hand, the Casimir pressure is $p_R\approx-1\times10^{-12}$ N/m$^2$. It is interesting to note that confinement of TE modes dominates that of TM modes, and that $N^{(2)}$ per cross-sectional area of a square-cylinder ($-0.0429968$ [@Milton2010]) is close to that of the circular-cylinder in Eq.  with deviation less than $4\%$. This brings a hope that the Casimir effect per cross-sectional area of circular-cylinder would be close to that of a square-cylinder as both are topologically same, and as the later specially is having 4-fold symmetry.

Casimir effect for a BEC inside the circular-cylinder {#Sec3}
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The Casimir force on an interacting Bose-Einstein condensate was studied by Biswas *et al* in Ref. \[\] for a film geometry. Here we generalize the same (in a nontrivial manner) for an interacting BEC confined to a cylindrical geometry.

Elementary excitations in the BEC inside the cylindrical tube
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Let us consider a Bose gas of $N$ identical particles be kept inside the same cylinder of radius $R$ and length $L$, and take the same set of cylindrical coordinates ($\rho,\theta,z$) to specify position $\textbf{r}$ as in the previous section. For $T\rightarrow 0$, almost all the particles ($N_0\approx N$) form the condensate, and the elementary (phononic as well as 