⎛ *C * oVs ⎞ ( *C *'- *G * p ' )*V * s '

α 1C + α 2 = - ⎜

⎜*L H*⎟

⎟

⎝ de ⎠

The full equation for sand deposition becomes

⎛ *C * 0 ⎞ ∂ *c *' ⎛ *C * 0V ⎞ ∂ *c *' ⎛ *C * 0V ⎞ ∂ *c *'

+⎜

+⎜

⎜

⎟

⎟*u *'

⎟*v*'

⎝ ∆ *t *⎠ ∂*t *' ⎝ *L *⎠ ∂*x *' ⎝ *L *⎠ ∂*y *'

⎛*DC *⎞ ∂

∂*c*' ⎛ *D*rC0 ⎞ ∂

∂*c*'

-⎜ r2 0 ⎟

-⎜ 2 ⎟

⎝ *L * ⎠ ∂*x*'

∂*x*' ⎝ *L * ⎠ ∂*y*'

∂*y*'

⎛ *C*oVs ⎞ (*C *'-*G * p ' )*V*s '

-⎜

⎜*L H*⎟

=0

⎟

⎝ de ⎠

where the first term is storage, the second and third terms are x- and y- advection, the

fourth and fifth terms are x- and y-diffusion, and the last term is the bed sink (sand

deposition)

These terms in the equation above represent the various aspects of the transport

process. Note that the terms with the primed variables have been scaled such that

these terms are non-dimensional, taking on the scale of unity. The bolded terms now

contain the dimensional characteristics of the process. The normal method for the

interpretation of these scaling factors is to develop the ratios of these terms.

There are several dimensional groupings that are independent of the bed source-sink

variables. The ratios of these terms will be addressed first.

For example, the ratio of the advective terms to the diffusion terms yields the Peclet

Number

⎛ *VL*

⎞

⎛ *C* 0Dr ⎞

⎛*C V *⎞

⎟= ⎜

⎟

⎜

⎜* D*

⎟

⎝* L* ⎠

⎝* L*

⎠

⎝

⎠

This parameter has been used extensively to deal with numerical oscillations caused

by over-advecting the solution.

Another basic parameter is the ratio of the advection and storage term

⎛ *C * o ⎞ ⎛ *V*∆*t *⎞

⎛ *C * oV ⎞

⎟=⎜

⎟

⎟

⎜

⎝ ∆*t *⎠ ⎝ *L *⎠

⎝* L* ⎠

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