Assessment of stagnant points in stratified laminar flow of two immiscible Phan Thein Tanner fluid films on a vertically upward moving plate

Analyzing the stratified laminar flow of two immiscible Phan Thein Tanner fluid films on a vertically upward moving flat plate can significantly improve predictive models for engineering and biomedical applications. This assessment focuses on the stagnant points in two immiscible fluid film flow on a flat plate, distinguishing between extraneous and real stagnant points, and establishing constraints for their existence. The exact solutions of a formulated set of ordinary differential equations are determined for shear stresses, normal stresses, average velocities, flow rates, extreme velocities, stagnant points, and velocity profiles. The constraints for the existence of extraneous and real stagnant points are determined. The extraneous stagnant points are either imaginary or appear outside the location of the fluid. The assessment delineated that when the density, elongation parameter, Deborah number, and Stokes number increase, the stagnant points shift closer to the plate. An increase in the Deborah number enhances drainage, which in turn shifts stagnant points closer to the plate and creates a stable elastic layer alongside it. Moreover, a comparison among exponential Phan Thein Tanner, linear Phan Thein Tanner, and Newtonian fluids is also present for stagnant points, which is significant as it provides insights into the behavior of immiscible fluids in different scenarios. The validity of the current results with the existing studies is also presented by reducing two immiscible Phan Thein Tanner fluid films to a single fluid film and hence to a Newtonian model, which coincides with existing literature.