## 2 Materials and Methods
### Application of Acoustic Streaming in SCC  
This section describes the method of applying acoustic streaming induced by LIFU. The process begins with analyzing acoustic waves in the frequency domain, followed by calculating the dissipation of acoustic energy. This dissipation generates a volumetric force that drives fluid velocity in the endolymph. Due to the asymmetric structure of the SCC [28], the fluid flow may vary depending on the location of the ultrasound's focal point.  
To model this effect, a nonlinear inhomogeneous Helmholtz equation is solved to compute the acoustic field generated by the transducer [39]. The parameters for the calculations are summarized in Table 1. The material properties of water were used for the endolymph parameters, given their similarity [3; 28]. The material properties of the temporal bone were derived from the literature [40; 41; 42]. The numerical model used a cap-sphere-shaped transducer designed to generate uniform pressure at its boundary [32; 43]. To determine the corresponding input pressure value, the following equation was used:  
\[p_{i}=\sqrt{\frac{\rho c}{\pi rh}\times P} \tag{1}\]  
where \(P\) is the radiated power of the transducer, \(\rho\) is the density of the surrounding water, \(c\) is the speed of sound in water, and \(r\) = 42 mm and \(h\) = 10 mm represent the radius and height of the transducer, respectively. For the given configuration, the input pressure was calculated as 33.632 Pa at \(P\) = 1 W, which was then used as the pressure boundary condition  
Figure 1: (**a**) 3D structure of the SCC model, including the cupula and endolymph. The red cross-section represents the interface between the cupula and endolymph, which will deform in subsequent analyses. The ultrasound radiation is emitted in the positive z-direction. (**b**) FEM domain of the numerical model in the xz-plane, highlighting a horizontal SCC filled with endolymph fluid and the cupula. The structure is centered at the coordinates (0, 0, 40).  
for the ultrasound transducer. Using this condition, the acoustic field was computed by solving [39]  
\[\
abla\cdot\left(-\frac{1}{\rho_{c}}\
abla p\right)-\frac{k_{c}^{2}p}{\rho_{c}}=0 \tag{2}\]  
where \(p\) is the acoustic pressure, \(k_{c}\) is the complex wave number, and \(\rho_{c}\) is the complex density. The wave number, speed of sound, and density for the attenuation coefficient \(\alpha\) are defined as follows:  
\[k_{c}^{2}=\left(\frac{\omega}{c_{c}}-i\alpha\right)^{2} \tag{3}\]  
\[c_{c}=\frac{\omega}{k_{c}} \tag{4}\]  
\[\rho_{c}=\frac{\rho c^{2}}{k_{c}} \tag{5}\]  
The time-averaged velocity of the endolymph at every position can be calculated by the time-averaged version of Navier-Stokes equations [39]:  
\[\rho(\overrightarrow{u}\cdot\
abla\overrightarrow{u})=-\
abla p+\mu\
abla^{ 2}\overrightarrow{u}+\overrightarrow{F} \tag{6}\]  
\[-\
abla\cdot\overrightarrow{u}=0 \tag{7}\]  
where \(\mathbf{u}\) is the fluid velocity, \(\rho\) is the fluid density, \(\mu\) is the dynamic viscosity, and \(\mathbf{F}\) is the volume force that causes the acoustic streaming. Equation (7) represents the continuity equation, which assumes the incompressibility of the fluid. The endolymph, which fills the canal except for the cupula, is considered an incompressible medium due to its similarity to water [3; 28]. The acoustic field derived from Equation (1) is then used to compute the volume force that induces fluid flow [32; 39; 43] as follows:  
\[F=\frac{2\alpha I}{c} \tag{8}\]  
where \(\alpha\) is the attenuation coefficient of the fluid and \(I\) is the time-averaged acoustic intensity.