Acoustics of a viscoelastic medium with encapsulated bubbles

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

“…In the theoretical analysis, the following assumptions are introduced for simplicity. (a-1)The elasticity of the liquid phase is considered; however, the viscoelastic shells [30] , [31] , [23] , [27] , [25] , [26] , [28] , [29] coating the bubbles are not. (a-2)The viscosity and elasticity of the gas phase are neglected because they are significantly smaller than those of the liquid phase.…”

“…(a-1)The elasticity of the liquid phase is considered; however, the viscoelastic shells [30] , [31] , [23] , [27] , [25] , [26] , [28] , [29] coating the bubbles are not.…”

“…When , Eq. (16) reduces to the Kelvin–Voigt viscoelastic model [32] , [33] , [34] , [35] , [36] , [38] , [19] , [37] , [39] , [30] , [31] . Further, when and , Eq.…”

“…Ultrasound also induces the oscillation or collapse of bubbles injected into blood vessels during drug delivery to brain cancer cells, which opens the blood–brain barrier [21] . Moreover, bubbles encapsulated by lipid shells improve the resolution of images [22] , [30] , [31] , [23] , [24] , [27] , [25] , [26] , [28] , [29] for ultrasound imaging as a real-time diagnostic technique.…”

“…Newtonian fluid is the most famous model in the field of fluid dynamics, but only viscosity is considered. The Kelvin–Voigt [32] , [33] , [34] , [35] , [36] , [38] , [19] , [37] , [39] , [30] , [31] and Zener models [41] , [42] , [45] , [46] , [43] , [44] , [47] have been used as mathematical models of liquid with viscoelasticity. The Kelvin–Voigt model corresponds to the generalized Newtonian fluid with the elasticity introduced while the relaxation effect is omitted.…”