Biomedical Engineering Reference
In-Depth Information
where the viscosity
is constant. Substitution of this constitutive equation into
the local momentum balance leads to the equation:
η
∇
v
T
+ ∇
v
∇
v
T
.
·
v
+
δ
v
− ∇
p
+
η
∇·
+
ρ
q
=
ρ
(13.29)
δ
t
The left-hand side of this equation can be simplified by using the following
identities (to be derived by elaboration in components):
∇
v
T
∇·
v
∇·
∇
=
(13.30)
∇
v
∇·∇
∇·
=
v
.
(13.31)
This leads to:
∇
v
T
.
∇
∇·
v
∇·∇
v
·
v
+
δ
v
δ
− ∇
p
+
η
+
+
ρ
q
=
ρ
(13.32)
t
Using the expression for incompressibility of the fluid:
∇·
v
=
0,
(13.33)
results in the so-called Navier-Stokes equation:
∇
v
T
.
∇·∇
·
v
+
δ
v
δ
− ∇
p
+
η
v
+
ρ
q
=
ρ
(13.34)
t
The last two relations, the incompressibility condition (continuity equation)
and the Navier-Stokes equation, together form a set that allows the deter-
mination of the velocity field
v
(
x
,
t
) and the pressure field
p
(
x
,
t
). For the
solution boundary conditions and initial conditions have to be supplied to the
equations.
13.3.3
Stationary flow of an incompressible Newtonian fluid
For a stationary flow the relevant field variables are only a function of the position
vector
x
within the volume
V
and no longer a function of time. To determine the
velocity field
v
(
x
) and the pressure field
p
(
x
) the set of equations that has to be
solved is reduced to:
∇·
v
=
0
(13.35)
∇·∇
∇
v
T
− ∇
p
+
η
v
+
ρ
q
=
ρ
·
v
.
(13.36)
In addition, it is necessary to specify a full set of boundary conditions. Initial con-
ditions do not apply for stationary problems. Note that the equation is non-linear
