Analytical solution for dynamic evaporation of liquid in isothermal condition

Alexander J. Wagner; Luiz Eduardo Czelusniak; Mathias J. Krause; Stephan Simonis; Tim Niklas Bingert

REVIEW 3 major objections 6 minor 17 references

Analytical solution for dynamic evaporation of liquid in isothermal condition

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Evaporation rate depends on fluid viscosity, model shows

desk verdict A transparent diffuse-interface derivation of an evaporation-rate formula with a real but approximate viscosity dependence; the paper overstates agreement in places but deserves serious refereeing. read the letter →

arxiv 2506.02270 v1 pith:6BMVLVIM submitted 2025-06-02 physics.flu-dyn physics.comp-ph

Luiz Eduardo Czelusniak , Tim Niklas Bingert , Stephan Simonis , Alexander J. Wagner , Mathias J. Krause This is my paper · ORCID

classification physics.flu-dynphysics.comp-ph

keywords diffuseinterfacemodelisothermalevaporationvelocityviscositydependencelatticeBoltzmannmethodanalyticalsolutionAbelequationsub-saturationpressure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives an analytical solution for the speed at which a liquid interface evaporates into vapor below the saturation pressure, using a diffuse interface model in which the interface is a thin region of continuously varying density rather than a sharp jump. Starting from mass and momentum conservation with a Korteweg-type pressure, the authors obtain an exact formula for the interface velocity in the inviscid limit and an approximate quadratic formula when viscosity is present. The central result is that the evaporation rate decreases with increasing fluid viscosity, a dependence that classical sharp-interface and kinetic-theory evaporation models do not contain. If correct, the formulas provide a benchmark for validating diffuse-interface codes and a starting point for extending evaporation modeling to non-isothermal and multi-component settings. The predictions are checked against lattice Boltzmann simulations over a range of density ratios, pressure ratios, and viscosities.

What carries the argument

The load-bearing device is the traveling-wave ansatz: after a transient, the interface moves at constant velocity $u_{\mathrm{int}}$ and every field becomes time-independent in the co-moving frame $x' = x - u_{\mathrm{int}}t$. This reduces the partial differential equations to ordinary differential equations in $x'$. Mass conservation in that frame gives the identity $\rho(u_{\mathrm{int}}-u) = \rho_l u_{\mathrm{int}}$, which removes the velocity from the momentum equation and leaves a single nonlinear ODE for the density profile, of the Abel second-kind form. In the inviscid case the ODE integrates directly, giving the exact interface velocity; in the viscous case the same ODE is closed by inserting the inviscid density profile, producing a quadratic equation for $u_{\mathrm{int}}$ that is the approximate solution. The Korteweg-type pressure, which adds surface-tension terms to the equation of state, enters through the density-gradient terms in that ODE.

What would settle it

Run a resolved diffuse-interface simulation of a flat liquid slab at fixed temperature, fixed sub-saturation pressure, and fixed density ratio while changing only the kinematic viscosity by, say, a factor of ten; if the measured interface velocity stays constant instead of decreasing with viscosity, the paper's central claim is falsified. The analogous experiment with two liquids of different viscosity and matched vapor pressure would test it in a laboratory.

Watch

Extended reading notes

Core claim

The paper claims that for a flat liquid–vapor interface in an isothermal fluid held below saturation pressure, the steady interface velocity $u_{\mathrm{int}}$ satisfies a closed ordinary differential equation derived without assuming local thermodynamic equilibrium at the interface. In the inviscid case the solution is exact: $u^{(0)}_{\mathrm{int}} = \sqrt{ \int_{\rho_v}^{\rho_l} \frac{p_{\mathrm{EOS}}-p_v}{\rho^2}\,d\rho } \big/ \sqrt{ \frac{\rho_l}{\rho_v}\left(\frac{1}{2}\frac{\rho_l}{\rho_v}-1\right) + \frac{1}{2} }$, coupled with the momentum balance and equation of state for the liquid density $\rho_l$. With viscosity, the density profile is approximated by the inviscid profile, turning the problem into a quadratic equation for $u_{\mathrm{int}}$ whose physically selected root decreases as viscosity grows. The paper argues that this viscosity dependence is a genuine feature of the diffuse-interface description, absent from sharp-interface models, and that it is confirmed by lattice Boltzmann simulations.

Load-bearing premise

The derivation rests on the assumption that, after a transient, the interface settles into steady motion at constant velocity with all fields stationary in the co-moving frame; if that steady traveling wave does not form, or if the viscous density profile departs too far from the inviscid one, the closed-form formulas do not apply.

Editorial extensions

If this is right

In the inviscid limit the interface velocity is independent of the surface-tension coefficient and depends only on the equation of state and the ratio of liquid to vapor density.
At finite viscosity the interface velocity falls below the inviscid value, and for large viscosity the quadratic term in the equation can be neglected so the velocity scales inversely with viscosity.
The viscous solution converges to the exact inviscid result as viscosity tends to zero, so the two formulas form a single continuous prediction.
Because the analytical profile can be used as an initial condition, diffuse-interface simulations reach the steady interface velocity with a shorter transient, and that final velocity is independent of domain size and initial condition.
The inviscid formula agrees with the sharp-interface solution across a wide parameter range, with the largest deviations at low density ratios and strongly reduced pressure ratios.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct experimental discriminator would be to evaporate different liquids with comparable saturation pressure but different kinematic viscosity at the same sub-saturation pressure: the model predicts slower interface motion for the more viscous liquid, while sharp-interface theory predicts equal speeds.
The same co-moving-frame reduction could be applied to non-isothermal evaporation by adjoining an energy equation, where viscosity would likely enter through viscous dissipation in the temperature profile as well.
The approximation's known weakness at low density ratio suggests a next-order correction in the viscosity parameter, obtained by perturbing the density profile away from the inviscid one, which would extend the formula's range.
The equations also provide a ready-made benchmark suite for any diffuse-interface solver: fixing density ratio, pressure ratio, and viscosity fixes a unique expected interface velocity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper is a useful benchmark for diffuse-interface evaporation codes, and the viscosity dependence it reports is a genuine prediction of the model—not an artifact. I checked the stress-test sign concern about Eq. (18a); it doesn't hold up. Substituting (4) into (16) and rearranging gives the plus sign as printed. That part is fine.

What's new: the derivation starts from the diffuse-interface equations, avoids local equilibrium at the interface, and produces an exact inviscid formula (28) and a documented viscous approximation (33). The inviscid formula reduces to the Jamet/Ishii sharp-interface result when you impose the Maxwell construction—the paper says this explicitly—so the genuinely new content is the diffuse-interface route and the viscous correction. The comparison with LBM simulations using Wagner's free-energy model is consistent, and the paper does report the validity limits: errors grow at low density ratio and high viscosity.

The soft spots are real but not disabling. Table I lists errors up to 24.8%, yet the abstract and conclusion say 'excellent agreement' without qualification. That's an overstatement; the agreement is excellent only in the low-ν, high-density-ratio regime. The validation is self-referential—same LBM model, same free-energy functional—so it confirms internal consistency, not physical accuracy. The paper would be stronger with an independent benchmark, e.g., a different multiphase method or experimental data. Also, the 'novel relationship' claim about viscosity should be checked against the phase-field literature on non-equilibrium evaporation; the citation list is thin on that point. The code is referenced via OpenLB but the specific simulation scripts are not provided, which makes exact reproduction harder.

Overall, the derivation is algebraically transparent, the assumptions (traveling wave, monotone profile) are stated, and the results are internally consistent. The central claim survives my reading. Who benefits: anyone implementing or validating diffuse-interface LBM codes, especially for fuel-injection or sub-saturation evaporation. It's a niche result, but a solid one. A serious referee should engage; the paper needs revisions to temper the agreement claims, add reproducibility details, and possibly broaden the comparison. Don't desk reject.

Referee Report

3 major / 6 minor

Summary. This paper derives an analytical expression for the interface velocity during isothermal liquid-vapor evaporation in a one-dimensional diffuse interface model with a Korteweg-type pressure. The mass and momentum balance equations are combined with a traveling-wave ansatz to obtain an exact inviscid solution (Eq. 28) and an approximate viscous solution (Eq. 33) that uses the inviscid density profile in the viscous correction. The results are validated against free-energy lattice Boltzmann simulations conducted with OpenLB. The paper's central claim is that the evaporation rate depends on viscosity, a dependence not captured by classical sharp-interface evaporation models.

Significance. The inviscid solution is an exact, parameter-free benchmark result for diffuse interface methods, and the comparison with Jamet's sharp-interface solution helps delineate the validity of the Maxwell construction. The disclosed viscosity dependence is a clear, testable prediction of the model. The paper is transparent about the approximate nature of the viscous solution and provides error tables. The use of an open-source library and the availability of the simulation code support reproducibility.

major comments (3)

[Abstract and Section VII] The abstract and conclusion state that the analytical approximation shows 'excellent agreement' with LBM simulations, but Table I lists relative errors up to 24.81% (for rsatρ=2, fp=0.97, ν*=6.4). The claims of agreement should be qualified by the valid parameter range (rsatρ≥8, fp≥0.90, low-to-moderate viscosity) that is actually documented in Section VI.B.
[Section IV.E, Eqs. (30)-(33)] The approximation z≈z^(0) used to derive Eq. (33) is formally a zeroth-order-in-viscosity perturbation, yet the paper uses Eq. (33) to predict the high-viscosity limit uint ∝ 1/ν. This high-ν branch is not justified by the derivation and should be presented as an extrapolation that happens to match the LBM results, unless a small-viscosity expansion of u is provided.
[Section VI.A] The verification of Eq. (16) in Figure 5 compares the analytical pressure balance against the same free-energy LBM model that is later used for validation. This is a consistency check of the derivation against a numerical solver of the same equations, not an independent physical validation; the paper should state this limitation in the conclusions.

minor comments (6)

[Section VI.A, Eq. (47c)] The definition of pfriction in Eq. (47c) appears to have a typo: '2ρ_l u_int ρ dρ/dx′' should be '2ρ_l u_int ν/ρ dρ/dx′' to be consistent with Eq. (16) and Eq. (47d).
[Section VI.A, Figure 5] The text specifies ν*=0.5 for the simulation, while the caption of Figure 5 states ν*=1; please reconcile the two values.
[Section V, Figure 4] The text states the mesh study used pv=0.99psat, while the caption of Figure 4 gives pv=0.95psat; the correct value should be indicated.
[Section IV.E] After Eq. (33), the variable 'ui' in the bullet point should read 'uint'.
[Section V, Figure 3] The caption says 'runned' and should say 'run'.
[Section VI.C] The statement that Eq. (50) 'is essentially the Maxwell construction' is terse; a brief explanation of why the integral equality corresponds to the Maxwell construction would help readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytical derivation is self-contained and the numerical validation is an independent consistency check.

full rationale

The paper's chain starts from the 1D mass and momentum conservation equations (1), the Korteweg pressure (4), and an explicit traveling-wave ansatz (Section IV, Assumptions 1-3). The interface velocity is not fitted to the simulations that it later predicts: in Eq. (28) it is obtained by integrating the inviscid ODE and imposing zero bulk gradients, and in Eqs. (32)-(33) the viscous correction is obtained by a stated perturbation step, inserting the inviscid profile z^(0) into the viscous integral. The target quantity u_int is the unknown solved for, not an input. The only self-references are to the Wagner free-energy LBM (Ref. 47), the Landau EOS (Ref. 65), and OpenLB (Ref. 43); these are code-reproduced, externally established numerical methods used for validation, and none of them is invoked to justify the analytical result. The comparison to Jamet's sharp-interface solution (Eq. 49) is an explicit cross-check and the paper acknowledges the Maxwell-construction limit under which the inviscid result coincides with it. The reviewer-flagged sign discrepancy between the viscous term in Eq. (16)/(18a) and Eq. (21)/(32c) is an internal algebraic consistency or correctness concern, but it is not a circularity: it does not make the claimed evaporation-rate expression equivalent to a fitted input by construction. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The solution rests on the stated diffuse interface equations and the traveling-wave ansatz. No constants are fitted to simulation data; the problem parameters (pressure ratio, density ratio, viscosity) are inputs. The new element is an approximate viscous correction built from the inviscid profile.

assumptions (7)

domain assumption The one-dimensional compressible Navier-Stokes equations with Korteweg pressure (Eqs. 1a, 1b, 3, 4) govern the two-phase flow.
The entire derivation is built on these continuum equations; if the diffuse interface model does not describe the intended physics, the solution is only a benchmark for that model.
domain assumption Isothermal conditions and fixed sub-saturation pressure at the vapor boundaries.
This defines the problem and justifies using a fixed temperature and a single EOS (Section II).
ad hoc to paper Traveling-wave ansatz: after a transient, the interface moves with constant velocity uint and all fields are stationary in the co-moving frame (Assumptions 1-3, Section IV).
This is motivated by simulations (Figure 2) but not derived from the equations. If it fails, Eqs. (28) and (33) do not follow.
domain assumption The density profile is monotonically increasing, dρ/dx' > 0, allowing transformation from x' to ρ (Section IV C).
Used to convert the second-order ODE into a first-order equation in z(ρ); for a single interface this is reasonable but not proven.
ad hoc to paper The viscous profile is approximated by the inviscid profile z ≈ z(0) inside the viscous integral (Eqs. 30-31).
This is the core approximation for the viscous solution; the paper shows it degrades at low density ratio and high viscosity.
domain assumption The stress tensor is τxx = 2νρ∂u/∂x rather than the full compressible form (Eq. 3).
Chosen to match the isothermal BGK lattice Boltzmann model. The paper acknowledges this is an assumption and that bulk viscosity is a complex issue.
domain assumption The free-energy LBM (Wagner 2006) is a consistent discretization of the continuum equations, so its steady interface velocity converges to the exact solution.
Used when treating LBM results as the reference for validation (Section VI B). This is a consistency check, not an independent physical test.

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Cite this review

Pith. "Pith review of Analytical solution for dynamic evaporation of liquid in isothermal condition." pith.science (2026). https://pith.science/paper/6BMVLVIM

@misc{pith2026250602270,
  author       = {Pith},
  title        = {Pith review of: Analytical solution for dynamic evaporation of liquid in isothermal condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BMVLVIM}},
  note         = {Machine review of arXiv:2506.02270}
}

read the original abstract

An analytical solution based on a diffuse interface model is presented for an isothermal evaporation problem under sub-saturation pressure. The macroscopic equations are derived from the free-energy method, widely recognized in the lattice Boltzmann literature, distinguishing our approach from conventional evaporation models that rely on jump conditions or pure kinetic theory. The interface behavior is fully described by differential equations, eliminating the need for assumptions such as local equilibrium at the interface. We derive an exact analytical solution for the inviscid case and propose an approximate solution when viscosity effects are considered. Our model unveils a novel relationship between evaporation rate and viscosity, providing new insights that have not been thoroughly explored in the literature. The analytical results are validated through numerical simulations using the open-source parallel library OpenLB, demonstrating excellent agreement in predicting the physical behavior of the evaporation phenomena within the framework of diffuse interface methods.

Figures

Figures reproduced from arXiv: 2506.02270 by the authors.

**Figure 2.** FIG. 2. Interface position [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Interface velocity [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Interface velocity [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Pressure [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Dependency of interface velocity [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Dependency of interface velocity [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

After a transient time, the solution reaches a state where the interface propagates with constant velocity uint

work page
[2]

The center of the drop will have a zero velocity by sym- metry and will retain a constant density

work page
[3]

∂ φ /∂ t′ = 0

The previous assumptions suggest that we can pick a frame of reference ( x′,t′) for half of the system that moves with the interface, where any ﬁeld φ is then in- dependent of time, i.e. ∂ φ /∂ t′ = 0. The transformation between the old and new frame of ref- erence is: x′ = x − uintt, t′ = t, x = x′ + uintt′, t = t′. (6) From the deﬁnitions in ( 5) and th...

work page
[5]

However, the friction does not con - tribute to change the pressure between the two bulk phases

The value of this quantity grows inside the interface due to the velocity grad i- ent across the interface. However, the friction does not con - tribute to change the pressure between the two bulk phases. The numerical results support our physical model. Any pos- sible physical inconsistency of this solution will be relat ed to the diffuse interface model...

work page
[6]

For very low viscosities, the solution converges to the inviscid limit given by ( 28)

The overall behavior of the solution is qualitatively similar a cross all cases. For very low viscosities, the solution converges to the inviscid limit given by ( 28). Conversely, when ν ∗ is suf- ﬁciently large, the interface velocity becomes small and th e quadratic term u2 int in ( 32a) can be neglected. In this high- viscosity regime, the interface ve...

work page
[10]

In Case 2, the system is initialized with zero velocity

In Case 1, the system is ini- tialized with the velocity proﬁle obtained from the analyti cal solution (33) and (10). In Case 2, the system is initialized with zero velocity. It is observed that, although the transient p eriod differs between the two cases, the ﬁnal interface velocity r e- mains entirely independent of the initial condition. There fore, w...

work page
[12]

Case 2: L∗ = 83, initial velocity equal to zero

Case 1: L∗ = 83, initial velocity equal to analytical solution ( 33). Case 2: L∗ = 83, initial velocity equal to zero. Case 3: L∗ = 166, initial velocity equal to zero. initialized equal to f eq i , which is only an approximation rather than the exact condition. As previously mentioned, the analytical solution depends solely on the boundary conditions and...

work page
[13]

Despite differences in the transient regime, the ﬁnal interface velocity is identi cal in both cases

Case 3 shares the same simulation conditions as Case 2 but with a domain size L∗ that is twice as large. Despite differences in the transient regime, the ﬁnal interface velocity is identi cal in both cases. FIG. 4. Interface velocity u∗ int dependency on interface resolution ξ /∆ x for LBM simulations in OpenLB43. Results at t∗ = 200 with L∗ = 83, rsatρ =...

work page

Show all 17 references

[17]

A theoretical study of interphase mass transfe r,

with the sharp- interface model given by ( 49). The results for different values of rsat ρ and fp are shown in Figure 7. Overall, the two solu- tions exhibit excellent agreement. Noticeable deviations are observed only when fp is signiﬁcantly reduced (e.g., to 0.7), or near th...

2022
[25]

(30) The problem with ( 30) is that we cannot use it to explicitly solve for z

for the viscid case is: z2(ρ ) ρ − z2 v ρ v = ∫ ρ ρ v [ 2 κ f (ρ ) ρ 2 − 4 ρ luint κ ν ρ 3 z(ρ ) ] dρ . (30) The problem with ( 30) is that we cannot use it to explicitly solve for z. To overcome this challenge and obtain an approx- imate solution, we consider that the viscosi...
[27]

The positive velocity is compatible with the orientat ion of the interface ( 26) (see Figure 1)

and isolate uint: u(0) int = √ ∫ ρ l ρ v pEOS−pv ρ 2 dρ √ ρ l ρ v ( 1 2 ρ l ρ v − 1 ) + 1 2 , (28) where u(0) int represents the interface velocity for the inviscid case. The positive velocity is compatible with the orientat ion of the interface ( 26) (see Figure 1). The invis...
[28]

The validatio ns are based on comparisons with numerical simulations using the free-energy LBM proposed by Wagner 47 and implemented in OpenLB43

and the viscid case approximate solution ( 33), we validate the physical equations that lead to our solution. The validatio ns are based on comparisons with numerical simulations using the free-energy LBM proposed by Wagner 47 and implemented in OpenLB43. Based on the results,...
[43]

All cases runned with rsatρ = 32, pv = 0.95psat and ν ∗ =
[45]

The system has a ﬁxed temperature T

We consider a system with two phases (one liquid and one vapor) separated by a ﬂat interface, whic h reduces to a one-dimensional case. The system has a ﬁxed temperature T . In equilibrium, this system would have a ho- mogeneous and constant pressure called saturation pressur ...
[46]

The article is organized as follows: In Section II, the phys- ical problem of isothermal evaporation under sub-saturati on pressure is introduced

In future works, our approach of constructing an analytical solution can be extended to non– isothermal and multi–component problems. The article is organized as follows: In Section II, the phys- ical problem of isothermal evaporation under sub-saturati on pressure is introduc...
[47]

position vs. time

Figure 1 shows a representation of the simulated system. The top plot shows the density proﬁle ρ ∗ at two in- stants of time ( t∗ = 200 and t∗ = 600). On the left and right boundaries (vapor phase) a pressure lower than the saturati on pressure pv = 0.99psat is imposed. The su...
[48]

The equilibrium velocity is a quantity used to deﬁne f eq i and its relation with the real ﬂuid velocity is introduced later

The equi- librium distribution function is a function of the ﬂuid dens ity ρ and equilibrium velocity ueq α 64: f eq i = wiρ ( 1 + ciα c2 s ueq α + ciα ciβ − c2 s δ αβ 2c4 s ρ ueq α ueq β ) , (35) 6 where wi are the lattice weights for each lattice direction i. The equilibrium...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

After a transient time, the solution reaches a state where the interface propagates with constant velocity uint

work page

[2] [2]

The center of the drop will have a zero velocity by sym- metry and will retain a constant density

work page

[3] [3]

∂ φ /∂ t′ = 0

The previous assumptions suggest that we can pick a frame of reference ( x′,t′) for half of the system that moves with the interface, where any ﬁeld φ is then in- dependent of time, i.e. ∂ φ /∂ t′ = 0. The transformation between the old and new frame of ref- erence is: x′ = x − uintt, t′ = t, x = x′ + uintt′, t = t′. (6) From the deﬁnitions in ( 5) and th...

work page

[4] [5]

However, the friction does not con - tribute to change the pressure between the two bulk phases

The value of this quantity grows inside the interface due to the velocity grad i- ent across the interface. However, the friction does not con - tribute to change the pressure between the two bulk phases. The numerical results support our physical model. Any pos- sible physical inconsistency of this solution will be relat ed to the diffuse interface model...

work page

[5] [6]

For very low viscosities, the solution converges to the inviscid limit given by ( 28)

The overall behavior of the solution is qualitatively similar a cross all cases. For very low viscosities, the solution converges to the inviscid limit given by ( 28). Conversely, when ν ∗ is suf- ﬁciently large, the interface velocity becomes small and th e quadratic term u2 int in ( 32a) can be neglected. In this high- viscosity regime, the interface ve...

work page

[6] [10]

In Case 2, the system is initialized with zero velocity

In Case 1, the system is ini- tialized with the velocity proﬁle obtained from the analyti cal solution (33) and (10). In Case 2, the system is initialized with zero velocity. It is observed that, although the transient p eriod differs between the two cases, the ﬁnal interface velocity r e- mains entirely independent of the initial condition. There fore, w...

work page

[7] [12]

Case 2: L∗ = 83, initial velocity equal to zero

Case 1: L∗ = 83, initial velocity equal to analytical solution ( 33). Case 2: L∗ = 83, initial velocity equal to zero. Case 3: L∗ = 166, initial velocity equal to zero. initialized equal to f eq i , which is only an approximation rather than the exact condition. As previously mentioned, the analytical solution depends solely on the boundary conditions and...

work page

[8] [13]

Despite differences in the transient regime, the ﬁnal interface velocity is identi cal in both cases

Case 3 shares the same simulation conditions as Case 2 but with a domain size L∗ that is twice as large. Despite differences in the transient regime, the ﬁnal interface velocity is identi cal in both cases. FIG. 4. Interface velocity u∗ int dependency on interface resolution ξ /∆ x for LBM simulations in OpenLB43. Results at t∗ = 200 with L∗ = 83, rsatρ =...

work page