Tham khảo tài liệu 'cavitation and bubble dynamics part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 3 - Cavitation and Bubble Dynamics - Christopher E. Brennen Trilling 1952 showed that one could use the approximation introduced by Kirkwood and Bethe 1942 to obtain analytic solutions that agreed with Schneider s numerical results up to that Mach number. Parenthetically we note that the Kirkwood-Bethe approximation assumes that wave propagation in the liquid occurs at sonic speed c relative to the liquid velocity u or in other words at c u in the absolute frame see also Flynn 1966 . Figure presents some of the results obtained by Herring 1941 Schneider 1949 and Gilmore 1952 . It demonstrates how in the idealized problem the Mach number of the bubble surface increases as the bubble radius decreases. The line marked incompressible corresponds to the case in which the compressibility of the liquid has been neglected in the equation of motion see Equation . The slope is roughly -3 2 since dR dt is proportional to R-3 2. Note that compressibility tends to lessen the velocity of collapse. We note that Benjamin 1958 also investigated analytical solutions to this problem at higher Mach numbers for which the Kirkwood-Bethe approximation becomes quite inaccurate. Figure The bubble surface Mach number - dR dt c plotted against the bubble radius relative to the initial radius for a pressure difference P-. -pi M- of bar. Results are shown for the incompressible analysis and for the methods of Herring 1941 and Gilmore 1952 . Schneider s 1949 numerical results closely follow Gilmore s curve up to a Mach number of . When the bubble contains some noncondensable gas or when thermal effects become important the solution becomes more complex since the pressure in the bubble is no longer constant. Under these circumstances it would clearly be very useful to find some way of incorporating the effects of liquid compressibility in a modified version of the Rayleigh-Plesset equation. Keller and Kolodner 1956 proposed the following modified form in the absence
