On this work we use COMSOL Multiphysics (model 5.three)22, a industrial finite-element based mostly software program bundle, to mannequin the thermo-viscous results round a single spherical silica particle (SiO2) in water, in a planar acoustic subject, utilizing the set-up proven in Fig. 1. The thermoviscous acoustics (TVA) frequency area module is used all through this work to calculate the temperature, strain, and velocity variations close to the partitions of the silica the place viscous and thermal losses are vital. The equations outlined by the TVA interface are the linearised Navier-Stokes equations in quiescent background circumstances fixing the continuity, momentum, and power equations. The mannequin is 2D axisymmetric and the TVA area is surrounded by a superbly matched layer (see Fig. 1) to keep away from reflections from the boundary. The simulations carried out on this work had been carried out utilizing a workstation with two Intel Xeon CPU E5-2630v3, 2.40 GHz processors and 256 GB of DDR4 RAM. On this part we current the simulation system.
The simulations mannequin an ultrasonic aircraft wave propagating within the z-direction utilized to a spherical silica particle in water. The system is 2D axi-symmetric.
The thermoviscous acoustic equations
The frequency-domain TVA mannequin relies upon the linearised Navier-Stokes equations for a harmonic subject of angular frequency ω of the shape exp (iωt). The Thermoviscous Acoustic Mannequin node is used to outline the mannequin inputs such because the background temperature (298.15 Okay) and the strain (we use the default worth of 1 atm), along with the equilibrium materials properties from the fabric mannequin (see part under). The equations of the system are as under (notice I is the identification matrix)):
$$iomega _bf=nabla cdot (-pbf+mu _S[nabla bf+(nabla bf)^]-[fracmu _bfS-mu _][nabla cdot bf]bf),$$
$$iomega rho +_nabla cdot bf=zero,$$
$$iomega __pT=nabla cdot (kappa nabla T)+iomega pT__,$$
the place ω is angular frequency, μS is shear viscosity, μB is bulk viscosity, Cp is warmth capability at fixed strain, κ is thermal conductivity, and ε0 is the quantity thermal enlargement coefficient at fixed strain. The time-harmonic perturbations in strain, velocity, temperature and fluid density (p, u, T, and ρ respectively) are written with out subscripts, these being superimposed upon their regular state values (indicated by subscript zero); thus ρ0 is regular state density. The equations (1–three) are solved within the thermo-visco-acoustic liquid area, along with the density variation ρ = ρ0(βTp − ε0T), the place βT is isothermal compressibility. The incident perturbing subject is outlined as a planar background strain subject propagating within the z-direction.
with the normalised wave path vector (hatbfz) and |pb| = zero.1 MPa. The simulation framework is linear and due to this fact all perturbations scale with the desired strain amplitude. The background velocity in response to the strain perturbation is,
and the background temperature is
Herein, the background acoustic subject wavenumber takes the shape,
$$b_television=fracfourthreemu _S+mu _+frac(gamma -1)boldsymbolkappa _p.$$
the place γ is the ratio of particular warmth capacities and c is the adiabatic wave pace for compressional waves. These background subject equations are taken from the COMSOL documentation, and are derived from the linearised time-harmonic subject equations (1–three) by expressing them in Helmholtz type utilizing a velocity potential, leading to wavenumber ok for every mode. The three wave modes (compressional/acoustic, thermal and shear) are denoted by subscripts c, T and S respectively; the wavenumber for the incident compressional background wave kb = kc. The scattered subject is superimposed on these background harmonic fields to acquire the complete perturbation subject values for every amount. The stable mechanics module is used to outline the properties of the silica particle which is isotropic and linearly elastic. The preliminary displacement and structural velocity fields are set to zero. Coupling between the stable particle and the TVA area is carried out within the Multiphysics Node of the software program. The coupling provides the connection between stable displacement and complete fluid velocity ut,fluid = iωusolid on the particle boundary, the place ut,fluid is the full fluid velocity together with the background subject and usolid is the displacement of the (boundary of the) silica sphere. Thus displacement is steady on the boundary; stress can also be steady on the boundary.
The mannequin construction
We have an interest within the shear and thermal fields across the particle, which have typical size scales which are orders of magnitude smaller than the compressional (acoustic) wavelength within the methods beneath investigation. These fields type thermal and viscous boundary layers, which have been studied because the early evaluation of Stokes23 and others, and later extensively by Schlichting24. The system should due to this fact be outlined when it comes to the wavelengths of the shear or thermal fields. Right here, since we examine a stable particle the place scattering is dominated by visco-inertial results, it’s the shear wavelength that’s used to outline the attribute scale of the system.
The shear and thermal decay fields have wavelengths,
$$lambda _S,=,2pi sqrtfrac,$$
$$lambda _T=2pi sqrt,$$
respectively, the place f is frequency. Axial symmetry is about alongside the z axis at R = zero (the cylindrical radial coordinate). The preliminary values of perturbed strain, velocity, and temperature are all set to zero. The issue is about across the particle in a area stretching 13 shear wavelengths (see Eqn. 9) from the particle centre in all instructions. The particle sits in an off-set place with its centre at (R, z) = (zero, 13λS). The axial boundaries are outlined as throughout the ranges [Rmin, Rmax] = [0, 19.5λS] and [zmin, zmax] = [−6.5λS, 32.5λS], together with the PML. The particle radius is denoted a.
The everyday wave pace for the PML is about right here to 1497 m/s (pace in water at 25 °C) and all temperatures are adjusted to 298.15 Okay. For stability the discretisation for strain is about one aspect order lower than for velocity and temperature. Linear aspect order is about for strain, and quadratic order for velocity and temperature. The order for temperature and velocity are stored the identical because of our requirement to mannequin thermal and viscous boundary layers. The dependent variables are strain p, velocity subject u, and the temperature variation T (all taking complicated values).
The mesh and completely matched layer
The mesh is “mapped” based on two distributions emanating from the road of axial symmetry contained in the particle and outdoors it (throughout the TVA area). Contained in the particle the symmetry axis is cut up into 400 parts of equal size 1.25 nm. Likewise the symmetry axis both facet of the particle within the TVA area is cut up into 1100 parts every of 12.49 nm size (which suggests roughly 86 parts slot in one shear wavelength). The result’s a radial cross meshing with tight sufficient sizing in order that radial unfold has restricted impact. The meshing contained in the TVA and stable mechanics domains will be seen in Fig. 2. Additional refinement of the mesh doesn’t enhance the simulations21.
A typical mesh, proven right here for a 500 nm diameter silica particle, in a water area. On the z-axis (vertical) there are 1100 parts exterior the particle (water space) every of 12.5 nm and 400 parts on the z-axis of the area of the particle every of 1.25 nm. The mannequin is axisymmetric. The silica particle is analysed for various diameters.
The peerlessly matched layer (PML) surrounds the TVA area as proven in Fig. 1 and is designed to soak up all outgoing wave power. This removes a difficulty of impedance mismatch on the boundary with the TVA. A predefined “wonderful” COMSOL triangular mesh is outlined for the PML. Within the PML circumstances we set the sort to spherical with a polynomial stretching sort. The PML is designed to use a fancy coordinate stretching in 1–three instructions, decided from the way it connects to the bodily area. The stretching is a operate of the wavelength related to the frequency of the utilized ultrasonic wave. The width of the PML is chosen to be 6.5 shear wavelengths (λS). The complicated displacement for stretching in a single path is Δx = λwfp(ξ) − Δwξ, the place λw is the wavelength (cwater/f) and Δw = 6.5λS. The ξ is a dimensionless coordinate that varies from zero to 1 over the PML. The polynomial stretching operate fp(ξ) = sξp(1 − i) the place p is a curvature parameter (set to 1 right here) and s is a scaling operate (additionally set to 1). Because the shear and thermal waves have virtually utterly decayed after the chosen propagation distance to the boundary with the PML a curvature issue of 1 is greater than ample to make sure passable mesh decision throughout the PML. The strain amplitude is discovered to be zero contained in the PML.
The bodily properties used within the modelling will be present in Desk 1. All different values are decided from these properties; isothermal compressibility, for instance, is set from the (adiabatic) sound pace and is βT = γ/ρ0c2 = four.51 × 10−10 Pa−1). The calculated Poisson’s ratio and shear wavespeed have additionally been checked for self-consistency. These properties outline the shear wavenumber which in flip determines the dimensions of the simulated area since we have an interest within the shear wave decay.
Desk 1 Bodily parameters of silica and water (at 25 °C) entered as materials values within the finite aspect modelling.