The Darcy law and the Brinkman equation for a Hele-Shaw flow
Let us consider an incompressible fluid of a constant viscosity η flowing be-tween parallel plates at a velocityv = (vx, vy , vz ), the plates being z = ±h/2. It is well-known and simple to check that the equations vy = vz = 0, 1 dp h2 vx(z) = − − z2 , (1.7) 2η dx 4 describe the planar Poiseuille flow driven by a constant pressure gradient dp/dx. At any gradient, this is a solution to the full stationary Navier– Stokes equations (NSE) with the no-slip conditions at the plates (we lay aside the question of its stability).
Now imagine that the flow for some reason is not unidirectional. The exact solution becomes prohibitively complicated, if feasible at all, because of the non-linear term in NSE. However, at low enough velocity the term is negligible throughout the entire flow domain (§2.7 of ), so that NSE simplifies to the linear Stokes equation: − ∇p + ηΔv = 0 . (1.8)
With some reservations, this equation admits an important reduction that we are going to describe now. It is remarkable that the pressure gradients are indeed independent of z. Besides, this relation proves thatu can be potential. We have demonstrated that Eqs.(1.9), (1.10) solve the incompressible Stokes equation (1.8). A z-independent potential force density can be added to Eq.(1.8) and absorbed into the pressure gradient, changing nothing in the analysis. Essentially, we have found that with such driving force, a three-dimensional laterally unbounded flow exactly reduces, by the substitution (1.9), to a two-dimensional one at a gap-averaged velocity. The unbounded-ness is essential for the following reason. Imagine a cylindric obstacle situated between the plates perpendicularly to them, filling the gap entirely; at the boundary of the obstacle some conditions are posed in terms of the velocityv (§4.8 of ). Then, apart from reducing the number of dimensions, the sub-stitution also eﬀectively reduces the order of the diﬀerential equation. Not all boundary conditions posed for the Stokes equation (1.8) can be satisfied at solving the reduced Eq.(1.10). (The exact analogy can also break at free surfaces and discontinuities.) The no-flux (non-permeability) condition can be satisfied, but not together with the no-slip condition. However, the flow “feels” the presence of the no-slip condition only within a thin belt of thickness ∼ h around the obstacle. Indeed, if vx and vy are required to vanish at the boundary as well as at the plates, their second derivatives in Eq.(1.8) across the gap and along the plates can be estimated to relate as (l/h)2, where l is a distance from the boundary. Therefore at l ≫ h the boundary condition has no impact on the pressure gradient and, by Eq.(1.12), on velocity. The same result is valid near the lateral sides of a Hele-Shaw cell .
Eq.(1.12) for the gap-averaged velocity is exactly (to within the coeﬃ-cient) a two-dimensional version of the Darcy law that is widely used to describe, at coarse enough scales, the groundwater flow  in porous media with permeability α = 12η/h2 (we will call it the friction coeﬃcient). Due to this direct analogy, the Hele-Shaw device (cell) was often employed to model the percolation processes. However, originally it was introduced by H. J. S. Hele-Shaw back in 1898 to model steady two-dimensional incom-pressible potential inviscid flows around various obstacles (§330 of ; see photos in ). At a suﬃciently narrow separation between the plates (com-pared to the dimensions L of the obstacle in the x, y plane, L ≫ h), the diﬀerence in the boundary conditions is negligible and the above approach can indeed be reverted. Of course, the pressure p of the Hele-Shaw flow will have nothing in common with the pressure piv of the inviscid flow (see §3.9 in ) that is calculated from the velocity through a non-linear relation: ρu( ∇u) = − piv .
The magnetic ponderomotive force
In this paragraph we will present the expressions for magnetic ponderomotive force and magnetic field that will be used throughout the following work. In the approximation of magnetostatics for a non-conducting ferrofluid the Maxwell equations give div B = 0 , (1.14) rot H = 0 . (1.15).
Relaxation of the magnetization M will be considered instantaneous, so that M H. The general formula for the ponderomotive-force density in a liquid mag-netic reads (Eq.(4.33) of  in Gaussian units3) H ∂(M v) fm = − dH + M ∇H , (1.16) ∂v 0 H,T.
Miscible interfaces in a Hele-Shaw cell: an overview and model
In a magnetic fluid, the transport processes allow an extra control parameter: the applied magnetic field. They attract scientific interest because of their specific cooperative nature, since the self-magnetic field of the colloid as a whole influences the magnetophoretic motion of colloidal particles and leads to the field-dependent anisotropic eﬀective diﬀusion.
However, as we have discussed in the Introduction, even if the external field is uniform, the self-magnetic field can give rise to a convective instabil-ity.1 In a thin plane layer with rigid transparent walls (a Hele-Shaw cell), miscible instabilities with MF’s can be observed directly with a microscope. In the experiment  MF and its pure carrier liquid were brought into con-tact in a Hele-Shaw cell forming a narrow straight “diﬀusion front.” In the perpendicular (to the cell) external field, the interface developed an intricate labyrinthine pattern, while a peak pattern was obtained in the normal field (i.e. in the field applied along the cell perpendicularly to the front). The pattern length scale was approximately as small as the layer thickness (less than 10−2 cm). Having quickly formed, the patterns were gradually blurred out by diﬀusion.
The miscible stability problem and the continuous spectrum2
Let us consider the linear stability of some one-dimensional concentration distribution c0(x, t0). The miscible basic flow is time-dependent due to dif-fusion so that a quasi-steady-state approximation (QSSA) must be adopted to study the linear stability by means of the normal-mode analysis at some moment t0 . Hereby we discard the further diﬀusion of the basic state as the flow perturbations evolve; their diﬀusion is taken into account, however. Technically, this amounts to “freezing” the time-dependent coeﬃcients in the linearized perturbation equations. QSSA is considered  valid for diﬀused enough interfaces. Non-QSSA attempts were an exception , though re-cently a quite general QSSA-free approach to the long-wave linear stability of miscible interfaces was suggested . Note also the boundary condi-tions introduced in  that render the basic flow with diﬀusion steady and can perhaps be used experimentally to create a controlled diﬀusion front (the solute spreads by diﬀusion upwards into a vertical Hele-Shaw cell whose open bottom side is immersed into a large reservoir, with a downward flow of the solvent opposing the spreading). In principle, a time-independent basic state can be maintained also by a chemical reaction .
Table of contents :
1 Governing equations
1.1 About ferrofluids
1.2 Gap-averaging, diffusion, and stresses
1.3 The Darcy law and the Brinkman equation
1.4 Magnetic ponderomotive force
2 Stability of a miscible interface
2.1 Miscible interfaces in a Hele-Shaw cell
2.2 Miscible problem and continuous spectrum
2.3 Labyrinthine instability
2.3.1 Derivation of the dispersion relation
2.3.2 Stability diagram and asymptotic analysis
2.3.3 Physical mechanism of oscillations
2.3.4 Labyrinthine instability of a diffused interface
2.4 Peak instability in a Hele-Shaw cell
2.4.1 Perturbations and sharp interface: normal field
2.4.2 A periodic stripe pattern in the normal field.
2.4.3 A diffused interface in the normal field
2.4.4 A comparison to FRS experiments
3 ST instability with an immiscible MF
3.1 The free-boundary problem
3.1.1 The context of the problem
3.1.2 Formulation of the problem
3.1.3 Alternative formulations
3.1.4 Integral equation
3.1.5 Magnetic force and finger
3.2 The numerical method
3.2.1 Modelling with BIE
3.2.2 Interface and curvature
3.2.3 Discretization of integrals
3.2.4 Characterization and validation
3.3 Numerical results for the perpendicular field
3.3.1 Magnetic Saffman–Taylor fingers
3.3.2 Dendritic patterns
3.4 Numerical results for the normal field
List of Figures
List of Tables