A general solution of the Hele–Shaw problem for flows in a channel

The reducibility of the anisotropic Hele-Shaw problem to the isotropic case

“…[3][4][5], including of fairly general form. 6 At the same time, for the case of an anisotropic medium there is obviously a unique exact solution of the problem with a free boundary of the Hele-Shaw problem type -the solution of the stationary problem of the influx of ground waters to a drain in a lock in an anisotropic ground. 7 The construction of this solution is based on the use of the well-known linear non-orthogonal coordinate transformation.…”

Unsteady motion of a bubble in a Hele-Shaw cell

“…We note the essential difference of the Polubarinova-Galin equation in this case from the equation for the Hele-Shaw flow in a channel-type cell (see, for example, [4,7]): the right-hand side of Eq. (2.6) is not constant in space.…”

“…At the same time, in [7] the form of the equation was obtained, which is valid for arbitrary Hele-Shaw flow:…”

“…We introduce the function r(ζ , t), which is the image of the Schwartz function S(z, t) in the auxiliary plane ζ [14,7]:…”

“…The geometry of the flows differs depending on the presence or absence of impermeable walls of the cell, as well as the law of suction (or supply) of the liquid on the periphery of the cell. To date, there are a number of exact solutions of the idealized Hele-Shaw problem: fingering on the interface in a channel-type slot, bubble growth in an infinite cell, evolution of a parabolic region boundary, and evolution of the "air-viscous fluid" interface in an unbounded slot cell, when the fluid occupies a region of half-space type [3][4][5][6][7][8][9]. In all cases considered, the interphase either advances onto the wetted part of the slot (air displaces a viscous fluid and the interface is unstable), or recedes from the wetted part of the slot (a viscous fluid displaces air, and the interface is stable) [2].…”