Page 241 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

240 CHAPTER 9 Application of microfluidics in cancer treatment

According to the definition of isentropic sound velocity,
 ∂ p 1 1
c =   = → k = (9.13)
2
s
2
2
c =∂p∂ρs=1ksρ → ks=1ρ c  ∂ ρ  s k ρ 0 ρ 0 c 2
s
0
0
ω
e−iwt It is assumed that all first-order fields have a harmonic time dependence e − it due
to the ultrasound field. There is a complete derivation of these equations assuming
the harmonic time dependence on the source [77] as follows.
Energy equation:
γ − 1
ω
iT + D ∇ 2 T = ∇ u 1 (9.14)
1
th
1
2
iwT +Dth∇ T = γ−1αp ∇u 1 α p
1
1
Momentum equation:

ω
iu +∇ 2 u + v β + i 1   ∇∇ u ) = α p ∇ T 1 (9.15)
(
v

1
1
1
2
iwu +v∇ u +vβ+i1vγρ wks∇(∇u ) =αpρ γks∇T 1  vγρ ω k s  ργ k s
0
0
1
1
0
1
0
The physical field of pressure and velocity field are obtained by the real part and
the imagining part.
9.4.2 Second-order equations
A number of acoustic wave effects were observed in the first-order fields obtained
by solving the first-order equations. The first-order theory is inadequate to describe
nonlinear behavior of the acoustic field in liquid since the time-averaged first-order
fields are zero due to harmonic time dependence. It is necessary to extend the second-
order equations of density, pressure, velocity, and temperature as follows:
ρ=ρ +ρ +ρ 2 ρ = ρ + ρ + ρ 2 (9.16)
1
0
0
1
p=p +p +p 2 p = p + p + p 2 (9.17)
1
0
1
0
u=u +u +u 2 u = u + u + u 2 (9.18)
1
0
1
0
which here symbolizes <x> denotes the time averaging of the function x (t) over a
complete period T.
< >= 1 τ ∫ dtx t() (9.19)
x
<x>=1τ∫0τdtx(t) τ 0
In contrast, the time averaging of the product of two first term, proportional to cos
1
2
2
cos (wt)=12 (wt) is nonzero because cos(ω t) = .
2
By applying the second-order acoustic field in the continuum, momentum, and
energy equations and by a series of mathematical operations and time averaging, the
second-order equations of a continuum and momentum are described as follows [78]:
∂tρ +ρ +ρ =−∇ρ +ρ +ρ v +v 2 ∂ t ( ρ + ρ + ρ 2 ) =−∇ ρ (( 0 + ρ + ρ 2 v )( 1 + v ) ) (9.20)
1
0
1
2
1
2
0
1
2 1
0
v )
∂tρ =−ρ ∇v −∇ρ v ∂ t ρ =− ρ ∇ v − ∇ ρ ( 1 1 (9.21)
2
2
0
2
1 1
2
0