Biomedical Engineering Reference
In-Depth Information
For a Newtonian luid, these coeicients can be calculated from irst principles. Note
that the stresses are, in efect, gradients of velocity (with the viscosity μ as a proportionality
coeicient):
+
∂
∂
u
x
∂
∂
u
x
∂
∂
u
y
∂
∂
u
z
+
∂
∂
u
x
y
x
x
x
z
2
+
∂
∂
u
x
∂
∂
u
y
∂
∂
u
z
∂
u
y
+
∂
∂
u
y
y
y
y
x
z
τ =
2
(
3.17
)
∂
∂
∂
u
z
u
u
x
u
y
u
z
∂
∂
+
∂
∂
+
∂
∂
∂
∂
y
x
z
z
z
2
z
Fortunately, for a rectilinear low,
u
y
=
u
z
= 0 and
u
x
does not vary downstream (with
y
or
z
),
so τ
xx
=
τ
yy
=
τ
zz
=
τ
yz
=
τ
zy
= 0 and the matrix is much simpler:
∂
∂
u
y
∂
∂
u
z
x
x
0
∂
∂
∂
∂
u
y
u
z
τ =
x
0
0
(
3.18
)
x
0
0
All the information is contained in this matrix, but we are not done yet because, in biology
and bioengineering, we are most concerned about the forces applied by the low onto the cell.
he stresses are what causes the forces. To ind the (vector) force applied onto cells that are cul-
tured at the loor of the microchannel (here, the plane
YZ
), we need to multiply our matrix by
the vector that represents the surface of our microchannel (“the normal,” or
Y
direction). he
result of this operation is a vector:
∂
∂
u
y
∂
∂
u
z
0
x
x
F n
∂
∂
u
y
∂
∂
u
y
x
x
= ⋅ =
τ
(
010
)
0
0
=
,
0 0
(
3.19
)
u
z
∂
∂
x
0
0
Matrix algebra shows clearly how, although the force is only in the direction of low (
x
), it
is caused by changes in low velocity in the
y
direction. In the case of the rectangular channel
(
Equation 3.7
), we have:
n z
h
n w
h
π
n y
h
π
2
3
cosh
cos
∞
∑
2
n
−
1
∂
∂
u
y
16
h
dp
dx
2
∂
∂
x
F
=
=
−
(
−
1
)
2
1
−
(
3.20
)
x
3
y
ηπ
π
n
cosh
n
=
1 3 5
,
, ...
2
y
=−
h
