Page 240 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

9.4 Governing equations 239

The thickness of the thermal and viscous boundary layers is usually 1000 times
smaller than the microchannel dimensions [76], which requires a fine mesh near
the boundaries plus high computational cost. The viscous interaction with this layer
drives the acoustic flow in the bulk fluid. The point to note here is that the thick-
ness of the viscous and thermal boundary layer changes with frequency, and as the
frequency increases, the boundary layer thickness decreases. It should be noted that
the ratio of two thicknesses also yields the Prandtl dimensionless number as follows,
which is also important in relation to the thickness of the boundary layer.

δ = µ c p = Pr (9.5)
δ th k δδth=µcp k=Pr

9.4.1 Equations of perturbation theory caused by the
acoustic field
Here, the basic perturbation equations are examined in the second order for heat
transfer and fluid flow. The first-order equations of fluid flow and heat transfer are not
covered, yet they are applied to reach the main governing equations.
In the first-order equations of the acoustic field due to SAWs, the energy equa-
tion for temperature (T ), the kinematic continuity equation for pressure (P ) and the
1
1
momentum equation for velocity field (u ) are described as follows [76]:
1
∂ T 1 = D ∇ 2 T + α p 0 p 1
T ∂
2
t ∂ th 1 ρ 0 c p t ∂ (9.6) ∂T ∂t=Dth∇ T +αpT ρ cp ∂p ∂t
1
0 0
1
1
∂ p 1 = 1  α ∂ T 1 −∇ u  (9.7)
t ∂ γ k   p t ∂ 1   ∂p ∂t=1γksαp∂T ∂t−∇u 1
s
1
1
u ∂
ρ 1 =−∇ p + µ∇ 2 u + βµ∇∇ u ) (9.8)
(
2
0 t ∂ 1 1 1 ρ ∂u ∂t=−∇p +µ∇ u +βµ∇(∇u )
1
1
1
0
1
where D , T , ρ , α , γ , k , and β are the thermal diffusivity, the wall temperature, T ρ α γ k Dth
p
s
p
0
0
s
th
0 0
mass density, thermal expansion coefficient, specific heat capacity ratio, isentropic
compressibility, and the viscosity ratios and are defined as follows:
ρ

α =− 1 ∂   (9.9)


p
ρ ∂ T  p αp=−1ρ∂ρ∂Tp
C α 2 T
γ = p = 1+ p 0 (9.10)
C v ρ 0 Ck γ=CpCv=1+αp T ρ Cpks
2
ps
0 0
µ 1
β = B + (9.11)
µ 3 β=µBµ+13
where the µ and µ are bulk viscosity and dynamic viscosity. µ
µB
B
1  ∂ ρ 
k =   (9.12)
s
ρ  ∂ p s ks=1ρ ∂ρ∂ps
0
0