Axisymmetric viscous flow between two horizontal plates

Abstract

The flow of viscous fluid injected from a point source into the space between two horizontal plates initially filled with a second fluid of lesser density and different viscosity is studied theoretically and numerically. The volume of the dense input fluid increases with time in proportion to tα. When the fluid has spread far from the source, lubrication theory is used to derive the governing equations for the axisymmetric evolution of the interface between the fluids. The flow is driven by the combination of pressure gradients associated with buoyancy and pressure gradients associated with the input flux. The governing equation is integrated numerically, and we identify that with a constant input flux, the flow is self-similar at all times with the radius growing in proportion to t1/2. In the regimes of injection-dominated and gravity-dominated currents, we obtain asymptotic approximations for the interface shape, which are found to agree well with the numerical computations. For a decreasing input flux (0 < α < 1), at short times, the flow is controlled by injection; the current fills the depth of the channel spreading with radius r ∼ tα/2. At long times, buoyancy dominates and the current becomes unconfined with the radius growing in proportion to t(3α+1)/8. The sequence of regimes is reversed in the case of an increasing input flux (α > 1) with buoyancy dominating initially while the pressure associated with the injection dominates at late times. Finally, we consider the release of a fixed volume of fluid (α = 0). The current slumps under gravity and transitions from confined to unconfined, and we obtain asymptotic predictions for the interface shape in both regimes.

Buoyancy segregation suppresses viscous fingering in horizontal displacements in a porous layer

“…In future work, these ideas could be extended to viscous displacements in horizontal channels, as is relevant to mantle plumes (Schoonman, White & Pritchard 2017). The base self-similar flow was found for two-dimensional and axisymmetric geometries by Zheng, Rongy & Stone (2015 b ) and Hinton (2020), but the stability has not yet been investigated. The analysis would be more complicated than the present work owing to the effect of the no-slip boundaries and the shear flow (Snyder & Tait 1998; John et al.…”

The evolution of a viscous gravity current in a confined geometry

“…The partial differential equation (2.6) thus reduces to the ODE and the boundary conditions (2.7)–(2.9) and (2.11) become where is the ratio of the scale height for an unconfined gravity current to the gap width of the cell and denotes derivatives with respect to . A similar dimensionless parameter , which reduces to when the density of the ambient fluid is neglected and represents a dimensionless volume flux, was identified by Hinton (2020). Although (2.15) and the boundary conditions at the leading edge (2.17 a , b ) apply to unconfined axisymmetric gravity currents, the boundary conditions (2.16 a , b ) are specific to confined flows.…”

“…In contrast to the model considered by Hinton (2020), the displaced air is assumed to be inviscid, which allows for the development of a grounding line separating the inner contact region from the outer annular region. We apply lubrication theory, which is valid once , where is the Reynolds number and is the smaller of the characteristic lengths corresponding to and .…”

“…Comparatively little research has been done to determine the conditions under which free-fluid, viscous gravity currents are impacted by confinement. Such flows have been investigated in two dimensions in the context of filling or cleaning of channels by injection of one viscous fluid to displace another (Taghavi et al 2009; Zheng, Rongy & Stone 2015) and in axisymmetry by Hinton (2020). A review of the effects of confining boundaries on viscous gravity currents has been conducted by Zheng & Stone (2022).…”

“…In two-dimensional flows in confined porous media (Pegler, Huppert & Neufeld 2014), it has been found that there is a transition from early times at which the injected fluid only partially fills the gap, to late times at which the flow domain has a region close to the source in which the injected fluid fills the gap completely and a region further away in which the injected fluid makes contact with only one of the boundaries. In the case of one viscous fluid displacing another (Taghavi et al 2009; Zheng et al 2015; Hinton 2020), there is a transition in propagation rates from early to late times though, as mentioned above, the ambient fluid cannot be completely evacuated. In the case of axisymmetric spreading of a viscous fluid into an inviscid ambient that we study here, there is a propagating grounding line but the proportion of the flow that makes contact with both boundaries is constant in time and we find a critical dimensionless parameter governing the flow that determines that proportion.…”