The overall objective of the current analysis is to examine how electroosmotic flow combined with peristaltic pumping phenomenon could ably contribute to intracellular fluid flow. The biofluid is taken as non-Newtonian Couple stress fluid, while micro-passage is approximated as two-dimensional inclined channel comprehending complex peristaltic walls. Following a traditional approach of peristaltic fluid flow problem balance of mass and momentum is utilized. Beside channel waves, flow is also generated by Lorentz force triggered by electric and magnetic fields. To integrate electric potential term Poisson-Boltzmann and Nernst Planck equation are utilized. Finally, a sixth order BVP in term of stream function is obtained by employing creeping flow, long wavelength and Debye-Hückel assumptions. A numerical solution is calculated and analyzed by plotting fluid velocity and level curves in MATLAB 2021b. It is observed that couple stress fluid flows with greater speed (in central region of the channel) as compared to Newtonian fluid. Moreover, electro-osmotic parameter and Debye-Hückel length are assistive factors to the fluid velocity in the lower half of the passage.

Cilia beating influences bio-fluid flow, and conduits with ciliated surfaces serve numerous purposes. Cilia are hair-like adjuncts that produce liquid drive and cell locomotion. This novel research integrates magnetohydrodynamics, Hall effects, and porous medium with the cilia-driven flow. The developed problem is further simplified by utilizing famous assumptions, i.e., low Reynolds number and large wavelength. The bvp5c function in MATLAB (2023a) is adopted to achieve a convergent solution. The graphical illustrations of streamlines, pressure drop, and axial velocity under the influence of pertinent parameters are thoroughly examined. The findings suggest that when a high magnetic field (Hall current) and porous media are combined with the Carreau–Yasuda fluid, the pumping equipment operates better than the viscous liquid. In addition, fluid movement in biological organs is improved by metachronal ciliary motion. Furthermore, it is predicted that this study will make a significant contribution to biomedical engineering in terms of the upgrading and advancement of various types of drug-delivery technologies.

Inspired by the prevalence of non-Newtonian liquids in biological flows, our aim in this article is to elaborate on the flow of Carreau-Yasuda fluid generated by wavy cilia inside the micro-channel. The impact of complex wave and magnetic field (magnetohydrodynamics) are also taken into account. The low Reynolds number and large wavelength approximation are employed to simplify the problem. The finite-difference approach (finite-difference method) is utilized to get the convergent solution to the problem in MATLAB (2022b). The impact of emerging parameters on the axial velocity and streamlines is examined through various graphs. Moreover, a comparison among the Carreau-Yasuda fluid, Carreau fluid, and Newtonian fluid is also expounded.

The determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

The thermal entrance problem (also known as the classical Graetz problem) is studied for the complex rheological Carreau fluid model. The solution of two-dimensional energy equation in the form of an infinite series is obtained by employing the separation of variables method. The ensuing eigenvalue problem (S–L problem) is solved for eigenvalues and corresponding eigenfunctions through MATLAB routine bvp5c. Numerical integration via Simpson’s rule is carried out to compute the coefficient of series solution. Current problem is also tackled by an alternative approach where numerical solution of eigenvalue problem is evaluated via the Runge–Kutta fourth order method. This problem is solved for both flat and circular confinements with two types of boundary conditions: (i) constant wall temperature and (ii) prescribed wall heat flux. The obtained results of both local and mean Nusselt numbers, fully developed temperature profile and average temperature are discussed for different values of Weissenberg number and power-law index through graphs and tables. This study is valid for typical range of Weissenberg number W e ≤ 1 $\left(We\le 1\right)$ and power-law index n < 1 $\left(n{< }1\right)$ for shear-thinning trend while n > 1 $\left(n{ >}1\right)$ for shear-thickening behaviour. The scope of the present study is broad in the context that the solution of the said problem is achieved by using two different approaches namely, the traditional Graetz approach and the solution procedure documented in M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.