solve each part separately and indicate your answer clearly with solutions Problem 2: Spherical Hole in an Incompres...
Problem 2: Spherical Hole in an Incompressible Fluid Consider an ideal incompressible luid p . A) u infinite extent. At tine-t 0 a spherical lik-of radius a exists in the luid. Assume that initially the fluid velocity f is zero everywhere and that at very large distance froti the hole the pressure is Pi and the hal elocity remains equal to zera at large distance. Iore the elects of gravity aud that the pressure in the hole is zero. For 0. tluid will flow and eventually fill the hole: In this problen yuu will caleulate the time it takes to fill the hote with flui a) Argue wly for U the subsequent Huid Bow will be spherically symnetrical and will nccessarily he of the form G(t) where G(t) is a function of time only and r is the radial coordinate measured from the center of the hole to a point in the fluid. [Hint: Consider the continuity lation or an inor "re İlle fluid. b) Explain brieilty wlhy the fluid velocity can be deriyed from a scalar velocity potential in this case. IVo, fid φ(ht) which will give the result in parta) end for which φ-r) as r-, oc. c) Apply Bernoulli's oquation for mcompressible time-dependent low to this problem. Recall that Bernoulli's eq. is given by: istant 1) Froni the houndary conditious at Iarge distanco from the hole evalhuate the constant in Bernoulli's equation. ii) For l > 0, the lole collapse in a ynu netric "ny, Let R() be the rudins of the hole at tinie t RIO) a Let l' bo the rnte of duLnge of the raditis or the hole. By evaluating Berti oulli's eq, at the surface of the hole. slhow that the equation of motion for R(t) can be written ns dt 2 ii) Eliminate t as thie independent variable and obtain a as a function of ft. lv) Flnd the time required for the hole to be iled with fluid. (Write your answer as a definite integral. The integral you obtain will not be lntegrable in terms of elementary functions)
Problem 2: Spherical Hole in an Incompressible Fluid Consider an ideal incompressible luid p . A) u infinite extent. At tine-t 0 a spherical lik-of radius a exists in the luid. Assume that initially the fluid velocity f is zero everywhere and that at very large distance froti the hole the pressure is Pi and the hal elocity remains equal to zera at large distance. Iore the elects of gravity aud that the pressure in the hole is zero. For 0. tluid will flow and eventually fill the hole: In this problen yuu will caleulate the time it takes to fill the hote with flui a) Argue wly for U the subsequent Huid Bow will be spherically symnetrical and will nccessarily he of the form G(t) where G(t) is a function of time only and r is the radial coordinate measured from the center of the hole to a point in the fluid. [Hint: Consider the continuity lation or an inor "re İlle fluid. b) Explain brieilty wlhy the fluid velocity can be deriyed from a scalar velocity potential in this case. IVo, fid φ(ht) which will give the result in parta) end for which φ-r) as r-, oc. c) Apply Bernoulli's oquation for mcompressible time-dependent low to this problem. Recall that Bernoulli's eq. is given by: istant 1) Froni the houndary conditious at Iarge distanco from the hole evalhuate the constant in Bernoulli's equation. ii) For l > 0, the lole collapse in a ynu netric "ny, Let R() be the rudins of the hole at tinie t RIO) a Let l' bo the rnte of duLnge of the raditis or the hole. By evaluating Berti oulli's eq, at the surface of the hole. slhow that the equation of motion for R(t) can be written ns dt 2 ii) Eliminate t as thie independent variable and obtain a as a function of ft. lv) Flnd the time required for the hole to be iled with fluid. (Write your answer as a definite integral. The integral you obtain will not be lntegrable in terms of elementary functions)