An analytical solution is found to exist for Equation (

28) and is given by the following

where

where

${J}_{0}(x)$,

${J}_{1}(x)$ are Bessel functions of the first kind ,

${Y}_{0}(x)$,

${Y}_{1}(x)$ are Bessel functions of the second kind and

${H}_{0}(x)$ is a Struve function.

${C}_{1},{C}_{2}$ and

${C}_{3}$ are constants taken to by each equal to unity. It is evident that the solution given by Equation (

33) with

${\kappa}_{1}$ through

${\kappa}_{4}$ as in Equations (

34)–(

37) is positive everywhere with the exception at

$r=0$ of being exactly zero. The convergence to zero is quite slow as

$r\to {0}^{+}.$ The dynamic viscosity

${\mu}_{2}$ of fluid 2 which we take to be for water as a test case is

$8.9\times {10}^{-3}$ dyn. s/cm

${}^{2}$ at a temperature of 25 degrees Celsius and at the same temperature the density

${\rho}_{2}$ of liquid water is

$9.97044\times {10}^{-1}$ gm/cm

${}^{3}$. Next we transform the center of the tube

$r=0$ to the interface as in

Figure 1. Mapping

r into

$r-{R}_{i}$ in Equation (

33), where

${R}_{i}$ is the radial value at the gas/water interface, it can be shown that

$f(r)=0$ at

$r={R}_{i}$. We choose

${R}_{i}=0.85$, and let

r run to

$r=1$. Results for the transformed function

$f(r)$ are shown in

Figure 2,

Figure 3,

Figure 4,

Figure 5 and

Figure 6. In

Figure 7 and

Figure 8 results are shown for composite velocity

$L(r)$ which is

$\overrightarrow{L}$ in the

r direction, i.e.,

${u}_{r}$, see

Figure 1. Finally we approach a hyperbolic tan function as

$\alpha $ becomes very large and negative for

$f(r)$. This is due to the fact that

$f(r)$ can be analytically continued to the interval closest to and to the left of

${R}_{i}$. It is evident that if we set

$d=r-{R}_{i}$ to be the standard distance function from the interface [

5,

6] we can solve numerically for

d in Equations (

33)–(

37) and verify that

d is oscillating hence implying that the interface is oscillating back and forth in the radial direction. In addition I have compared to [

16] for verification of a tan hyperbolic representation near the interface which is used there in a numerical approach. The solution given by Equation (

33) can also be written in the form

$1-\frac{\mathrm{sin}(\beta d)}{\beta d}$ , where

$\beta $ is chosen to be large and negative (not equal to

$\alpha $). It was verified that

$\varphi $ could be shifted by scalar addition to avoid a velocity

$\overrightarrow{L}$ that is undefined at zero as in Equation (

18). The governing equations were extended by multiplying by the result of

$\varphi $ added to a nonzero constant. This results in a finite integral for

$\overrightarrow{L}$ over the tube radius and meaningful comparisons to velocity measurements can be made as found in the literature [

17], we have,

where

and

${\varphi}_{l}$ is lifting of

$\varphi $ by a constant

A. I have used data presented in [

17] for

${\mu}_{2}$,

${\rho}_{2}$,

$\frac{{\mu}_{2}}{{\rho}_{2}}=0.002\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$ and in Equation (

39),

$A+1=-0.002689328$. Comparing Equation (

33), to Equation (

39) for

$r=1/2\phantom{\rule{4pt}{0ex}}\mathrm{m}$ [

17],

$Fz=0.3$,

$\alpha =-1\times {10}^{6}$, constants

${C}_{1},{C}_{2}$ and

${C}_{3}$ equal to unity and

${\mu}_{2}$ and

${\rho}_{2}$ as in [

17], a solution for

$\beta $ is obtained.