Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

J. Lorenzo-Trueba, V. R. Voller

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 ≤ Ra b ≤ 1-the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0 ≤ θ → ∞, the similarity solution shows that when Ra b < 0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0 ≤ θ ∼ 2, reasonable agreement in predictions extents up to Ra b ∼ 0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.

Original languageEnglish
Pages (from-to)538-549
Number of pages12
JournalJournal of Mathematical Analysis and Applications
Volume366
Issue number2
DOIs
StatePublished - 15 Jun 2010

Fingerprint

Stefan Problem
Moving Boundary
Diffusivity
Ocean
Similarity Solution
Analytical Solution
Numerical Solution
Slope
Exponent
Sediment Transport
Linear Diffusion
Prediction
Dimensional Analysis
Dimensionless
Range of data
Sediment transport
Governing equation
Power Law
Model
Enthalpy

Keywords

  • Dual moving boundaries
  • Enthalpy solution
  • Sediment delta
  • Similarity solution
  • Stefan problem

Cite this

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title = "Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation",
abstract = "A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 ≤ Ra b ≤ 1-the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0 ≤ θ → ∞, the similarity solution shows that when Ra b < 0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0 ≤ θ ∼ 2, reasonable agreement in predictions extents up to Ra b ∼ 0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.",
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AB - A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 ≤ Ra b ≤ 1-the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0 ≤ θ → ∞, the similarity solution shows that when Ra b < 0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0 ≤ θ ∼ 2, reasonable agreement in predictions extents up to Ra b ∼ 0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.

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