Let bubble of radius $x$ expands and pushes water of density $\rho$ by a distance $dx$.

Then, kinetic energy of water of strip is $dx$ is 

$dK=\frac{1}{2}dm·v_x^2\Rightarrow dK=\frac{1}{2}4\pi x^2·dx·\rho ·v_x^2$

or $dK=2\pi R^2·\rho ·v_x^2·dx$   $...\left(\mathrm{i}\right)$

Considering laminar flow, for section $1$ and $2$ shown in figure, by equation of continuity

$A_1{v}_1=A_2{v}_2\Rightarrow 4\pi x^2{v}_x=4\pi R^{2}v$

$\Rightarrow v_x=\frac{R^2}{x^2}.v$  $...\left(\mathrm{iii}\right)$

From Eqs. $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we have

$dK=2\pi x^2\rho {\left(\frac{R^2}{x^2}·v·\right)}^{2}dx$

$\Rightarrow dK=2\pi R^4\rho v^2.\left(\frac{dx}{x^2}\right)$

So, kinetic energy of complete volume of

water, 

$K={\int }_R^{\infty }dK=2\pi R^4\rho v^2{\int }_R^{\infty }\frac{dx}{x^2}$

$\Rightarrow K=2\pi R^4·\rho ·v{\left[-\frac{1}{x}\right]}_R^{\infty }$

$=2\pi \rho R^3{v}^2$