Exact solutions for thermal problems: buoyancy, marangoni, vibrational and magnetic-field-controlled flows

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Abstract

In general, the thermal‐convection (Navier‐Stokes and energy) equations are nonlinear partial differential equations that in most cases require the use of complex algorithms in combination with opportune discretization techniques for obtaining reliable numerical solutions. There are some cases, however, in which such equations admit analytical solutions. Such exact solutions have enjoyed a widespread use in the literature as basic states for determining the linear stability limits in some idealized situations. This review article provides a synthetic review of such analytical expressions for a variety of situations of interest in materials science (especially crystal growth and related disciplines), including thermogravitational (buoyancy), thermocapillary (Marangoni), thermovibrational convection as well as “mixed” cases and flow controlled via static and uniform magnetic fields.
LanguageEnglish
Pages1-14
Number of pages14
JournalReview of Applied Physics
Volume1
Issue number1
Publication statusPublished - 2012

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Heat problems
Buoyancy
Magnetic fields
Mixed convection
Materials science
Crystal growth
Nonlinear equations
Partial differential equations
Hot Temperature
Convection

Keywords

  • thermal convection
  • analytic solutions
  • navier-stokes equations

Cite this

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title = "Exact solutions for thermal problems: buoyancy, marangoni, vibrational and magnetic-field-controlled flows",
abstract = "In general, the thermal‐convection (Navier‐Stokes and energy) equations are nonlinear partial differential equations that in most cases require the use of complex algorithms in combination with opportune discretization techniques for obtaining reliable numerical solutions. There are some cases, however, in which such equations admit analytical solutions. Such exact solutions have enjoyed a widespread use in the literature as basic states for determining the linear stability limits in some idealized situations. This review article provides a synthetic review of such analytical expressions for a variety of situations of interest in materials science (especially crystal growth and related disciplines), including thermogravitational (buoyancy), thermocapillary (Marangoni), thermovibrational convection as well as “mixed” cases and flow controlled via static and uniform magnetic fields.",
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