This study investigates the thermal behavior for Jeffery–Hamel flow of a couple stress fluid under the influence of a magnetic field, employing an artificial neural network (ANN) methodology. The research evaluates how the variations in Reynolds number (Re), Eckert number (Ec), Prandtl number (Pr), couple stress parameter (S), magnetic parameter (M), and channel angle (α) alter the temperature distributions within the fluid. The governing coupled nonlinear differential equations for couple stress fluid in Jeffery–Hamel flow and heat transfer are solved using an artificial neural network (ANN) approach. The ANN is trained to satisfy the boundary value problem by minimizing the residuals of the governing equations and boundary conditions, providing an efficient mesh-free solution. The analysis reveals that increasing the magnetic parameter amplifies the Lorentz force, which leads to an increase in the fluid temperature across both divergent and convergent channels. It is observed that when the couple stress parameter is very low, an increase in channel angle raises fluid temperature. Conversely, for a large value of couple stress parameter, increasing the channel angle decreases fluid temperature in the divergent channel but increases it in the convergent channel. In the absence of a magnetic field (M = 0), enhancing the couple stress parameter results in a decrease in the fluid temperature for a divergent channel, while in a convergent channel, it leads to an increase in the fluid temperature. These insights offer a comprehensive understanding of the interplay between couple-stress fluid properties, magnetic field, heat transfer, and channel geometries, contributing to the optimization of thermal management in systems utilizing couple-stress fluids.

A theoretical analysis of the piezo-viscous dependence of micropolar fluid flow between parallel annular disks has been conducted. It is assumed that viscosity changes exponentially with pressure. Pressure is analytically approximated using a small perturbation method with the viscosity coefficient as the perturbation parameter. Squeeze film time, pressure, and load carrying capacity are calculated and graphically demonstrated across various viscosity–pressure characteristics. The study examines how an increase in cross-viscosity correlates with notable growth in pressure, response time, and bearing load. According to the results, micropolar fluid offers an increase in the load-bearing capacity and therefore lengthens the response time to prevent the contact of parallel annular disks. These observations are useful for bearings, seals, and lubrication systems, where controlling viscosity and pressure is crucial for smooth operation and productivity. As the limiting case of micro-polar fluid when the coupling number tends to zero or the couple stress parameter approaches infinity, without the annular ring, the present derivation agrees well with the Newtonian viscous case of parallel planar squeezing disks described by Bird et al. [Transport Phenomena (John Wiley & Sons, 2006)].

The investigation of the couple stress fluid flow behaviour between two parallel plates under sudden stoppage of the pressure gradient is considered. Initially, a flow of couple stress fluid is developed between the two parallel plates under a constant pressure gradient. Suddenly, the applied pressure gradient is stopped, and the resulting unsteady flow is studied. This type of flow is known as run-up flow in the literature. Now the flow is expected to come to rest in a long time. Usually, these types of problems are solved by using the Laplace transform technique. There are difficulties in obtaining the inverse Laplace transform; hence, many researchers adopt numerical inversions of Laplace transforms. In this paper, the problem is solved by using the separation of variables method. This method is easier than the transform method. The velocity field is analyti-cally obtained by applying the usual no-slip condition and hyper-stick conditions on the plates, and hence the volumetric flow rate is derived at subsequent times. The steady state solution before the withdrawal of the pressure gradient is matched with the initial condition on time. The rest time, i.e. the time taken by the fluid to come to rest after the pressure gradient is withdrawn is calculated. The graphs for the velocity field at different times and different couple stress parameters are drawn. In the special case when a couple stress parameter approaches infinity, couple stress fluid becomes a viscous fluid. Our results are in good agreement with this special case.

Experimental evidence shows that for certain fluids, at high pressures, the viscosity of the fluid depends on pressure. Such fluids are referred to as piezo-viscous fluids. The present paper investigates the piezo-viscous dependence of coupled stress fluid flow between parallel annular disks. The study assumes an exponential change in viscosity with pressure and analytically approximates the pressure using a perturbation method, with the viscosity parameter as a small perturbation parameter. The results, which include the calculation and visual demonstration of squeeze film time, load-bearing capacity, and pressure over various viscosity-pressure characteristics, reveal a significant relationship between the increase in piezo-viscosity and the growth in pressure, response time, and bearing load. These findings have direct implications for practical applications in bearings, seals, and lubrication, where the control of viscosity and pressure is crucial for maintaining smooth operation and enhancing productivity.