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arxiv: 2604.09393 · v1 · submitted 2026-04-10 · ⚛️ physics.flu-dyn

HOC simulations of miscible viscous fingering of a finite slice: A new insight

Mijanur Rahaman , Jiten C. Kalita , Satyajit Pramanik This is my paper

Pith reviewed 2026-05-10 17:01 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords viscous fingeringmiscible fluidsporous mediaboundary conditionsnumerical simulationmixing lengthfinite slice
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0 comments X

The pith

Permeable boundaries allow solute mass to increase and strengthen fingering instabilities over long times in finite miscible slices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper simulates viscous fingering when a finite slice of fluid displaces another in porous media, with viscosity set by local solute concentration. It solves the coupled Darcy flow and advection-diffusion equations using a high-order compact scheme under three transverse boundary types. Early fingering onset and initial growth remain the same across periodic, impermeable, and permeable cases. Permeable walls that admit normal velocity but block diffusive solute flux let total solute mass rise, which then produces more vigorous fingers, longer mixing zones, and irregular interface stretching at later times. These boundary-dependent effects are relevant to separation techniques such as chromatography.

Core claim

Although the onset of viscous fingering and early-time behavior are independent of boundary type, long-time dynamics, solute mixing, and interface evolution depend on the choice of transverse boundaries. Permeable boundaries permit an increase in solute mass, which drives stronger fingering instabilities, larger mixing lengths, and non-trivial changes in interfacial lengths compared with periodic or impermeable boundaries.

What carries the argument

Fourth-order compact finite difference discretization of the stream-function formulation of Darcy's law coupled to an advection-diffusion equation for solute transport, with viscosity an exponential function of concentration.

Load-bearing premise

The chosen numerical discretization and viscosity-concentration relation accurately capture the long-time physics of the system.

What would settle it

Laboratory visualization of a finite miscible slice between permeable transverse walls that tracks total solute mass and mixing length over time would confirm or refute the predicted mass increase and stronger instabilities.

Figures

Figures reproduced from arXiv: 2604.09393 by Jiten C. Kalita, Mijanur Rahaman, Satyajit Pramanik.

Figure 1
Figure 1. Figure 1: Schematic representation of the problem under consideration. A finite sample of viscosity 𝜇2 is displaced by another fluid of viscosity 𝜇1 in a homogeneous, isotropic porous medium (not to be scaled). The displacing fluid is injected at a velocity (𝑈, 0). viscosity 𝜇1 and solute concentration 𝑐 = 0 is injected at a constant speed 𝑈 along the 𝑥-direction to displace the finite sample (see figure 1 for a sch… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The 9-point HOC stencil with the associated coefficients in the matrix equation and (b) The structure of the coefficient matrix along with the locations of the non-zero coefficients. 3. Numerical method A closed-form analytical solution is not available for the nonlinear system of two-way coupled advection-diffusion equations (7)-(8). We seek numerical solutions of these equations utilising a higher-or… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the approximation of periodic boundary condition. entries clustered around the principal diagonal, another three entries following 𝑁𝑥 − 3 zeros, and the final three entries located after (𝑁𝑥 − 1)(𝑁𝑦 − 3) − 2 zeros. The coefficient matrix of the global system largely preserves the structure shown in figure 2(b), except at those locations where the coefficients forming the clusters of diagona… view at source ↗
Figure 4
Figure 4. Figure 4: Spatio-temporal distribution of the finite sample of width 𝑙 = 256 exhibiting fingering dynamics and solute spreading at various time levels for (a) 𝑅 = 3, (b) 𝑅 = −3 corresponding to the Type-I boundary conditions. Other parameter are 𝐿𝑥 = 4096, 𝐿𝑦 = 512. respectively. These measures help quantify the effects of diffusion in the longitudinal direction. As anticipated, the solute sample diffuses isotropica… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Spatio-temporal distribution of concentration for 𝑅 = 0 at different time levels corresponding to the Type-I boundary conditions, (b) the corresponding transverse-averaged concentration. (c) Rescaled longitudinal variances, (𝜎 2 𝑥 (𝑡)∕𝜎 2 𝑥,0 − 1), corresponding to the Type-I (red), the Type-II (green), and the Type-III (blue) boundary conditions are shown. It is observed that in the absence of viscous… view at source ↗
Figure 6
Figure 6. Figure 6: Spatio-temporal distribution of a more viscous slice of width 𝑙 = 256 for 𝑅 = 3 corresponding to (a) Type-I, (b) Type-II, and (c) Type-III boundary conditions. Other parameters are the same as in the figure 4. (a) (b) (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Spatio-temporal distribution of a less viscous slice of width 𝑙 = 256 for 𝑅 = −3 corresponding to (a) Type-I, (b) Type-II, and (c) Type-III boundary conditions. Other parameters are the same as in the figure 4. dynamics does not depend strongly on the choice of the boundary conditions. Post-breakthrough dynamics vary with the choice of the boundary conditions. To better understand the spreading and mixing … view at source ↗
Figure 8
Figure 8. Figure 8: Temporal evolution of the normalized mass 𝑚(𝑡) for (a) 𝑅 = 3, (b) 𝑅 = −3 corresponding to Type-III boundary conditions. The solid line represents the mass accumulating through area integration, whereas the dashed line represents the mass from flux calculation. Other parameters are the same as in the figure 4. a feature of diffusive spreading. However, for 𝑡 > 𝑡𝑏𝑘, the fate of the stable interface depends o… view at source ↗
Figure 9
Figure 9. Figure 9: Transverse-averaged concentration profile for (a) 𝑅 = 3, and (b) 𝑅 = −3 corresponding to Type-I boundary conditions. Other parameters are the same as in the figure 4. (a) (b) 1000 2000 3000 4000 0 0.2 0.4 0.6 0.8 1 500 1500 2500 3500 4000 0 0.2 0.4 0.6 0.8 1 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Transverse-averaged concentration profile for (a) 𝑅 = 3, and (b) 𝑅 = −3 corresponding to Type-II boundary conditions. Other parameters are the same as in the figure 4. increases with time before it experiences an instantaneous decrease followed by a subsequent increase in time. This temporal decrease is caused by a dilution of the advancing finger, and the temporal increase is again due to a fingering gro… view at source ↗
Figure 11
Figure 11. Figure 11: Transverse-averaged concentration profile for (a) 𝑅 = 3, and (b) 𝑅 = −3 corresponding to Type-III boundary conditions. Other parameters are the same as in the figure 4. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of mixing lengths (a) rear unstable (𝑅 > 0) interface (backward mixing length, 𝐿 − 𝑑 ) and (b) frontal stable interface (forward mixing length, 𝐿 + 𝑑 ) for 𝑅 = 3 corresponding to different types of boundary conditions. The stable diffusive mixing length is represented by 𝑅 = 0. It clearly depicts the domination of mixing length for Type-III boundary conditions. gradient near the stable interfac… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of mixing lengths (a) frontal unstable (𝑅 < 0) interface (forward mixing length, 𝐿 + 𝑑 ) and (b) rear stable interface (backward mixing length, 𝐿 − 𝑑 ) for 𝑅 = −3 corresponding to the different types of boundary conditions. 𝑅 = 0 corresponds to the diffusive mixing length. It measures the temporal variation of the concentration gradient across the domain, and captures the onset of the viscous f… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of mixing lengths at the rear unstable interface for 𝑅 = 3 (backward mixing length, 𝐿 − 𝑑 ) and at the frontal unstable interface for 𝑅 = −3 (forward mixing length, 𝐿 + 𝑑 ) for different boundary conditions: (a) Type-I, (b) Type-II, and (c) Type-III. Diffusive mixing is represented by 𝑅 = 0. flow conditions considered here. This is because viscous fingers carry the less viscous sample along wit… view at source ↗
Figure 15
Figure 15. Figure 15: Temporal evolutions of the interfacial lengths for (a) 𝑅 = 3 and (b) 𝑅 = −3 corresponding to the different type of boundary conditions. Other parameters are the same as in the figure 4. the late-time dynamics–particularly the fingering behavior and solute transport after breakthrough–depend on the imposed boundary conditions. Permeable boundaries (Type-III boundary conditions) allows mass enhancement and … view at source ↗
Figure 16
Figure 16. Figure 16: Temporal evolution of interfacial lengths for 𝑅 = 3 and 𝑅 = −3 corresponding to (a) Type-I, (b) Type-II, and (c) Type-III boundary conditions. Diffusive interfacial length is represented by 𝑅 = 0. Other parameters are the same as in the figure 4. Kim, M.C., 2012. Linear stability analysis on the onset of the viscous fingering of a miscible slice in a porous medium. Advances in water resources 35, 1–9. Kim… view at source ↗
Figure 17
Figure 17. Figure 17: Temporal evolution of (a) distance of the mean position from its initial value, (b) variance normalized by its initial value for different flow conditions: Type-I (blue), Type-II (black), and Type-III (red). It is observed that the mean position and the variance of a less viscous sample (𝑅 = −3, dashed lines) are higher than the corresponding more viscous sample (𝑅 = 3, solid lines). Other parameters are … view at source ↗
Figure 18
Figure 18. Figure 18: Temporal evolution of skewness of more (𝑅 = 3, solid lines) and less (𝑅 = −3, dashed lines) viscous samples for different flow conditions: Type-I (blue), Type-II (black), and Type-III (red). The grey dash-dotted line represents the stable scenario (𝑅 = 0). Other parameters are the same as in the figure 4. Rousseaux, G., De Wit, A., Martin, M., 2007. Viscous fingering in packed chromatographic columns: Lin… view at source ↗
read the original abstract

We investigate the dynamics of viscous fingering (VF) in miscible slices in homogeneous, isotropic porous media. The fluid flow is governed by incompressible Darcy's law, whereas the solute transport is described using an advection-diffusion equation. The viscosity of the miscible system depends on the solute concentration, creating a viscosity contrast between the displacing fluid and the finite sample. When expressed in terms of stream function, the flow is described by a system of nonlinear, two-way coupled advection-diffusion type equations. We consider three types of boundary conditions: (a) periodic, (b) impermeable (zero normal velocity) and no-flux (solute), and (c) permeable (allowing non-zero normal velocity) and no diffusive flux (solute) transverse boundaries. This initial boundary value problem is solved numerically using a fourth-order compact finite difference method, while the Crank-Nicolson technique is used for time integration. Although the onset of viscous fingering and early time behavior are independent of the choice of boundary types, long-time behavior, solute mixing and spreading depend on the boundary conditions. In particular, it is observed that the permeable boundaries allow solute mass to increase, leading to stronger fingering instabilities, larger mixing lengths and non-trivial evolution of interfacial lengths. The findings of this study have implications in chromatography separation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript numerically studies viscous fingering of a finite miscible slice in porous media governed by Darcy's law and advection-diffusion, with viscosity depending on concentration. It solves the stream-function formulation using fourth-order compact finite differences in space and Crank-Nicolson in time, comparing periodic, impermeable, and permeable transverse boundary conditions. The central claim is that while onset and early-time behavior are insensitive to boundary type, permeable boundaries permit net solute mass increase at long times, producing stronger fingering, larger mixing lengths, and non-monotonic interfacial length evolution, with implications for chromatography.

Significance. If the mass-increase observation is confirmed to be physical, the work supplies a useful distinction between boundary-condition effects on long-time mixing and spreading in miscible displacements. The chosen high-order compact scheme is standard and appropriate for the problem; the forward-simulation nature avoids circularity. However, the quantitative long-time claims rest on unverified numerical transport through open boundaries.

major comments (1)
  1. [Numerical results (long-time behavior)] The headline result that permeable boundaries produce net solute mass growth (and thereby stronger instabilities and larger mixing lengths) is load-bearing for the abstract and conclusions. No grid-convergence study, discrete mass-flux integral, or conservation check for the permeable case is reported. Because the fourth-order compact FD + Crank-Nicolson discretization is not locally conservative by construction, the observed mass increase could be a long-time truncation artifact at the open boundaries rather than a feature of the continuous model.
minor comments (2)
  1. [Abstract] The abstract and introduction supply no validation against known VF benchmarks, no error bars, and no convergence data; adding a short statement on these points would improve reader confidence in the quantitative mixing-length and interfacial-length results.
  2. [Governing equations] The viscosity-concentration functional form and its numerical implementation should be stated explicitly (including any parameters) so that the reported sensitivity to boundary conditions can be reproduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the recognition of the work's potential significance and address the major comment point by point below. We will incorporate revisions to strengthen the numerical validation as outlined.

read point-by-point responses

  1. Referee: [Numerical results (long-time behavior)] The headline result that permeable boundaries produce net solute mass growth (and thereby stronger instabilities and larger mixing lengths) is load-bearing for the abstract and conclusions. No grid-convergence study, discrete mass-flux integral, or conservation check for the permeable case is reported. Because the fourth-order compact FD + Crank-Nicolson discretization is not locally conservative by construction, the observed mass increase could be a long-time truncation artifact at the open boundaries rather than a feature of the continuous model.

    Authors: We agree that the manuscript lacks an explicit grid-convergence study and discrete mass-flux verification for the permeable-boundary simulations, and that the chosen scheme is not locally conservative. This is a legitimate concern for long-time results. However, the mass growth is consistent with the underlying continuous model: the permeable transverse conditions allow non-zero normal velocity (with zero diffusive flux), so the advection term permits net solute transport across the boundaries depending on the instability-induced flow. To resolve the issue, the revised manuscript will add a dedicated numerical-validation subsection. This will include (i) total solute mass histories on successively refined grids (e.g., 256×256 to 1024×1024) demonstrating convergence of the long-time growth, (ii) direct evaluation of the discrete advective mass-flux integrals at the open boundaries, and (iii) a comparison showing that the observed dM/dt matches the boundary flux to within the truncation error of the scheme. These additions will confirm that the mass increase is a physical consequence of the permeable conditions rather than a numerical artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: forward numerical simulation of stated PDE system

full rationale

The paper solves the incompressible Darcy flow coupled to an advection-diffusion equation for concentration, with viscosity as a given function of concentration, using a fourth-order compact finite-difference spatial scheme and Crank-Nicolson time stepping. All reported quantities (onset time, mixing length, interfacial length, mass evolution) are direct outputs of the discrete solver applied to the initial-boundary-value problem under three different transverse boundary conditions. No parameters are fitted to data, no auxiliary quantities are defined in terms of the target observables, and no self-citation is invoked to justify a uniqueness theorem or ansatz that would close the derivation. The central observations therefore follow from the numerical integration of the stated continuous model rather than from any reduction to the inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The study rests on standard continuum assumptions for porous-media flow and transport; no new entities are postulated and the only free parameters are the usual dimensionless groups (viscosity ratio, Péclet number, etc.) typical of VF simulations.

free parameters (2)
  • viscosity ratio
    The contrast between displacing and displaced fluid viscosities is a central control parameter whose specific value is chosen for the simulations.
  • Péclet number
    Ratio of advection to diffusion that governs the sharpness of the concentration front.
axioms (2)
  • domain assumption Flow obeys incompressible Darcy's law
    Standard modeling assumption for slow flow in homogeneous isotropic porous media.
  • domain assumption Viscosity is a function of local solute concentration
    Common constitutive relation for miscible VF; functional form not specified in abstract.

pith-pipeline@v0.9.0 · 5550 in / 1364 out tokens · 53210 ms · 2026-05-10T17:01:07.503802+00:00 · methodology

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Reference graph

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