Lie point symmetries for biological magneto-Jeffrey fluid flow in expanding or contracting permeable walls: a blood vessel model

Abstract

This article addresses biological flow in expanding or contracting blood vessels. The alternate contraction and expansion of vessels is known to act as a physiological pump to generate flow. When muscles compress blood vessel walls, the valve at the end of the source is closed, and the downstream end opens; for this reason, blood is pumped in the downstream direction. To study this situation, a model has been developed here that consists of the unsteady two-dimensional flow of an incompressible magneto-Jeffrey fluid in a porous semi-infinite channel with expanding or contracting walls; the channel is closed from one end by a compliant membrane. The Lie group analysis method was used to transform a system of nonlinear partial differential equations into nonlinear ordinary differential equations that were solved using the perturbation method. The effects of Jeffrey parameters ( $ \lambda _1 $ , which is the ratio of relaxation time to retardation time, and Deborah parameter $ D_e $ ) and other physical parameters are plotted and discussed. We find that the axial velocity of a Newtonian fluid is greater than that of a non-Newtonian fluid if the walls are in contraction and vice versa if the walls are in expansion. If the walls are expanded, the blood is distributed in a large area, and thus, the normal pressure decreases, while if the walls are contracted, blood is distributed in less space, and thus, the pressure increases. Normal pressure $ \Delta p_n $ in expanding blood vessels is less than that in contracting blood vessels.

Keywords

• Lie point symmetries
• Jeffrey fluid
• porous channel
• magnetic field
• expanding and contracting walls