Journal of King Saud University: Science (Jan 2024)
Jaffrey-Hamel flow features of Oldroyd-B model through intersecting plates
Abstract
We evaluate the steady Jaffrey-Hamel flow of a viscoelastic fluid using Oldroyd-B model in a deformable channel formed by two intersecting plates. To be more precise, we offer a mathematical structure for computing the leading-order impacts of the fluid viscoelasticity on the flow in the setting of relaxation and temporal retardation interactions between the fixed walls of the channel. The typical dimensionless variables influencing the interaction of fluid and structure in both wider (divergent) and narrower (convergent) channels are primarily identified. The flow originates from a source located at the apex, travel from convergent to divergent zone, and exist at the outlet to the reservoir. Only radial component of velocity contributes to the fluid velocity while the azimuthal component is zero. The fluid attributes are independent of hydraulic pressure and temperature. We highlight the respective contributions of various components of momentum equation coupled with pressure gradient along the radial and tangential direction. The pressure gradient is omitted, since gradients of viscoelastic shear stresses predominantly cause the contribution for narrower/expanding geometries. We further demonstrate that, although the pressure is minimal along the midline line for narrow geometries, viscoelastic stresses are equal to or greater than shear stresses across the domain. Applying the principle of momentum and mass conservations in a cylindrical polar framework, the system of governing equations are constructed. The computer based MATLAB code (bvp4c tool) is used to numerically solve the consequent set of modelled equations. The results pertaining to a Navier-Stokes fluid, and a Maxwell fluid exist as limiting instances of our formulations. Effect of inertial forces 20≤Re≤140 and channel opening have similar effects on converging and diverging section of the channel. A higher strain delaying time and a shorter stress relaxation phase produce an improved velocity profile, but both viscoelastic times have the opposite effect.
