Pressure acoustics and laminar flow modelling
The pressure distribution in a system can be calculated by inhomogeneous
Helmholtz equation:
\(\nabla.\ \left[\ -\frac{1}{\rho_{c}}\left(\nabla P\right)\right]-\ \frac{\omega^{2}}{\rho_{c}{C_{c}}^{2}}\text{\ P}=\ 0\)[Eq. 1]
where P is the acoustic pressure of the media. The angular
frequency \(\omega\) is defined as \(\omega=2\pi f\), where fis the ultrasound frequency.
\(\rho_{c}\) and Cc is the complex density and
complex sound speed, respectively. They can be expressed by the
following equation as –
\(\rho_{c}\) = \(\frac{\rho{C_{s}}^{2}}{{C_{c}}^{2}}\) and [Eq. 2]
Cc = \(\frac{\omega}{k_{c}}\) [Eq. 3]
Where ρ and Cs are density of the medium and sound
speed, respectively. Complex wave number kc can be
expressed as:
kc = \(\frac{\omega}{C_{s}}\) -i \(\alpha\) [Eq. 4]
Where Cs and α are the sound speed and
absorption coefficient in the media respectively.
The input pressure is calculated by equation [5] -
\(P_{0}\) = \(\sqrt{}(2\ \rho C_{s}I\ )\) [Eq.5]
Where ‘I’ is the intensity of the sound. It is assumed that all
the power enters the tube through the tip of the transducer. The power
input Pw is in watts. Intensity is given by,
I = \(\frac{P_{w}}{A}\) [Eq. 6]
Where A is the area of tip (30.2mm2 for the standard
1/8” or 3mm diameter sonication tip). In this study the input power is
set at 5.5 W because this is corresponding to power delivered by a
Qsonica 125 with 50% amplitude with 3.1 mm diameter sonication tip
which is the usual sonication parameter for cell lysis. In cavitating
media, the absorption coefficient is very high and difficult to
determine (Louisnard, 2012). For the purpose of this study the
absorption coefficient of the media is considered 1
m-1 for water adopted from Xu (Xu et al., 2013). All
the input parameters are listed in (Supplement table 1).
Solving the stationary Helmholtz equation [Eq.1] gives pressure at
each point of the simulated tube. From the resulting pressure field, the
intensity field can be calculated using equation [5].
The stationary laminar flow field is modelled by two equations- 1. The
momentum balance equation [Eq. 7] and 2. The continuity equation
[Eq. 8]:
\(\rho(\overrightarrow{u}.\nabla)\overrightarrow{u}=\nabla.\left[-PI+k\right]+F\)[Eq. 7]
\(\rho\nabla.\overrightarrow{u}=0\) [Eq. 8]
Where \(\rho,\ \overrightarrow{u},\ P,I,\ k\text{\ and\ }F\ \)are fluid
density, velocity vector, fluid pressure, identity tensor, fluid
viscosity and volumetric force respectively.