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Asymmetric collapse by dissolution or melting in a uniform flow
Chris H Rycroft1, Martin Z Bazant2
1Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA; Department of Mathematics, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA.
Proceedings. Mathematical, Physical, and Engineering Sciences
|March 22, 2016
Summary
This study presents a dissolution model where objects erode in a 2D flow, collapsing to a point. A novel method reveals an exact relationship for the collapse point, offering insights into finite-time singularities.
Area of Science:
- Mathematical Physics
- Fluid Dynamics
- Computational Mathematics
Background:
- Modeling object erosion in fluid flow is crucial for understanding various physical and chemical processes.
- Previous models often lack analytical solutions for complex geometries and flow conditions.
- Advection-diffusion-limited dissolution presents unique challenges due to evolving boundaries and flow interactions.
Purpose of the Study:
- To introduce and analyze an advection-diffusion-limited dissolution model for objects in 2D potential flow.
- To develop a numerical method leveraging conformal invariance to track boundary evolution.
- To investigate the finite-time collapse of dissolving objects and the underlying mathematical principles.
Main Methods:
- Developed a dissolution model based on advection-diffusion limitations in a 2D potential flow.
- Utilized conformal invariance to track object boundary evolution using a time-dependent Laurent series.
- Employed residue calculus for analytical derivation and a generalized Newton-Raphson algorithm for numerical root-finding.
Main Results:
- Objects simulated using the model shrink and collapse to a single point in finite time.
- Discovered an exact relationship between the collapse point, flow velocity, and initial shape described by Laurent series coefficients.
- Identified potential model breakdown before complete collapse due to topological singularities, analogous to droplet pinch-off.
Conclusions:
- The model provides a powerful framework for studying finite-time singularities in dissolution processes.
- The derived exact relationship and numerical methods offer practical tools for predicting collapse points.
- The study highlights fundamental mathematical questions regarding broken symmetries in dynamical systems.
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