Generalization of Joukowski’s Solution for a Bubble in a Channel

Abstract—

A new exact solution for the problem of potential flow of a capillary fluid past a two-dimensional bubble in a rectilinear channel is derived; the solution generalizes the known particular Joukowski’s solution. It is shown that in one of the limiting cases of an infinitely small bubble this solution coincides with the exact McLeod’s solution for a bubble in an infinite flow.

Appendices

APPENDIX А

In the case in which ${{a}_{1}} \ne 0$ we can immediately derive an expression of the parameter $A$ in terms of ${{a}_{1}}$ from Eq. (4.10)

$$A = - a_{1}^{{ - 1}}.$$

(А.1)

Then rewriting expression (4.11) in terms of Eq. (4.10) yields

$$B = 3 - a_{1}^{2} - {{a}_{1}}(\xi _{0}^{2} + \xi _{0}^{{{\kern 1pt} - 2}}).$$

(А.2)

Substituing this expression into Eq. (4.12) and taking Eq. (А.1) into account we arrive at the expression

$$(\xi _{0}^{2} + \xi _{0}^{{ - 2}}) = F({{a}_{1}}),$$

(А.3)

where the auxiliary function $F({{a}_{1}})$ is used (cf. Eq. (4.13)).

Relation (А.3) is a quadratic equation in $\xi _{0}^{2}$. Solving this equation and discarding its smaller in absolute value root we obtain

$$\xi _{0}^{2} = 0.5\left[ {F({{a}_{1}}) + \sqrt {{{F}^{2}}({{a}_{1}}) - 4} } \right]$$

(А.4)

In order for the quantity $\xi _{0}^{2}$ be real the following inequality must be fulfilled

$$F({{a}_{1}}) > 2.$$

(А.5)

It can restrict the boundaries of the interval on which ${{a}_{1}}$varies: $- 1 < {{a}_{1}} < 1$. In fact, let us write the quadratic equation

$$3a_{1}^{2} - 2{{a}_{1}} - 1 = 0,$$

which represents the denominator of the function $F({{a}_{1}})$. This equation has two roots ${{a}_{1}} = - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}$ and ${{a}_{1}} = 1$ and only on the interval

$$ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3} < {{a}_{1}} < 1$$

(А.6)

the condition $(1 + 2{{a}_{1}} - 3a_{1}^{2}) > 0$ is fulfilled. Then the verification of inequality (А.5) with account for (4.13) reduces to the verification of inequality

$$ - {{a}_{1}}{{\left[ {(a_{1}^{{ - 1}} + {{a}_{1}}) - 2} \right]}^{2}} > 0.$$

(А.7)

This inequality is not fulfilled on the entire interval (А.6) but only on the interval

$$ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3} < {{a}_{1}} < 0.$$

(А.8)

Thus, the solution is physically meaningful only on interval (А.8), where $\xi _{0}^{2}$ is real, and using Eqs. (А.2)–(А.4) the parameters $B$ and ${{\xi }_{0}}$ can be expressed in terms of the parameter ${{a}_{1}}$

$$B = 3 - a_{1}^{2} - {{a}_{1}}F({{a}_{1}}),\quad {{\xi }_{0}} = \sqrt {0.5\left[ {F({{a}_{1}}) + \sqrt {{{F}^{2}}({{a}_{1}}) - 4} } \right]} $$

(А.9)

The physical condition $B > 0$ can also correct the boundaries of the interval for ${{a}_{1}}$ which is now determined by inequality (А.8). In view of Eqs. (4.13) and (А.9) and the fact that $(1 - 3a_{1}^{2} + 2{{a}_{1}}) > 0$, it is necessary to verify the fulfillment of the inequality

$$B(1 - 3a_{1}^{2} + 2{{a}_{1}}) = (1 - 3a_{1}^{2} + 2{{a}_{1}})(3 - a_{1}^{2}) - 6{{a}_{1}} + {{(1 + a_{1}^{2})}^{2}} + 2a_{1}^{3} > 0.$$

It is easy to verify that this reduces to the verification of the inequality $4{{(1 - a_{1}^{2})}^{2}} > 0$, which is always fulfilled. Thus, on interval (А.8) we have always B > 0.

APPENDIX В

To estimate the bubble area $S$ we will use the formula

$$S = \int\limits_{BA} {xdy - ydx} = - \frac{1}{{2i}}\int\limits_0^{2\pi } {{{{\left. {z\overline {\left( {\frac{{d{\kern 1pt} z}}{{d\sigma }}} \right)} } \right|}}_{\Gamma }}d\sigma } .$$

(В.1)

In the $\zeta$ plane the boundary $\Gamma$ is associated with $\zeta = {{e}^{{i\sigma }}}$; on this boundary the differential $d\sigma$ can be replaced by ${{(i\zeta )}^{{ - 1}}}d\zeta$ [14]. Using the operator $\mathcal{P}$ introduced in Section 3 we can rewrite Eq. (В.1) in the form of the contour integral:

$$S = - \frac{1}{{2i}}\mathop{\int\mkern-20.8mu \circlearrowleft}\limits_{{{C}_{\zeta }}} {z(\zeta )\overline {\left( {i\zeta \frac{{{\kern 1pt} d{\kern 1pt} z}}{{d\zeta }}} \right)} \frac{{d\zeta }}{{i\zeta }}} = \frac{1}{{2i}}\mathop{\int\mkern-20.8mu \circlearrowleft}\limits_{{{C}_{\zeta }}} {\frac{{z(\zeta )}}{\zeta }\mathcal{P}\left[ {\zeta \frac{{{\kern 1pt} d{\kern 1pt} z}}{{d\zeta }}} \right]d\zeta } ,$$

(В.2)

where the contour ${{C}_{\zeta }}$ is the unit circle ${\text{|}}\zeta {\text{|}} = 1$. Taking Eqs. (3.6) and (5.1) into account we obtain

$$\mathcal{P}\left[ {\zeta \frac{{dz}}{{d\zeta }}} \right] = \frac{{\gamma ({{\xi }_{0}} + \xi _{0}^{{{\kern 1pt} - 1}}){{{(1 - {{a}_{1}}{{\zeta }^{2}})}}^{2}}\zeta }}{{\pi ({{\zeta }^{2}} - \xi _{0}^{2})({{\zeta }^{2}} - \xi _{0}^{{ - 2}})}}.$$

(В.3)

Substituting Eqs. (5.2) and (В.3) into expression (В.2) we obtain the expression

$$S = \frac{{\gamma ({{\xi }_{0}} + \xi _{0}^{{{\kern 1pt} - 1}})}}{{2\pi i}}\left[ {\mathop{\int\mkern-20.8mu \circlearrowleft}\limits_{{{C}_{\zeta }}} {\frac{{{{\Phi }^{ - }}(\zeta )d\zeta }}{{({{\zeta }^{2}} - \xi _{0}^{{ - 2}})}}} - \mathop{\int\mkern-20.8mu \circlearrowleft}\limits_{{{C}_{\zeta }}} {\frac{{{{\Phi }^{ + }}(\zeta )d\zeta }}{{({{\zeta }^{2}} - \xi _{0}^{2})}}} } \right],$$

(В.4)

where two auxiliary functions ${{\Phi }^{ - }}(\zeta )$ and ${{\Phi }^{ + }}(\zeta )$ are introduced (cf. Eq. (5.5)).

The integrand of the first integral (В.4) is regular within the unit circle ${\text{|}}\zeta {\text{|}} < 1$ everywhere except for the simple poles at points $\zeta = \pm \xi _{0}^{{{\kern 1pt} - 1}}$. At the same time, the integrand of the second integral (В.4) is regular outside the unit circle ${\text{|}}\zeta {\text{|}} > 1$ everywhere, except for the simple poles at points $\zeta = \pm {{\xi }_{0}}$, while at infinity it behaves as follows:

$$\zeta \to \infty {\text{:}}\quad \frac{{{{\Phi }^{ + }}(\zeta )}}{{({{\zeta }^{2}} - \xi _{0}^{2})}} = \left[ {\frac{{\gamma a_{1}^{4}({{\xi }_{0}} + \xi _{0}^{{{\kern 1pt} - 1}})}}{\pi } - \frac{{2a_{1}^{2}{{D}_{1}}}}{{{{\xi }_{0}}}}} \right]{{\zeta }^{{ - 1}}} + O({\text{|}}\zeta {{{\text{|}}}^{{ - 3}}}).$$

Then the contour integrals (В.4) can be calculated using theory of residues [14]

$$S = \gamma ({{\xi }_{0}} + \xi _{0}^{{ - 1}})\left[ {\sum\limits_{\zeta = \pm \xi _{0}^{{ - 1}}} {{\text{Res\;}}\frac{{{{\Phi }^{ - }}(\zeta )}}{{({{\zeta }^{2}} - \xi _{0}^{{ - 2}})}}} + \sum\limits_{\zeta = \pm {\kern 1pt} {{\xi }_{0}},\infty } {{\text{Res\;}}\frac{{{{\Phi }^{ + }}(\zeta )}}{{({{\zeta }^{2}} - \xi _{0}^{2})}}} } \right].$$

As a result, we obtain a closed expression for the bubble area S (cf. Eq. (5.6)).