Peristaltic Transport of a Couple Stress Fluid: Some Applications to Hemodynamics

The present paper deals with a theoretical investigation of the peristaltic transport of a couple stress fluid in a porous channel. The study is motivated towards the physiological flow of blood in the micro-circulatory system, by taking account of the particle size effect. The velocity, pressure gradient, stream function and frictional force of blood are investigated, when the Reynolds number is small and the wavelength is large, by using appropriate analytical and numerical methods. Effects of different physical parameters reflecting porosity, Darcy number, couple stress parameter as well as amplitude ratio on velocity profiles, pumping action and frictional force, streamlines pattern and trapping of blood are studied with particular emphasis. The computational results are presented in graphical form. The results are found to be in good agreement with those of Shapiro et. al \cite{r25} that was carried out for a non-porous channel in the absence of couple stress effect. The present study puts forward an important observation that for peristaltic transport of a couple stress fluid during free pumping when the couple stress effect of the fluid/Darcy permeability of the medium, flow reversal can be controlled to a considerable extent. Also by reducing the permeability it is possible to avoid the occurrence of trapping phenomenon.

💡 Research Summary

The paper presents a theoretical investigation of peristaltic transport of a couple‑stress fluid through a porous channel, motivated by the need to model blood flow in the micro‑circulatory system where particle‑size effects are significant. By treating blood as a couple‑stress fluid, the authors incorporate the rotational and couple‑moment contributions of suspended cells (e.g., erythrocytes) that are absent in classical Newtonian models. The porous nature of the vessel wall is represented using the Darcy‑Brinkman formulation, introducing a permeability parameter (Darcy number, Da) and a porosity ratio (η).

The governing equations are derived in a wave‑fixed frame (ξ = x − ct) and expressed in terms of a stream function ψ to automatically satisfy continuity. The momentum equation becomes a fourth‑order partial differential equation containing the couple‑stress parameter α, the Darcy term (1/Da)ψ, and the pressure gradient ∂p/∂ξ. Boundary conditions reflect the sinusoidally moving walls y = ±h