Page 14 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

1.8 Principles of heat and fluid flow 11

In which pressure (P) and velocity components (u, v, and w) are dependent vari-
ables. Eq. (1.7) is the general form of momentum equation and is used for both com-
pressible and incompressible fluids. For incompressible fluids (ρ = const) third term
from the left-hand side of Eq. (1.7) vanishes and it reduces to [47]:
∂V
)
V
∇⋅−  pI + µ(∇ +∇V) T ) +=g ρ + ρ ( ⋅∇ V (1.8)
V
(


∂t ∇⋅−PI+µ∇V+∇VT+g=ρ∂V∂t+ρV⋅∇V
In general, to solve Navier-Stokes equation initial and boundary conditions must
be available. The initial boundary condition is the condition of the system at time
zero. Typical boundary conditions in fluid dynamic problems are solid boundary con-
ditions, inlet and outlet boundary conditions, and symmetry boundary conditions and
are specifically defined for each problem [47].
1.8.1.1 Euler and Bernoulli equations
In the case of inviscid fluids (µ = 0) the equation of motion (Eq. 1.4) is simplified
to [49]:
∂ V
∇+ ρ + ρ V ( ⋅∇ V ) (1.9)
Pg =
t ∂ ∇P+g=ρ∂V∂t+ρV⋅∇V
Eq. (1.9) is commonly referred to Euler’s equation of motion. Under steady-state
condition, the first term of the right-hand side of the equation is to be zero and the
Euler’s equation is simplified as follows [49]:

Pg =
∇+ ρ V ( ⋅∇ V ) (1.10)
∇P+g=ρV⋅∇V
By integrating this equation along some arbitrary streamline and also assuming
incompressible fluid we obtain [49]:

P + V 2 + z = const (1.11)
γ 2 g Pγ+V 2g+z=const
2
where γ is the specific weight of fluid and z is the elevation of the point above a ref-
erence plane. Eq. (1.11) is called Bernoulli equation and is valid for inviscid, steady,
incompressible flow along a streamline or in the case of irrotational flow along with
any two arbitrary points. In another word, Bernoulli indicates that the pressure stays
constant during the flow when the tube cross-section and height do not change.

1.8.1.2 Nondimensional parameters
A fundamental nondimensional parameter in analyses of fluid flow is the Reynolds
number (Re) [49]:
ρUL
Re = (1.12)
µ Re=ρULµ
U is velocity scale and L denotes a representative length. The Reynolds number
represents the ratio between inertial and viscous forces. At low Reynolds numbers,