2.25 Advanced Fluid Mechanics

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1 MIT Depatment of Mechanical Engineeing 2.25 Advanced Fluid Mechanics Poblem 4.27 This poblem is fom Advanced Fluid Mechanics Poblems by A.H. Shapio and A.A. Sonin u(,t) pg Gas Liquid, density Conside a bubble of high-pessue gas exploding in an incompessible liquid in a spheically-symmetical fashion. The gas is not soluble in the liquid, and the liquid does not evapoate into the gas. At any instant R is the adius of the bubble, dr/ is the velocity of the inteface, p g is the gas pessue (assumed unifom in the bubble), u is the liquid velocity at the adius, and p is the liquid pessue at a geat distance fom the bubble. Gavity is to be neglected. The following questions petain to the fomulation of an analysis which will lead to the details of the pessue and velocity distibutions and to the ate of bubble gowth in the limit of inviscid liquid flow. (a) Show that at any instant R 2 dr u = 2 (4.27a) (b) Show that the ate of gowth of the bubble is descibed by the equation R d2 R dr 2 + 2σ R = p g p (4.27b) whee σ is the suface tension at the gas-liquid inteface. (c) What additional infomation o assumptions would be necessay in ode to establish the bubble adius R as a funciton of time? Explain how you would use this infomation Advanced Fluid Mechanics 1 Copyight c 2010, MIT

2 Inviscid Flows A.H. Shapio and A.A. Sonin 4.27 Solution: CV u(,t) CS1 CS2 (a) If the contol volume is chosen coectly, it is possible to detemine u using eithe Fom A o B of the integal mass consevation equation, howeve, hee we will use Fom A. Fom A is witten d dv + (u u CS ) nda ˆ = 0 (4.27c) CV CS We choose a contol volume taking the shape of a hollow sphee whose inne contol suface, CS 1, has adius and whose oute contol suface, CS 2, has abitay adius. Futhemoe, the intenal suface CS 1 is selected to move adially outwad at exactly the ate of expansion of the bubble dr/. Let us fist evaluate the volume integal tem, by noting that the density is constant and the total volume of ou CV is 4 V = π( 3 3 ). Accodingly, 3 d 4 d dr dv = π 3 3 = 4πR 2 (4.27d) CV 3 Next, we evaluate the suface flux integals. At CV 1 we note that the liquid velocity is exactly equal to the velocity of the gas-liquid inteface, dr/ which is also the speed at which CV 1 is moving, hence thee is no elative velocity between the liquid and CV 1 so u u CS = 0, and CS 1 (u u CS ) nda ˆ = 0 (4.27e) Convesely, at CS 2, may be abitay, but it is fixed in time so that u CS2 = 0, and povided R <, then the liquid velocity at this -value is u(, t), so the suface flux integal at CS 2 is CS 2 (u u nda = u(, t)ê ê 4π 2 = 4π 2 CS ) ˆ u(, t) (4.27f) Substituting Eq. (4.27d), (4.27e) and (4.27f) into Eq. (4.27c), we find 4πR 2 dr + 4π 2 u(, t) = 0 (4.27g) 2.25 Advanced Fluid Mechanics 2 Copyight c 2010, MIT

3 Inviscid Flows A.H. Shapio and A.A. Sonin 4.27 and hence u(, t) = R2 dr 2 (4.27h) pg 1 2 p σ (b) Neglecting gavity, and povided the flow is inviscid and iotational, we may apply the unsteady Benoulli equation along a steamline with station 1 located just on the liquid side of the gas-liquid inteface and station 2 located at a geat distance fom the bubble whee p = p and the liquid velocity is appoximately zeo, u 2 0. The unsteady Benoulli equation along this steamline is s2 du 1 ds + p2 = p u (4.27i) s 1 2 The pessue p 1 diffes fom the pessue in the bubble p g by the Laplace pessue such that p 1 = p g 2σ, whee σ is the suface tension of the gas-liquid inteface. Substituting ou esult fo u in Eq. (4.27h) into Eq. (4.27i) and setting ds = d, s 1 = and s 2 =, we obtain ( ) d R 2 ( ) dr 2σ 1 dr d + p = pg + (4.27j) 2 R 2 2 Expanding the time deivative in the integand, we find ( ( ) 2 2R R(t 2 ) dr ) R 2 d 2 ( ) 2 R 2σ 1 dr d + p = pg + (4.27k) R 2 Completing the integal, we obtain which educes to ( ( ) 2 ) = 2 2R dr R 2 d 2 ( ) R 2σ 1 dr 2 + p = p g + (4.27l) R 2 = 2.25 Advanced Fluid Mechanics 3 Copyight c 2010, MIT

4 Inviscid Flows A.H. Shapio and A.A. Sonin 4.27 ( 2 ) 2 ) dr d 2 ( ) R 2σ 1 dr + R + p = p 2 g + R 2 ( 2 (4.27m) Finally, we eaange this esult to obtain the govening equation fo R in its desied fom ( ) 2 d 2 R 3 dr 2σ p p R + + = g (4.27n) 2 2 R (c) Since ou govening equation, Eq. (4.27h) is second ode in time, we equie two initial conditions, such as the initial bubble adius R 0 and intefacial velocity dr/ t=0. Futhemoe, we equie a elationship between bubble pessue p g and bubble adius, which could be obtained fom the ideal gas law, p = RT and some easonable assumption about the natue of the bubble expansion (e.g. adiabatic o isothemal). Poblem Solution by TJO, Fall Advanced Fluid Mechanics 4 Copyight c 2010, MIT

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