Consider the shape a Liquid drop assumes when placed on a smooth horizontal surface, as depicted in Fig. $3.$ The liquid drop's shape will be axisymmetric about the vertical $u-$axis and is such that the total energy of the drop is minimized. The two forms of energy associated with the liquid drop are:

Potential energy, arising from the gravitational field, given by:

$E_p={\displaystyle \underset{0}{\overset{R}{}}}[g{r}^{2}u(r)u^{\text{'}}(r)]dr$

Surface energy, caused by surface tension at the liquid$-$gas interface, expressed as:

$E_s={\displaystyle \underset{0}{\overset{R}{}}}[2r\sqrt[\phantom{2}]{1+[u^{\text{'}}(r){]}^2}]dr$

$2$

Here, $$ is the density of the liquid, $g$ is the acceleration due to gravity; $u(r)$ is the vertical distance to the interface from the surface, $$ is the surface tension $($surface energy per unit area$),$ and $R$ is the base radius of the drop.

The objective is to determine the axisymmetric drop shape $u(r)$ that minimizes the total energy $E[u(r)],$ subject to a constraint. The constraint is that the total volume $V$ of the liquid drop is fixed and given by:

$V={\displaystyle \underset{0}{\overset{R}{}}}[r^{2}u(r)]dr$

Additionally, the problem involves the following boundary conditions for $u(r)$;

$u(R)=0,u^{\text{'}}(0)=0,$ and $,u^{\text{'}}(R)=-tana$

where $$ is the contact angle.

Note that while only two boundary conditions are typically required, here, a third boundary condition is provided. This additional boundary condition can be used to determine the unknown base radius $R$ of the bubble.

Derive the Euler equation that must be satisfied to minimize the total energy of the liquid drop, considering the given constraints and boundary conditions.