Biomedical Engineering Reference
In-Depth Information
Fig. 5.6
Deformed fluid element due to the action of the surface forces, in the form of normal and
tangential stresses
(5.9)
F
=
g
body
Surface forces are forces that act on the surface of the fluid element causing it to
deform (Fig.
5.6
).
This includes the normal stress
σ
xx
,
which are a combination of pressure
p
ex-
erted by the surrounding fluid and normal viscous stress components
τ
xx
that both
act perpendicular to fluid element, and tangential stresses
τ
yx
and
t
zx
that act on the
surfaces of the fluid element. The sum of these surface forces on a control volume
in the
x
-direction can be written in terms of the pressure gradient and the viscous
forces (known as the diffusion term)
∂
=− +∇
∂
P
F
·
σ
(5.10)
surface
ij
x
diffusion
pressure
where
i
σ
is the stress tensor that accounts for the viscous forces acting on the fluid
element. Combining Eqs. 5.8-5.10 gives
D
U
D
ρ
= ∂ +∇ +
P
·
σρ
g
ij
i
(5.11)
pressure
gravity
diffusion
inertia
Equation (5.11) is the fluid momentum equation in the form of Cauchy's equation
momentum and can be used as a base for determining deformation in structures as
well (see Sect. 5.3). This can be expanded out to see its individual terms for the
x
-
y-
and
z-
components of momentum. The term
D
U
/D
t
is a material derivative and is
defined as the local and advection inertial force through its acceleration as
D
Dt
U
=
∂
∂
u
t
+∇⋅
u
ij
(5.12)
advection
local
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