Spatial Pattern Formation of Motile Microorganisms: From Gravitactic Bioconvection to Protozoan Culture Dynamics - Porous Media: Applications in Biological Systems and Biotechnology - page 544

V c H

D c be the dimensionless cell velocity and rewrite equation
(14.8) as n = nP ee P H Y
e Pe

Let Pe =

− 1 from which we readily deduce the concentrations at
the bottom ( Y = 0) and the top ( Y = H ) of the cavity

Pee Pe
e Pe

Pe
e Pe

n 0 =

1 n ; n 1 = 1 n

− −

∆ n =( n 1 − n 0 )= nP e (14.10)

14.2.2.1 Nondimensional Equations

Nondimensional governing equations may be expressed in the following form,
by choosing H and H 2 /D c as length and time scales, respectively.

V = 0

∇· (14.11)

D c K

V =

νH 2

− ∇ P − RaNe 2 (14.12)

∂N
∂t

+ ∇· ( N V )+ ∇· ( N − Pe )= ∇

2 N

(14.13)

In terms of the stream function ψ , they become

2 ψ = Ra ∂N
∂x

∇ (14.14)

∂N
∂t

( N − Pe )= ( N V )+

2 N

+ ∇· ∇· ∇ (14.15)

where

V = ∂ψ

∂ψ
∂x ∂y ,

−

N = n

n 0
∆ n

−

and the Rayleigh number

Ra = gKHβnP e
νD c

(14.16)

with initial and boundary conditions

= e Pe N = N = n

n 0
∆ n

− − Pe − 1

at t = 0 (14.17)

( e Pe − 1) Pe

ψ =0 ,∂N/∂x =0 at x =0 ,F and
ψ =0 ,∂N/∂y = PeN + Pe/ ( e Pe − 1) at y = 0 (14.18)

The diffusion state (14.8) can be expressed in the nondimensional form

N d = e Pey

− 1

(14.19)

e Pe − 1