Two-fluid confined flow in a cylinder driven by a rotating end wall

Abstract

The flow of two immiscible fluids completely filling an enclosed cylinder and driven by the rotation of the bottom end wall is studied numerically. The simulations are in parameter regimes where there is significant advection of angular momentum, i.e., the disk rotation rate is fast compared to the viscous diffusion time. We consider two classes of scenarios. The first consists of cases that are straightforward to reproduce in physical experiments where only the rotation rate and the viscosity ratio of the fluids are varied. Then we isolate different forces acting on the system such as inertia, surface tension, and gravity by studying variations in individual governing parameters. The viscosity ratio determines how quickly the upper fluid equilibriates dynamically to the flow in the lower fluid and plays a major role in determining how vortex lines are bent in the neighborhood of the interface between the two fluids. This in turn determines the structure of the interfacial layer between the two swirling fluids, which is responsible for the flow in the upper fluid. The simulations show that even when there is significant interfacial deformation, both the dynamics and the equilibrium flow are dominated by vortex bending rather than vortex stretching. The simulations show that for the range of immiscible fluids considered, surface tension effects are significant. Increased surface tension reduces the degree to which the interface is deformed and the limit of zero surface tension is not an appropriate approximation.

Vortex breakdown in a water-spout flow

“…The conditions at the interface are similar to those formulated by Brady et al 4,7 The numerical procedure, used in this paper, is described in detail in our previous study 10 where comparison with the results by Brady et al 4 is made, which validates the code.…”

“…The TCL development is common for the air-water flow and for the flow of the two fluids studied in Ref. 4 ( Fig. 10).…”

“…Other control parameters are the Reynolds number, Re = ωR 2 /ν a , characterizing the swirl strength, the Froude number, Fr = ω 2 R/g, which is a centrifugal-to-gravity acceleration ratio, and the Weber number, We = ρ w ω 2 R 3 /σ , characterizing the effect of surface tension σ at the interface; g = 9.81 m 2 /s is the gravity acceleration. We first focus on the air-water flow and then consider the fluids studied by Brady et al 4,7 In the air-water case, ν a = 15 × 10 −6 m 2 /s is the kinematic viscosity of air, ρ w = 1000 kg/m 3 is the water density, and σ = 0.0715 kg/s 2 at T = 300 K. We assume that pressure on the interface at rest has its atmospheric value and the air density is ρ a = 1.22 kg/m 3 . Imagine a physical experiment where R = 1 mm, a = ν a 2 /(gR 3 ) = 0.0225, and b = ρ w ν a 2 /(Rσ ) = 0.00315 are fixed, while ω eventually increases.…”

“…To explore, whether the TCL development occurs for other fluids as well, and to observe large deformations of interface in a steady axisymmetric flow, we address now the media used in Ref. 4, where the light-to-heavy fluid density ratio is ρ r = 0.5284 and dynamic viscosity ratio μ r = 0.2. We keep the same values of the aspect ratio (H = 1) and the height of the heavy fluid at rest (H h = 0.5).…”

Bifurcations of a creeping air–water flow in a conical container

“…The exploration of topological transformations and VB development in two-fluid flows was initiated by the pioneer paper of Brady et al [19]. This paper contributes to a new field which can be referred to as "Bifurcating multi-fluid flows" which in turn can be viewed as a part of the topological fluid mechanics whose origin can be traced back to the paper by Arnold [20].…”