The objective of the present investigation is to provide a theoretical analysis of the flow of an incompressible couple stress fluid through a Reiner–Rivlin liquid which is covered by a permeable medium. On the surface of the sphere, the boundary conditions, such as nil couple stress condition, impenetrability condition, continuity of velocities, and continuity of stresses, are applied. This research aims to analyze the flow behavior which has been evaluated under the influence of dimensionless parameter, couple stress parameter, and parameter impacting viscosity. The essential components that influenced the couple stress parameter in relation to couple stress viscosity are physically interpreted and presented. The analysis concludes with a graphical representation of flow pattern and drag under the effects of the couple stress parameter, couple stress viscosity ratio, permeability parameter, etc. The findings demonstrate that the Reiner–Rivlin liquid parameter (S) has an increasing function in the non-dimensional drag case but a decreasing function in the pressure case. Moreover, the stress viscosity of the couple is a significant component that impacts the drag and the pressure. Also, the streamline pattern shows that the couple stress fluid was completely penetrated by the Reiner–Rivlin liquid.

This article explores the micropolar fluid flow past a fluid sphere enclosing a solid core which is embedded in a porous medium. Flow fields of the porous region and nonporous region are governed by Brinkman equation and Stokes equation, respectively. Explicit expressions for stream functions, velocities, and microrotation components are examined. The drag force accomplished by a fluid sphere placed inside the porous medium is evaluated. The drag coefficient and its dependence are analyzed numerically and graphically with variation of permeability parameter and viscosity ratio. Some well-known results are validated with past known cases.

The present article investigates the axisymmetric flow of a steady incompressible Reiner–Rivlin liquid sphere enveloped by a spherical porous layer using the cell model technique. The Brinkman-extended Darcy model is deployed for the porous medium hydrodynamics, and isotropic permeability is considered. The stream function solutions of the governing equations are obtained, which involves the Gegenbauer functions and the modified Bessel functions. An asymptotic series expansion in terms of the Reiner–Rivlin liquid parameter S has been employed to determine the expression of the flow field for the Reiner–Rivlin liquid. Boundary conditions on the cell surface corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Analytical expressions are derived for dimensionless pressure, tangential stress, and the couple stress components using the method of separation of variables and Gegenbauer functions/polynomial. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the porous zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the Reiner–Rivlin fluid on the porous sphere are derived with corresponding expressions for the drag coefficient. Mathematical expression of the drag coefficient, pressure distribution, velocity profile, and separation parameter is established. On the basis of viscosity ratio, permeability parameter, dimensionless parameter, and the volume fraction, variations of the drag coefficient, velocity profiles, fluid pressure, and separation parameter (SEP) are investigated with their plots. The effects of the streamline pattern make the flow more significant for the Mehta–Morse when compared to the other models. Additionally, the mathematical expression of the separation parameter (SEP) is also calculated, which shows that no flow separation occurs for the considered flow configuration and is also validated by its pictorial depiction. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behavior often arises in addition to porous media effects.

This study presents an analytical solution for the creeping flow of a couple stress fluid within a porous medium that encloses a semipermeable cylinder containing an embedded solid core. The flow field is modeled with a uniform velocity far from the cylinder, ensuring the proper application of boundary conditions. The fluid motion within the porous medium is described by the Brinkman and Darcy equations, each accounting for distinct permeability characteristics. Specifically, the Brinkman equation with permeability k1 governs the region outside the semipermeable cylinder, while the Darcy equation with permeability k2 applies within the semipermeable region. To improve the accuracy and applicability of the model, the governing equations are formulated using cylindrical polar coordinates, reflecting the structural configuration of the system. The stream function approach is employed to reduce the complexity of the governing equations, resulting in a sixth-order differential equation for the Brinkman region and a second-order differential equation for the Darcy region. The analytical solutions, derived via the separation of variables technique, are expressed in terms of modified Bessel functions. These solutions provide explicit mathematical representations for essential flow characteristics, including normal and tangential velocities, couple stress components, pressures, stresses, and drag forces on the semipermeable cylinder. A graphical analysis examines the variations in radial velocity, pressure, and drag force with respect to parameters such as permeability (ξ1, ξ2), separation parameter (γ), and couple stress parameter (λ). Additionally, streamline patterns are investigated to gain deeper insights into flow behavior. This study contributes to fluid mechanics by advancing the understanding of couple stress fluid flows in porous media and their implications for complex porous structures.

Micropolar Fluid Past a Sphere Coated with a Thin Fluid Film

The current study pivoted on the numerical simulation of magnetohydrodynamic nanofluid flow through the irregular boundary over the Riga plate with heat radiation, suction/injection. The governing equations of continuity, momentum, and energy are constructed to be nonlinear in nature, based on the current problem. The combination of Quasilinear technique with finite difference method using irregular boundaries has been used to solve the governing equations. The velocity profile is enhanced for incremented values of Hartman number. The thermal radiation parameter due to Rosseland approximations is enhanced for temperature and concentration profiles whereas it declines the heat transfer and mass transfer profiles. Temperature, concentration, and heat transfer rates increase for augmented magnetic parameter values, but velocity and skin friction coefficient profiles show the opposite behaviour. The enhanced values of Biot number along with suction/injection parameter reduces the values of skin friction coefficient. The radiation parameter helps to drop the energy rate in the fluid flow as it reduced the skin friction coefficient values. The upsurged values of the thermal radiation parameter along with electromagnetic parameter decreases the values of the Nusselt number.