流體中柔性圓管之波傳研究

Abstract

為探究超音波對血管的影響，本文研究置於流體中的柔性管與流體之間交互作用的頻散現象與波傳模態。 根據Chebyshev-Gauss-Lobatto插值法與微分矩陣原理，採用Spectral數值方法離散彈性力學理論的波動方程，建立軸對稱縱向、非軸對稱周向和彎曲模態對應的廣義特徵值問題。此法克服Bessel函數的發散問題，成功解出頻譜、模態中各物理量的振形與質點軌跡圖。 第一、第二頻散曲線不論在低頻或高頻區域皆對應於彈性管表面波模態，分別為反對稱與對稱模態；第三、第四頻散曲線在兩相速度尚未收斂至相當接近的區域時，皆為管內外流體表面波模態，且在管面上流體振幅比管振幅大，沿兩頻散曲線移動，隨頻率增加兩相速度值逐漸靠近，第三頻散曲線最終收斂至管外流體表面波模態，第四頻散曲線收斂至管內流體表面波模態。 對應於EKOS的2MHz高頻模式，能量會集中在血管表面，血管內部振幅極小；而操作在45kHz的低頻模式時，血管內部會承受較大的振幅。因此以2MHz高頻模式操作對血管的傷害較低。

The wave propagation in flexible tubes immersed in fluid is studied in order to explore the effect of ultrasonic waves incident on blood vessels. Based on Chebyshev-Gauss-Lobatto interpolation and differential matrix, spectral method discretizes the Helmholtz equation to establish generalized eigenvalue problems. These generalized eigenvalue problems are formulated for axial symmetric, circumferential, and bending modes respectively. The numerical method overcomes the divergent problem of Bessel functions and the spectrum, mode shapes, and motion of particles are found successfully. Mode shapes corresponding to the first and second dispersion curves display surface wave patterns of the tube in a very wide frequency range. The first dispersion curve gives antisymmetric modes while the second one offers symmetric modes. Before convergence, the third and fourth dispersion curves provide interface waves of fluid on both inside and outside of tubes. As frequency is increased, mode shapes corresponding to the third dispersion curve converge to surface waves of fluid outside the tube and that corresponding to the fourth dispersion curve converge to surface waves of fluid inside the tube. For EKOS operated at high frequency mode of 2MHz, surface wave modes will be excited most due to the ultrasonic source location. Therefore, the ultrasonic energy is mainly concentrated in the vicinity of vessel surface. When it is operated at low frequency mode of 45kHz, there is more ultrasonic energy penetrating into the vessel wall than that of 2MHz mode. That is, high frequency operation mode causes less damage to the vessel than low frequency operation mode.