Biomedical Engineering Reference
In-Depth Information
14.2.2.2
Linearized Equations
Let the diffusion state expressed by subscript “
d
” be perturbed by quantities
denoted by “
1
”, we will have
V
(
x, y, t
)=
V
1
N
(
x, y, t
)=
N
d
+
N
1
;
ψ
(
x, y, t
)=
ψ
1
;
(14.20)
V
being the disturbed state and
N
1
N
d
=
e
Pey
−
1
e
Pe
with
N
,
ψ
,
−
1
.
By substituting equation (14.20) in equations (14.14-14.15), neglecting
second-order terms of perturbations, and then substituting the diffusion state
(14.19) into the obtained system, we have
∂
2
ψ
1
∂x
2
+
∂
2
ψ
1
∂y
2
=
Ra
∂N
1
∂x
(14.21)
∂N
1
∂t
−
G
(
y
)
∂ψ
1
∂x
+
Pe
∂N
1
∂y
2
N
1
=
∇
(14.22)
with
G
(
y
)=
Pe
e
V
c
y
e
Pe
(14.23)
−
1
The required boundary conditions are
ψ
1
=
∂N
1
∂x
=0 at
x
=0
,F
and
ψ
1
=
∂N
1
∂y
−
PeN
1
=0 at
y
=0
,
1
(14.24)
The linear system (14.21-14.22) with boundary conditions (14.24) deter-
mines the initial evolution of perturbations and the criterion for the onset
of bioconvection. Numerical solutions are presented in the next section. Note
that by making the change of variable
N
1
=
T
1
and substituting it into
equations (14.21-14.22), we readily obtain the following system:
−
∂
2
ψ
1
∂x
2
+
∂
2
ψ
1
∂y
2
Ra
∂T
1
∂x
=
−
(14.25)
∂T
1
∂t
+
G
(
y
)
∂ψ
1
∂x
+
Pe
∂T
1
∂y
2
T
1
=
∇
(14.26)
with the boundary conditions
ψ
1
=
∂T
1
∂x
=0 at
x
=0
,F
and
ψ
1
=
∂T
1
∂x
−
PeT
1
=0 at
y
=0
,
1
(14.27)
1, the above system of equations
reduces formally to the equations governing the fixed-flux Benard problem
(Kimura et al. 1995; Prud'homme and Nguyen 2002).
When the cell velocity
Pe
→
0,
G
(
y
)
→
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