A fluid of constant density and viscos- ity is contained in a very long pipe of length L and radius R. Initially the fluid is at rest. At time t = 0, a pressure gradient (Po-P)/L is imposed on the system. Determine how the ve- locity profiles change with time. V Vmax 1.0 0.8 0.6 0.4 0.2 Tube center 0 1.0 0.8 0.6 0.4 0.2 vt R 0.2 0.15 0.1 0.05 81 Tube wall 0.5 0.4. 0.3 0 0.2 0.4 0.6 0.8 1.0 r/R-> Fig. 4D.2. Velocity distribution for the unsteady flow re- sulting from a suddenly impressed pressure gradient in a circular tube [P. Szymanski, J. Math. Pures Appl., Series 9, 11, 67-107 (1932)]. (a) Show that the relevant equation of motion can be put into dimensionless form as follows: 26 26 = 4 + T 1 a } a (5. (, ) = (1 ?) 8 #-1 ok in which & = r/R, T = t/pR, and = [(P-P)R/4L]v. (b) Show that the asymptotic solution for large time is = 1 - . Then define o, by (, 7) = ..(E) - (E, T), and solve the partial differential equation for o, by the method of separation of variables. (c) Show that the final solution is Jo(,) 1a J () (4D.2-1) (4D.2-2) -exp(-ar) in which J() is the nth order Bessel function of , and the a,, are the roots of the equation Jo(a) = 0. The result is plotted in Fig. 4D.2.