### 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 ≤ R_{a 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 R_{a 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 R_{a 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 language | English |
---|---|

Pages (from-to) | 538-549 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 366 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jun 2010 |

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### Keywords

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

### Cite this

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*Journal of Mathematical Analysis and Applications*, vol. 366, no. 2, pp. 538-549. https://doi.org/10.1016/j.jmaa.2010.01.008

**Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation.** / Lorenzo Trueba, Jorge; Voller, V. R.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

AU - Lorenzo Trueba, Jorge

AU - Voller, V. R.

PY - 2010/6/15

Y1 - 2010/6/15

N2 - 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.

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.

KW - Dual moving boundaries

KW - Enthalpy solution

KW - Sediment delta

KW - Similarity solution

KW - Stefan problem

UR - http://www.scopus.com/inward/record.url?scp=76749139847&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2010.01.008

DO - 10.1016/j.jmaa.2010.01.008

M3 - Article

VL - 366

SP - 538

EP - 549

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -