Fickian Diffusivity#

Fickian Diffusivity = {model_name} <species> {float_list}

Description / Usage#

This card allows the user to select a Fickian diffusion mode when the model in the Diffusivity card is HYDRO. There are two {model_name} options for this mode; definitions of the input parameters are as follows:

ANISOTROPIC

an anisotropic Fickian diffusion.

  • <species> - an integer designating the species equation.

  • <float1> - the value of the diffusivity for the X direction, Dx.

  • <float2> - the value of the diffusivity for the Y direction, Dy.

  • <float3> - the value of the diffusivity for the Z direction, Dz.

EXP_DECAY

an exponential decay of flux.

  • <species> - an integer designating the species equation.

  • <float1> - the coefficient to the exponential decay, Do

  • <float2> - the exponent value for exponential decay, D1

Examples#

Following are two sample cards:

Fickian Diffusivity = ANISOTROPIC 0 2.e-6 2.e-6 0.

Fickian Diffusivity = EXP_DECAY 0 0.01 1.e-3

Technical Discussion#

In modeling suspension flow, often a sharp concentration gradient is encountered, and the numerical convergence becomes very poor. This card should be used for numerical stability (smooth out the wiggles) and should only be introduced as a last resort. The magnitudes should remain small relative to shear rate and viscosity diffusivities.

As the name implied, anisotropic Fickian diffusivity defines an additional flux contribution much like a classic Fickian diffusion term; i.e.,

\[J_x=D_x\frac{\partial C}{\partial x},\quad J_y=D_y\frac{\partial C}{\partial y},\quad J_z=D_z\frac{\partial C}{\partial z}\]

If the exponential decay option is used, the flux vector has the form,

\[J_i=D_o(\exp(-D_1C)+\exp(-D_1(C_{max}-C)))\]

where C and Cmax are volume fractions of suspension locally and at maximum packing.

References#

No References.