Consider a gas bubble rising in a liquid$-$filled tube. If the bubble volume $(V_B)$ and tube radius $(R_T)$ are such that $V_B^{\frac{1}{3}}{R}_T,$ then the bubble will be highly elongated. Assume that the bulk of the bubble forms a cylinder of radius $R_B$ and length $L,$ as shown in Fig. P$7-15($a$).$ The liquid film between the bubble and tube wall is expected to be very thin $)=(R_T-R_B{R}_B.$ Consequently, local rectangular coordinates can be used in the film, as shown in Fig. P$7-15($b$).$ It is convenient to fix the coordinate origin on the wall.

Assuming that the flow in the liquid film is fully developed and that the bubble dimensions are known, derive an expression for the steady rise velocity of the bubble, $U.$ State any additional assumptions that are needed. $($Hint: The bubble velocity is not uniform, so that the velocity at the gas$-$liquid interface differs from $U,$ the mean value for the bubble.$)$

X$-$momentum: $0=-\frac{dP}{dx}+\frac{d^2{U}_x}{d{y}^2}.$ the y$-$momentum: $0=\frac{-dP}{dy}.$ The boundary conditions are $,$y$=$H$)=0$ and $,$y$=0)=0.$