Math 2.0

Fluid mechanics Definition

Grashof number

Builds on Dimensionless numbers in fluid mechanicsThermal expansion coefficientViscosityBoussinesq approximation for buoyancy

Definition

Let a surface at temperature TsT_s be in a fluid at temperature T∞T_\infty under gravity of acceleration g>0g > 0. Let the fluid have thermal expansion coefficient β\beta and kinematic viscosity ν>0\nu > 0. Let L>0L > 0 be a characteristic length, such as the height of a vertical plate. The Grashof number is the dimensionless number Gr=g β (Ts−T∞) L3ν2.\mathrm{Gr} = \frac{g \, \beta \, (T_s - T_\infty) \, L^3}{\nu^2}.

Remarks

It arises when the momentum equation with the Boussinesq approximation is made dimensionless. Some authors use ∣Ts−T∞∣\abs{T_s - T_\infty} so that Gr≥0\mathrm{Gr} \ge 0.

Example: air with β≈1/300\beta \approx 1/300 per kelvin and ν≈1.5⋅10−5\nu \approx 1.5 \cdot 10^{-5} square metres per second, at a plate L=0.5L = 0.5 metres high and 3030 kelvin warmer. Then Gr≈9.81⋅0.1⋅0.125/(2.25⋅10−10)≈5.4⋅108\mathrm{Gr} \approx 9.81 \cdot 0.1 \cdot 0.125 / (2.25 \cdot 10^{-10}) \approx 5.4 \cdot 10^8.

Non-example: gβΔTL3/(να)g \beta \Delta T L^3 / (\nu \alpha), with thermal diffusivity α\alpha, is not the Grashof number. It is the Rayleigh number, equal to Gr⋅Pr\mathrm{Gr} \cdot \mathrm{Pr}.

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Boussinesq approximation for buoyancy, concept, no evidence levelBoussinesq approximat…conceptDimensionless numbers in fluid mechanics, concept, no evidence levelDimensionless numbers…conceptThermal expansion coefficient, concept, no evidence levelThermal expansion coe…conceptViscosity, concept, no evidence levelViscosityconceptGrashof number, definition, no evidence levelGrashof numberdefinition
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  1. Boussinesq approximation for buoyancy concept
  2. Dimensionless numbers in fluid mechanics concept
  3. Thermal expansion coefficient concept
  4. Viscosity concept
  5. Grashof number definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Grashof_number
About this page
Tag7YGN4K82VT
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WritingW0 stub
MSC76R10, 80A19
DefinesGrashof number
UsesDimensionless numbers in fluid mechanics, Thermal expansion coefficient, Viscosity, Boussinesq approximation for buoyancy
Changed3 November 2026

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