[@Milton2010] for the higher orders in condensate density. Finally, we summarize our results in the conclusion.

Casimir effect on the hollow cylinder {#Sec2}
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Before discussing the Casimir effect for a Bose-Einstein condensate enclosed in a circular-cylinder geometry, let us review the conventional Casimir effect on a hollow circular-cylinder. This will also enable us to establish the condensate-density dependence on the Casimir force, and the validity of our approximations later. Let us consider an infinitely long hollow circular-cylinder of radius $R$ and length $L$, whose axis is along the $z$-direction, and whose radial and angular coordinates are represented by $(\rho, \theta)$. The zero point or vacuum energy of the electromagnetic field inside the cylinder is given by [@Casimir] $$\begin{aligned}
\label{eqn:11}
E=\frac{1}{2}\sum_{\bm{k}}\hbar\omega_{\bm{k}}~,\end{aligned}$$ $\bm{k}$ represents all possible (TM and TE) modes of the vacuum fluctuations compatible to the boundary conditions imposed by the waveguide *i.e.* the circular-cylindrical conductor. While in free space or in the space between two parallel conductors, the two polarization states of the electromagnetic field are equivalent; in the space inside a cylindrical waveguide, the two polarization states known as transverse magnetic (TM) and transverse electric (TE) modes are not equivalent. For the conducting cylindrical shell surface, the TM 