Falling film on a conical surface (see Fig. 2C.5)⁷ a fluid flows upward through a circular tube and then downward on a conical surface. Find the film thickness as a function of the distance s down the cone. 

(a) Assume that the results of S2.2 apply approximately over any small region of the cone surface. Show that a mass balance on a ring of liquid contained between s and s + ∆s gives: 

(b) Integrate this equation and evaluate the constant of integration by equating the mass rate of flow w up the central tube to that flowing down the conical surface at s = L. Obtain the following expression for the film thickness:   

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$ = downstream distance along cone surface, measured from the cone apex Film thickness is 8 (s) S = L fFluid in with mass flow rate w Part (a) (s8³) = 0 (s8{v)) = 0 ds or %3D ds Part (b) 3μυ V πρgL sin 2β