### Definitions and Explanations of Characteristic Numbers

In this section a short overview of the definitions for characteristic numbers used in CrysMAS is given. In addition a short explanation of the physical meaning is added. For a more detailed description pleas refer to the literature. The units of the physical quantities used are given in the Appendix.

Characteristic number

Definition

Description

Prandtl number Pr

Pr = vd*cp/l

vd = dynamic viscosity,

cp = heat capacity,

l = thermal conductivity

Material related characteristic number: Describes the ratio of sensitivity to convective or diffusive heat transfer in a material. High Pr means that in such a material it is expected that fluid flow will lead to significant changes even if the flow velocities are not to high.

Grashof number Gr

Gr = (g*dT*b*L3*rho2)/(vd2)

L= characteristic length,

g = gravity

b = thermal expansion coefficient

rho = density

Ratio between thermal buoyancy forces and viscous forces.

Rayleigh number Ra

Ra = (g*dT*b*L3)/(vk*diffT)

vk = kinematic viscosity

diffT = thermal diffusivity (l/(cp*rho))

Ratio between thermal buoyancy forces and viscous forces; Please note: Ra = Gr*Pr

Reynolds number Re

Re = v*L*rho/vd

v = characteristic velocity

Ratio between dynamic and viscous forces. In the case of tube flow, if Re > 2000 the flow is considered to be turbulent.

The Reynolds number for forced convection is computed here. Its value will depend on the prescribed fluid velocity at the inlet point. For the buoyancy driven convection the Reynolds number is computed as square root from the Grashof number. .

Schmidt number Sc

Sc = vd/(rho*D)

D = diffusion coefficient

Material related characteristic number: Characterizes the ratio of convective and diffusive species transport for different materials with given flow field.

Lewis number Le

Le = l/(cp*rho*D)

Material related characteristic number: Characterizes the ratio of energy to species transport for different materials with given flow field. Le = Sc/Pr;

Solutal Peclet number PeC

PeC =v*L/D

Ratio between diffusive and convective species transport in a flow field.

Thermal Peclet number PeT

PeT = v*L/ diffT

Ratio between diffusive and convective energy transport in a flow field.

Magnetic Taylor number Tam

Tam = B02* R4*s*f/(2*rho*vk*p)

B0 = Magnetic induction at the wall

s = electrical conductivity of the sample

f = rotation frequency

p = Number of pole pairs

Magnetic Taylor number for rotating magnetic fields. Give the ratio of the Lorentz-force to the viscous forces. High Ta indicates a time-dependent or turbulent flow. The actual critical Ta for the transition to time dependent flows depends on the aspect ratio of the sample.

Hartmann number

Ha = B*L*sqrt(s/(vk*rho))

Describes the effective strength of the magnetic induction not depending on the material used as sample. Used for static magnetic fields.