Consider an incompressible fluid flowing in space (no

Chapter 14, Problem 33

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Consider an incompressible fluid flowing in space (no sources or sinks) with variable density r(x, y, z, l) and velocity field v(.{, ), z, t). Let B be a small ball with radius r and spherical surface S centered at the point (xo, yo,zo). Then the amount of fluid within S at time t is and differentiation under the integral sign yields (a) Consider fluid flow across S to get lt es):- // ov.nas, '.t.t where n is the outer unit normal vector to S. Now apply the divergence theorem to convert this into a volume integraal. (b) Equate your two volume integrals for Q'Q), apply the mean value theorem for integrals, and finally take limits as r -+ 0 to obtain the continuity equation 33. ot,t : [[[ aav. o'o= lll"ffav 29. 30. ff*o.tu"y-0.