh (h_{2} \kappa )} \right) + a_{6} \left( {\cosh (h_{1} \kappa ) - \cosh (h_{2} \kappa ) - \sinh (h_{1} \kappa ) + \sinh (h_{2} \kappa )} \right)}}{{\kappa^{4} - M^{2} \kappa^{2} }} \\ \end{aligned}$$

$$\begin{aligned} a_{8} = & \frac{{a_{3} (h_{1} - h_{2} )}}{{M^{2} }} - \frac{{a_{4} \left( {\cosh (a_{2} h_{1} ) - \cosh (a_{2} h_{2} ) - \sinh (a_{2} h_{1} ) + \sinh (a_{2} h_{2} )} \right)}}{{a_{2}^{3} - M^{2} a_{2} }} \\ & \quad + \;\frac{{a_{5} \left( {\cosh (h_{1} \kappa ) - \cosh (h_{2} \kappa ) + \sinh (h_{1} \kappa ) - \sinh (h_{2} \kappa )} \right) - a_{6} \left( {\cosh (h_{1} \kappa ) - \cosh (h_{2} \kappa ) - \sinh (h_{1} \kappa ) + \sinh (h_{2} \kappa )} \right)}}{{\kappa^{3} - M^{2} \kappa }}, \\ \end{aligned}$$

$$a_{9} = \cosh (Mh_{2} ) - \cosh (Mh_{1} ) - \sinh (Mh_{1} ) + \sinh (Mh_{2} ) + M\left( {\cosh (Mh_{2} ) + \sinh (Mh_{2} )} \right)(h_{1} - h_{2} )$$

$$a_{10} = \cosh (Mh_{2} ) - \cosh (Mh_{1} ) + \sinh (Mh_{1} ) - \sinh (Mh_{2} ) - M\left( {\cosh (Mh_{2} ) - \sinh (Mh_{2} )} \right)(h_{1} - h_{2} )$$

$$\begin{aligned} a_{11} = & a_{7} - \frac{{\left( {a_{5} (\cosh (h_{2} \kappa ) + \sinh (h_{2} \kappa )) - a_{6} (\cosh (h_{2} \kappa ) - \sinh (h_{2} \kappa )} \right)(h_{1} - h_{2} )}}{{M^{2} \kappa - \kappa^{3} }} - \frac{{a_{3} h_{2} (h_{1} - h_{2} )}}{{M^{2} }} \\ & \quad + \;\frac{{a_{4} (h_{1} - h_{2} )\left( {\cosh (a_{2} h_{2} ) - \sinh (a_{2} h_{2} )} \right)}}{{a_{2} M^{2} - a_{2}^{3} }}, \\ \end{aligned}$$

$$a_{12} = q_{w} + h_{2} - h_{1} - a_{11} - \frac{Nb + Nt}{{Nb\,(h_{1} - h_{2} )}},$$

$$a_{13} = - M\left( {\cosh (Mh_{1} ) - \cosh (Mh_{2} ) + \sinh (Mh_{1} ) - \sinh (Mh_{2} )} \right),$$

$$a_{14} = M\left( {\cosh (Mh_{1} ) - \cosh (Mh_{2} ) - \sinh (Mh_{1} ) + \sinh (Mh_{2} )} \right),$$

$$\begin{aligned} C_{1} = & \frac{{a_{3} h_{2}^{2} }}{{2M^{2} }} - C_{3} \left( {\cosh (Mh_{2} ) + \sinh (Mh_{2} )} \right) - C_{4} \left( {\cosh (Mh_{2} ) - \sinh (Mh_{2} )} \right) - C_{2} h_{2} + \frac{{q_{w} }}{2} \\ & \quad - \;\frac{{a_{4} \left( {\cosh (a_{2} h_{2} ) - \sinh (a_{2} h_{2} )} \right)}}{{a_{2}^{4} - a_{2}^{2} M^{2} }} - \frac{{a_{5} \left( {\cosh (h_{2} \kappa ) + \sinh (h_{2} \kappa )} \right) + a_{6} \left( {\cosh (h_{2} \kappa ) - \sinh (h_{2} \kappa )} \right)}}{{\kappa^{4} - M^{2} \kappa^{2} }}. \\ \end{aligned}$$

$$\begin{aligned} C_{2} = & \frac{{a_{7} }}{{h_{1} - h_{2} }} + \frac{{(a_{8} a_{9} + a_{12} a_{13} )(\cosh (Mh_{1} ) - \cosh (Mh_{2} ) + \sinh (Mh_{1} ) - \sinh (Mh_{2} ))}}{{(a_{9} a_{14} - a_{10} a_{13} )(h_{1} - h_{2} )}} + \frac{Nb + Nt}{{Nb\,(h_{1} - h_{2} )^{2} }} \\ & \quad - \;\frac{{(a_{8} a_{10} + a_{12} a_{14} )(\cosh (Mh_{1} ) - \cosh (Mh_{2} ) - \sinh (Mh_{1} ) + \sinh (Mh_{2} ))}}{{(a_{9} a_{14} - a_{10} a_{13} )(h_{1} - h_{2} )}}, \\ \end{aligned}$$

$$C_{4} = \frac{{a_{8} a_{9} + a_{12} a_{13} }}{{a_{10} a_{13} - a_{9} a_{14} }},\quad C_{3} = \frac{{a_{8} a_{9} + a_{12} a_{13} }}{{a_{9} a_{14} - a_{10} a_{13} }},$$