Selective withdrawal of a two-layer viscous fluid

Selective withdrawal of a three-layer viscous fluid

“…The chosen geometry, shown in Fig. 1, was largely motivated by the findings of Cosgrove and Forbes [7]. In their work they concluded that two line sinks placed symmetrically on the base of the tank was the optimal positioning for the withdrawal of the lower, denser fluid layer to be maximised.…”

“…In our formulation of the linearised model solutions were obtained by solving Laplace's equation in each of the three fluid regions. When extending to the nonlinear model, we adopt the strategy outlined by Cosgrove and Forbes [7] and derive solutions by considering the entire tank to be a single fluid region with an initial prescribed density profile. With the two-dimensional flow q ˜= ui ˜+ v j ˜, where u and v denote the horizontal and vertical velocity components, Boussinesq theory allows us to express the conservation of mass in terms of an incompressible part…”

“…in which the time-independent Fourier coefficients A n and B n are given by (2.10). Following Farrow and Hocking [6] and Cosgrove and Forbes [7], slip conditions are imposed along the side walls and on the base of the tank so that the boundary layers on these surfaces can be ignored. As the boundary conditions (2.3), (2.4) and (2.5) are satisfied by the steady component b , the unsteady correction term has to account for the viscosity of the fluid and we write…”

“…Cosgrove and Forbes [7] built on the studies of Farrow and Hocking [6], Forbes and Hocking [5], Hocking [8] and others, but extended the analysis to allow for weakly compressible, viscous and miscible fluids that flow rotationally. The motivation for this work was the suggestion that such non-ideal fluids might align more closely with real world situations.…”

“…One of these is located on the base of the rectangular tank while the other two are positioned at the bottom of each side wall. Although different drain configurations are considered, the problem closely follows solutions presented in [7], [6] and [5] with the key modification that we have three fluid layers of distinct densities which are separated by two sharp interfaces. The fluids are assumed to be weakly compressible, viscous and flow rotationally.…”