Bifurcation Analysis of Two-Dimensional Laminar Flow Through Sudden Expansion Channel

1. Introduction

The problem of the development of the velocity profile of a Newtonian fluid flows through a tube having larger diameter to a smaller one and vise-versa during abrupt change in geometry is one of the widely studied BVP in hydrodynamics. As an example we can take phenomena of fluid from a reservoir through a circular tube. Practically, these types of flow are found in many cases like infinite spinning, tabular heat exchangers and capillary-tube viscometer. The analysis of Newtonian fluid flow from a reservoir into circular tube having varying cross sectional area has been the subjects of numerous works done in last 50 years. At the onset the works were analytical in nature but later on both experimental and numerical work in the field have been done. It is to be mentioned that such flows encountered frequently in industrial processes involving fluid transmission. Another interesting feature of such flows is the presence of recirculation zone. For fluid flowing through a suddenly expanded passage, the separation occurs in the downstream of the expansion region.Flow of Newtonian Fluid in sudden contraction-expansion and sudden expansion channels have many important applications in fluid mechanics (Cantwell et al., 2010; Cherdron et al., 1978; Drikakis, 1997; Liou & Liu, 1986; Liou et al., 1987; Mishra & Jayaraman, 2002; Nashta & Garde, 1988; Saha et al., 2020) as well as in many manufacturing processes like mold filling, creams, mixing vessel (Ginghina, 2007; Pitton et al., 2017; Saha et al., 2021; Saha et al., 2019a; Saha et al., 2020) and on pharmaceutical science (Saha et al., 2019a). Willey et al. (1965) mentioned that increase in the velocity causes decrease in the pressure near the wall. Durst et al. (1974) shown that in a two-dimensional planar expansion channel, Coanda effect forms asymmetric vortices. Moffatt et al. (1964) stated that as Re increases, eddies form in both lower and upper corner sections, called Moffatt eddies. Mishra et al. (1959) obtained the asymmetric solution for expansion ratio 16, using continuation method. It is concluded in their work that the solution is stable for Re£ Re_cr. It is also observed that in asymmetric jet (Re>Re_cr) flow becomes stronger and complicated. Mishra et al. (1959) and Revuelta () stated how the ER affects the flow dynamics through expansion channel. Other authors (Revuelta, 2005; Sobey & Drazin, 1986) found that flow patterns changed to symmetric along the lower and upper corner vortices with the increase in Re. For Re>Re_cr, Revuelta. [16], Sobey et al. (1986) and Sandip et al. (2019b) studied the steady symmetric solution and found that recirculation zone expands with the increases in the values of Re which results in Hopf bifurcation and the flow becomes unsteady.Generally, for a given value of ER the bifurcation point cannot be calculated easily. Fearn et al. (1990) tried to find that in a two-dimensional sudden expansion channel but their attempt turn in vain. In their experimental work, they concluded that 3D effects prevent Hopf bifurcation. Linear stability analysis was performed to analyze the bifurcation phenomena and concluded that bifurcation is caused by pitchfork bifurcation. Biswas et al. (2004) numerically studied the laminar flow situation for the backward facing step flow and found that the flow shows a strong two-dimensional behavior in the plane of symmetry for . They also observed that Moffatt eddies are formed as Re tends to zero and the evolution of the vortices at the corner of the walls. Lima et al. (2008) considered the nature of three recirculation zones formed during a flow through a backward-facing step. But the outcomes of their work show good agreement with earlier works in these field for smaller values of Re only, but the laminar-turbulent transition in the flow effects the results in an inexplicable way for higher value of Re, where the instability arises in the flow along with the third recirculation region. In another study, Balakrishna with his coauthors (2010) presented results for minor loss coefficients and pressure loss in the flow of oil-water mixture through sudden expansion and contraction geometries. These studies have been performed keeping in mind the industrial need of pumping of fluids through channels having varying cross-section across the world. The present work illustrates the bifurcation characteristics in two-dimensional sudden expansion and sudden contraction-expansion channels as well as analyzes the Re_cr analytically for different values of expansion ratio. This work can be treated as an extension of the work of Saha and Das (2021) (2021). Saha et al. (2021) (2021) numerically studied the Shear-thinning flow bifurcation characteristics, but the present work analytically investigates the Newtonian fluid flow bifurcation characteristics. Among all the cited works, it is clear that all the works done earlier are on only sudden expansion channel and sudden contraction expansion channel separately. Till now no work has been done on both sudden expansion and sudden contraction expansion channels. The results of this work will become very much helpful in many manufacture processes such as mold filling, creams, mixing vessels, and in many practical applications on pharmaceutical sciences and sustainable industries.