Fluid mechanics and living organisms
In this chapter we discuss the basic laws of fluid
flow as they apply to life processes at various size
scales. For example, fluid dynamics at low
Reynolds’ number dominates the universe of uni
cellular organisms: bacteria and protozoa. Fluid
flow—both uniform and pulsatile—is the domi
nant process by which both air and blood circulate
in land animals, including humans.
1.
Euler’s equation
We consider the form of Newton’s second law
appropriate to a fluid. The mass of a small volume
of fluid is
∆m = ρ ∆V ;
its acceleration is therefore determined by the
forces acting upon it,
∆m
dv
→
dt
≡ ρ ∆V
∂v
→
∂t
+ (v
→
⋅∇)v
→
= ∆F
→
.
The forces are generally of two kinds:
1. external, longrange forces—gravity, elec
tromagnetism;
2. internal forces—especially pressure and
viscosity.
Thus, e.g., if the fluid is in a gravitational field of
local acceleration g
→
,
∆F
→
= ρ∆V g
→
.
On the other hand, consider a pressure gradient
in—say—the xdirection, as shown to the right.
Clearly the net xcomponent of force is
∆F
x
=
p(x) − p(x+dx)
A ≈ −
∂p
∂x
∆V ;
In the absence of viscous forces the equation of
motion is therefore
ρ
∂v
→
∂t
+ (v
→
⋅∇)v
→
= − ∇p + ρ g
→
.
This is sometimes known as Euler’s equation
(L. Euler, 1755).
Example
We first apply Euler’s equation to two problems in
hydrostatics. In hydrostatics the fluid is not mov
ing, hence v
→
= 0 and we obtain
−∇p + ρ g
→
= 0 .
Defining g
→
= −g z
∧
we easily see that
p(x,y,z;t) ≡ p(z)
so that
dp
dz
= −ρ g .
We now have two cases to consider:
1. The fluid is incompressible (ρ
.
= 0).
2. The fluid is a perfect gas:
ρ =
µ
RT
p ,
p(x) p(x+dx)
F
x
= A
p(x) − p(x+dx)
Physics of the Human Body 37
Chapter 4:Fluid mechanics and living organisms
Example*
Another interesting case is a pipe of rectangular
cross section. Here the equation of steady flow is
∂
2
v
z
∂x
2
+
∂
2
v
z
∂y
2
= −
λ
η
whose solution may be written (take the origin in
the xy plane at a corner of the rectangle)
v
z
=
∑
m, n
a
mn
sin
mπx
L
x
sin
nπy
L
y
.
For this solution the fluid flow rate is
dm
dt
= ρ
∫
0
L
x
dx
∫
0
L
y
dy v
z
(x, y)
= ρ
4L
x
L
y
π
2
∑
m, n odd
a
mn
mn
.
We use the NavierStokes equation, together with
the orthogonality of the functions sin
(2n+1)θ
to
determine a
mn
:
a
mn
=
16
π
4
mn
m
2
L
x
2
+
n
2
L
y
2
−1
.
Comparing square and circular pipes of equal
crosssectional areas we find the central flow
speeds in the ratio
v
L
⁄
2
,
L
⁄
2
v(r = 0)
≈ 0.93
and the rates of mass transport in the ratio
(dm
⁄
dt)
(dm
⁄
dt)
=
512
π
5
∑
m, n odd
1
(mn)
2
(m
2
+ n
2
)
≈ 0.88 .
It should not surprise us to find that for pipes of
equal area, the circular cross section poses the least
resistance.
*for advanced students
44 Chapter 4:Fluid mechanics and living organisms
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