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# Grashof number

The Grashof number is a dimensionless number in fluid dynamics which approximates the ratio of the buoyancy to viscous force acting on a fluid. It is named after the German engineer Franz Grashof.

$Gr = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}\,$

where

g = acceleration due to Earth's gravity
β = volumetric thermal expansion coefficient
Ts = source temperature
T = quiescent temperature
L = characteristic length
ν = kinematic viscosity

The product of the Grashof number and the Prandtl number gives the Rayleigh number, a dimensionless number that characterizes convection problems in heat transfer.

There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems.

$Gr_c = \frac{g \beta^* (C_{a,s} - C_{a,a} ) L^3}{\nu^2}$

where

$\beta^* = -\frac{1}{\rho} \left ( \frac{\partial \rho}{\partial C_a} \right )_{T,p}$

and

g = acceleration due to Earth's gravity
Ca,s = concentration of species a at surface
Ca,a = concentration of species a in ambient medium
L = characteristic length
ν = kinematic viscosity
ρ = fluid density
Ca = concentration of species a
T = constant temperature
p = constant pressure