Thin film hydrodynamics of second- and third-order fluids confined in a Hele–Shaw cell

Abstract

The thin film coating developed by penetration of a long gas bubble through a Hele–Shaw cell initially filled with a viscoelastic fluid is theoretically investigated using the third-order fluid model. The work presented in this paper is directed at identifying effects of fluid elasticity on the liquid fraction deposited on the walls of a Hele–Shaw cell. The singular perturbation method is used to determine the residual liquid film thickness of a viscoelastic fluid on the wall of a planar geometry when displaced by another immicible fluid. Inner and outer expansions are developed in terms of small parameters Ca 1/3, δ ( δ = De/ Ca 1/3) and δ 2. The method of matched asymptotic expansions is used to match the inner and outer solutions by means of transition region between the advancing meniscus and the entrained film where the fluid rheology has its greatest effect. An understanding of the role of fluid elasticity comes from a consideration of both the second- and third-order fluid models, a detailed analysis indicates that the residual liquid film thickness of the viscoelastic fluid tends to decrease while the pressure drop across the bubble front tends to increase as the fluid becomes more viscoelastic. While the force due to the present of a normal stress gradient in the flow direction was evaluated from the second-order fluid model and acts to decrease the film thickness and increase the pressure jump, whereas the force due to shear stress thinning was evaluated from the third-order fluid model and tends to increase the film thickness and decrease the pressure jump. Theoretical results presented in this paper are in agreement with other the experimental and theoretical studies.