### Abstract

We analyze the effect of various thermal boundary conditions on the linear stability of surface-tension-driven flow in an unbounded liquid layer subject to a longitudinal temperature gradient. An original approach is devised to estimate the critical instability parameters. The order of magnitude estimates are used to solve the problem asymptotically for small Prandtl numbers. The instability is shown to be essentially determined by the thermal boundary conditions. For insulating boundaries the critical wavenumber scales as k_{c} ∼ Pr^{1/2} meaning that the most unstable wave is considerably longer than the depth of the layer. When the bottom is well conducting, the critical wavelength is comparable to the depth of the layer. For the case of insulating bottom and non-adiabatic free surface the critical wavenumber depends on the Biot number as k_{c} ∼ Bi^{1/2}. Even a weak thermal coupling between the free surface and the ambient medium such that Bi ∼ Pr can significantly influence the instability threshold.

Original language | English |
---|---|

Pages (from-to) | 1621-1634 |

Number of pages | 14 |

Journal | Physics of Fluids |

Volume | 9 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 1997 |

Externally published | Yes |

### Fingerprint

### Keywords

- Boundary value problems
- Free surface
- Flow instabilities
- Thermocapillary convection
- Convection

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Computational Mechanics
- Mechanics of Materials
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*9*(6), 1621-1634. https://doi.org/10.1063/1.869282

**Influence of thermal boundary conditions on the stability of thermocapillary-driven convection at low Prandtl numbers.** / Priede, Janis; Gerbeth, Gunter.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 9, no. 6, pp. 1621-1634. https://doi.org/10.1063/1.869282

}

TY - JOUR

T1 - Influence of thermal boundary conditions on the stability of thermocapillary-driven convection at low Prandtl numbers

AU - Priede, Janis

AU - Gerbeth, Gunter

PY - 1997/6

Y1 - 1997/6

N2 - We analyze the effect of various thermal boundary conditions on the linear stability of surface-tension-driven flow in an unbounded liquid layer subject to a longitudinal temperature gradient. An original approach is devised to estimate the critical instability parameters. The order of magnitude estimates are used to solve the problem asymptotically for small Prandtl numbers. The instability is shown to be essentially determined by the thermal boundary conditions. For insulating boundaries the critical wavenumber scales as kc ∼ Pr1/2 meaning that the most unstable wave is considerably longer than the depth of the layer. When the bottom is well conducting, the critical wavelength is comparable to the depth of the layer. For the case of insulating bottom and non-adiabatic free surface the critical wavenumber depends on the Biot number as kc ∼ Bi1/2. Even a weak thermal coupling between the free surface and the ambient medium such that Bi ∼ Pr can significantly influence the instability threshold.

AB - We analyze the effect of various thermal boundary conditions on the linear stability of surface-tension-driven flow in an unbounded liquid layer subject to a longitudinal temperature gradient. An original approach is devised to estimate the critical instability parameters. The order of magnitude estimates are used to solve the problem asymptotically for small Prandtl numbers. The instability is shown to be essentially determined by the thermal boundary conditions. For insulating boundaries the critical wavenumber scales as kc ∼ Pr1/2 meaning that the most unstable wave is considerably longer than the depth of the layer. When the bottom is well conducting, the critical wavelength is comparable to the depth of the layer. For the case of insulating bottom and non-adiabatic free surface the critical wavenumber depends on the Biot number as kc ∼ Bi1/2. Even a weak thermal coupling between the free surface and the ambient medium such that Bi ∼ Pr can significantly influence the instability threshold.

KW - Boundary value problems

KW - Free surface

KW - Flow instabilities

KW - Thermocapillary convection

KW - Convection

UR - http://www.scopus.com/inward/record.url?scp=0031171421&partnerID=8YFLogxK

U2 - 10.1063/1.869282

DO - 10.1063/1.869282

M3 - Article

VL - 9

SP - 1621

EP - 1634

JO - Physics of Fluid

JF - Physics of Fluid

SN - 1070-6631

IS - 6

ER -