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Capillary flow in corners slowed down by gravity and evaporation

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Published/Copyright: March 6, 2025
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Chemical Product and Process Modeling
From the journal Chemical Product and Process Modeling Volume 20 Issue 2

Abstract

Capillary flow in corner geometries in the presence of gravity and evaporation is relevant for numerous natural phenomena and industrial applications. In the absence of gravity, the length of the rivulet in the corner follows the t 1/2 asymptotic law (Lucas-Washburn kinetics), where t is the time. If the liquid flows against gravity, the propagation of the rivulet tip decelerates to follow the t 1/3 asymptotic law. In this paper, we present a model for simulation of the rivulet shape evolution in a corner with an arbitrary cross-section shape. Gravity and evaporation are taken into account. Several exact and asymptotic solutions are presented. In particular, a simple expression for the proportionality coefficient in the t 1/3 asymptotic law is derived, as well as an expression for the cross-over time moment corresponding to change from the t 1/2 to t 1/3 asymptotic behavior. In the presence of evaporation, the rivulet length reaches a maximal value, at which the rate of evaporation is balanced by the rate of the capillary flow. We derive expressions for the maximal rivulet length in the limiting cases of “strong” and “weak” evaporation. In the case of “strong” evaporation, the maximal rivulet length behaves as E −1/2, where E denotes the dimensionless evaporation rate. In the case of “weak” evaporation, uniform evaporation rate and triangular groove geometry, the maximal rivulet length is proportional to E −1/5 Bo −3/5, where Bo denotes the Bond number.

Keywords: imbibition; capillary rise; evaporation; corner geometry
Corresponding author: Tatiana Gambaryan-Roisman, Institute for Technical Thermodynamics, Technische Universität Darmstadt, Peter-Grünberg-Strasse 10, 64287 Darmstadt, Germany, E-mail:

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: Project ID 265191195

Award Identifier / Grant number: Project ID 422792679

Acknowledgments

The author gratefully acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the framework of the Priority Program “Dynamic Wetting of Flexible, Adaptive and Switchable Surfaces” (SPP 2171), Project ID 422792679, as well as in the framework of CRC 1194, Project ID 265191195, subproject A04.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The author states no conflict of interest.

  6. Research funding: The study was financially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the framework of the Priority Program “Dynamic Wetting of Flexible, Adaptive and Switchable Surfaces” (SPP 2171), Project ID 422792679, as well as in the framework of CRC 1194, Project ID 265191195, subproject A04.

  7. Data availability: The data that support the findings of this study are available from the author, T. G.-R., upon reasonable request.

A Appendix: Derivation of long-time asymptotic behavior of rivulet propagation in the presence of gravity and in the absence of evaporation

At the advanced stage of rivulet propagation we approximate the function R(ψ, τ) in the following form:

(A.1) R ( ψ , τ ) = R ∞ ( ψ , τ ) = 1 B o λ ψ + R m − 1 , 0 < ψ < ψ ˜ ,

(A.2) R ( ψ , τ ) = R lin ( ψ , τ ) = λ Θ d λ d τ 1 − ψ , ψ ˜ ≤ ψ < 1 .

The requirement that both the function R(ψ, τ) and its derivative, ∂R/∂ψ, are continuous at ψ = ψ ˜ can be expressed as follows:

(A.3) R ∞ ( ψ ˜ , τ ) = R lin ( ψ ˜ , τ ) ,

(A.4)   ∂ R ∞ ∂ ψ ψ = ψ ˜ = ∂ R lin ∂ ψ ψ = ψ ˜ .

Combining those two equations, we get:

(A.5) 1 R ∞ ∂ R ∞ ∂ ψ ψ = ψ ˜ = 1 R lin ∂ R lin ∂ ψ ψ = ψ ˜ ,

or

(A.6) 1 ψ ˜ + R m B o λ − 1 = 1 1 − ψ ˜ .

The solution of this equation is

(A.7) ψ ˜ = 1 2 − 1 2 R m B o λ .

We consider the advanced stage of the rivulet propagation, where the rivulet is long enough, so that R m Boλ(τ) ≫ 1, and R ∞ ( ψ ˜ , τ ) ≪ R m . Then Eq. (A.7) is simplified to

(A.8) ψ ˜ = 1 2 .

This means that, at long times, the function R(ψ, τ) approximately follows the asymptotic solution R ∞(ψ, τ) (Eq. (29)) from the base of the rivulet up to the half of its length. Over the upper half length of the rivulet, the curvature radius, R, decreases linearly until it reaches zero at ψ = 1.

Substituting the solution ψ ˜ = 1 / 2 into any of the conditions (A.3) and taking into account that R m Boλ(τ) ≫ 1, we obtain the following relation:

(A.9) λ 2 d λ d τ = 4 Θ B o .

Integration of this equation with the initial condition λ(0) = 0 leads to the expression

(A.10) λ ( τ ) = 12 Θ B o 1 / 3 τ 1 / 3 .

B Appendix: Derivation of an expression for a maximal length of rivulet in the presence of gravity and of evaporation

In this appendix, the case of “weak” evaporation is considered, i.e. when the maximal rivulet length is reached after the gravity-governed stage, in which l ∝ t 1/3. We approximate the solution of the equation (37) in the form:

(B.1) R st , B o ( ψ ) = R ∞ ( ψ , τ ) = 1 B o λ max ψ + R m − 1 , 0 < ψ < ψ ˜ ,

(B.2) R s t , B o ( ψ ) = R tip ( ψ ) = 1 − ψ 2 / 3 λ max 2 / 3 3 E 2 Φ * 1 / 3 , ψ ˜ ≤ ψ < 1 .

We require that both the function R st,Bo (ψ) and its derivative are continuous at ψ = ψ ˜ . This means that also the following equation should be satisfied:

(B.3) 1 R ∞ ∂ R ∞ ∂ ψ ψ = ψ ˜ = 1 R tip d R tip d ψ ψ = ψ ˜ ,

which after substituting of Eqs. (B.1) and (B.2) leads to

(B.4) 1 ψ ˜ + R m B o λ max − 1 = 2 3 1 1 − ψ ˜ .

The solution of this equation is:

(B.5) ψ ˜ = 3 5 − 2 5 R m B o λ max .

Under a condition of “weak” evaporation, the maximal length of the rivulet is reached during an advanced gravity-driven phase, for which the inequality R m Boλ max ≫ 1 holds. Therefore, the solution (B.5) can be simplified to

(B.6) ψ ˜ = 3 5 .

Applying the condition of continuity of the function R st,Bo (ψ) or its derivative at ψ = ψ ˜ = 3 / 5 , we derive the relation for λ max:

(B.7) λ max = 5 3 1.5 Φ * 1 / 5 E − 1 / 5 B o − 3 / 5 .

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Received: 2024-07-20
Accepted: 2025-01-02
Published Online: 2025-03-06

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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  2. Editorial
  3. CPPM special issue in honor of Professor Eugeny Kenig
  4. Review
  5. Capture and catalytic conversion of CO2 from marine and offshore applications – A review
  6. Research Articles
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  12. Capillary flow in corners slowed down by gravity and evaporation
https://doi.org/10.1515/cppm-2024-0070
Keywords for this article
imbibition; capillary rise; evaporation; corner geometry

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. CPPM special issue in honor of Professor Eugeny Kenig
  4. Review
  5. Capture and catalytic conversion of CO2 from marine and offshore applications – A review
  6. Research Articles
  7. Deep learning based hybrid POD-LSTM framework for laminar natural convection flow in a rectangular enclosure
  8. Application of CFD simulation to predict the fluid dynamics of tube columns with stacked packings for gas-liquid systems
  9. High order moment conserving method of classes in CFD code
  10. A framework for validating efficiency models of thermal separation columns with tray internals
  11. A systematic selection and decision matrix for energy-efficient intensified distillation technologies
  12. Capillary flow in corners slowed down by gravity and evaporation
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