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Nonlinear sloshing of liquid in rigid cylindrical container with a rigid annular baffle: free vibration

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Abstract

In this paper, the free nonlinear sloshing of liquid in a rigid partially liquid-filled cylindrical container with a rigid annular baffle is investigated. The liquid domain is firstly divided into four simple sub-domains so that the liquid velocity potential in each sub-domain has continuous boundary conditions of class \(\hbox {C}^{1}\). Then, based on the Bateman–Luke variational principle, the equivalent variational formulation of the free boundary problem for the liquid in a baffled container is derived. By introducing the generalized time-dependent coordinates, the surface wave height and the velocity potential are expanded in the form of generalized Fourier series. Substituting the series expression of surface wave height and velocity potential into the variational formulation enables us to obtain the infinite dimensional modal system. Finally, based on the Moiseev asymptotic relations, the infinite dimensional modal system is reduced to a finite dimensional modal system. Good agreements have been achieved by comparing the present results with those obtained from the Gavrilyuk’s solutions. The effects of baffle parameters and initial conditions on the nonlinear sloshing of liquid are discussed in detail.

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Acknowledgments

The financial support from National Natural Science Foundation of China, Grant No. 11172123 is gratefully acknowledged.

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Authors and Affiliations

  1. College of Civil Engineering, Nanjing Tech University, Nanjing, 210009, People’s Republic of China

    D. Zhou, J. D. Wang & W. Q. Liu

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  1. D. Zhou
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  2. J. D. Wang
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  3. W. Q. Liu
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Correspondence to D. Zhou.

Appendices

Appendix 1: Coefficients \(E_{ij}\,(i=1-5, j= 1-7)\) in Eq. (74)

$$\begin{aligned}&E_{11} =E_{21} =\pi \int _0^{r_{2}} {\left. {\varPhi _{11}^{2}} \right| _{z=z_{2}} r\hbox {d}r},\\&E_{12} =E_{22} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{01} \frac{\partial \varPhi _{11}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{13} =E_{23} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{21} \frac{\partial \varPhi _{11}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r} , \\&E_{14} =E_{24} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{3} \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{15} =E_{25} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{01}^{2} \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r} , \\&E_{16} =E_{26} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{21}^{2} \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r},\\&E_{17} =E_{27} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{01} \varPhi _{21} \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r} , \\&E_{31} =\pi \int _0^{r_{2}} {\left. {\varPhi _{01}^{2}}\right| _{z=z_{2}} r\hbox {d}r},\\&E_{32} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{01}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{33} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{01}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{34} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{01}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r}, \\&E_{35} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \varPhi _{01} \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{36} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \varPhi _{21} \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r} , \\&E_{37} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{3} \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{38} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01} \varPhi _{21}^{2} \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r}, \\&E_{41} =E_{51} =\pi \int _0^{r_{2}} {\left. {\varPhi _{21}^{2}} \right| _{z=z_{2}} r\hbox {d}r},\\&E_{42} =E_{52} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{21}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r} , \\&E_{43} =E_{53} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01} \varPhi _{21} \frac{\partial \varPhi _{21}}{\partial z}}\right) }\right| _{z=z_{2}} r\hbox {d}r},\\&E_{44} =E_{54} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \varPhi _{21} \frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r}, \end{aligned}$$
$$\begin{aligned}&E_{45} =E_{55} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \varPhi _{01} \frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r},\\&E_{46} =E_{56} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \varPhi _{21} \frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}}\right) } \right| _{z=z_{2}} r\hbox {d}r}, \\&E_{47} =E_{57} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{3} \frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}}\right) }\right| _{z=z_{2}} r\hbox {d}r}. \end{aligned}$$

Appendix 2: Coefficients \(H_{i}\,(i=1-28)\) in Eq. (75)

$$\begin{aligned}&H_{1} =\pi \left\{ \int _0^{r_{2}} {\left. {\left[ {\varPhi _{01} \left( {\frac{\partial \varPhi _{11}}{\partial z}}\right) ^{2}}\right] } \right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad +\int _0^{r_{2}} \left. {\left( {\frac{\varPhi _{01} \varPhi _{11}^{2}}{r}}\right) }\right| _{z=z_{2}} \hbox {d}r\\&\quad \left. +\int _0^{r_{2}} {\left. {\left[ {\varPhi _{01} \left( {\frac{\partial \varPhi _{11}}{\partial r}}\right) ^{2}}\right] }\right| }_{z=z_{2}} r\hbox {d}r \right\} ,\\&H_{2} =\frac{\pi }{2}\left\{ \int _0^{r_{2}} {\left. {\left[ {\varPhi _{21} \left( {\frac{\partial \varPhi _{11}}{\partial z}}\right) ^{2}}\right] } \right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad -\int _0^{r_{2}} {\left. {\frac{\varPhi _{11}^{2} \varPhi _{21}}{r}}\right| _{z=z_{2}}} \hbox {d}r\\&\quad \left. +\int _0^{r_{2}} {\left. {\left[ {\varPhi _{21} \left( {\frac{\partial \varPhi _{11}}{\partial r}} \right) ^{2}}\right] }\right| _{z=z_{2}}} r\hbox {d}r \right\} ,\\&H_{3} =\frac{\pi }{4}\left[ 3\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad + \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{3} \frac{\partial \varPhi _{11}}{r\partial z}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&\quad \left. + 3\int _0^{r_{2}} {\left. {\left( {\frac{\partial \varPhi _{11} }{\partial r}\frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} \varPhi _{11}^{2} r\hbox {d}r\right] ,\\&H_{4} =\frac{\pi }{4}\left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{11} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad + 3\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{3} \frac{\partial \varPhi _{11}}{r\partial z}} \right) }\right| _{z=z_{2}}} \hbox {d}r \end{aligned}$$
$$\begin{aligned}&\quad \left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{5} =\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{11} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad +\int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \varPhi _{11} \frac{\partial \varPhi _{11} }{r\partial z}}\right) }\right| _{z=z_{2}} \hbox {d}r} \\&\quad \left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{6} =\frac{\pi }{2}\left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{11} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{21}^{2} \frac{\partial \varPhi _{11} }{r\partial z}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&\quad \left. +\int _0^{r_{2} } {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{7} =\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{01} \varPhi _{21} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{11} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad -\int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \varPhi _{01} \varPhi _{21} \frac{\partial \varPhi _{11} }{r\partial z}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&\quad \left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{01} \varPhi _{21} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{8} =\frac{\pi }{2}\left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}\frac{\partial \varPhi _{11} }{\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad -\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{3} \frac{\partial \varPhi _{11}}{r\partial z}} \right) }\right| _{z=z_{2}}} \hbox {d}r \end{aligned}$$
$$\begin{aligned}&\quad \left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}\frac{\partial \varPhi _{11}}{\partial r}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{9} =\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial \varPhi _{01}}{\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\quad \left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial \varPhi _{01} }{\partial r}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{10} =\pi \left\{ \int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{01} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}\frac{\partial \varPhi _{01}}{\partial z}\right. \right. \right. \\&\left. \left. \left. +\frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}\right) \right] \right| _{z=h} r\hbox {d}r+\int _0^{r_{2}}\\&\left. {\left. {\left[ {\varPhi _{11} \varPhi _{01} \left( {\frac{\partial ^{2}\varPhi _{11}}{\partial z\partial r}\frac{\partial \varPhi _{01}}{\partial r}+\frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{01}}{\partial z\partial r}}\right) }\right] } \right| _{z=h}} r\hbox {d}r \right\} ,\\&H_{11} =\frac{\pi }{2}\left\{ \int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{21} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}\frac{\partial \varPhi _{01}}{\partial z}\right. \right. \right. \\&\left. \left. \left. +\frac{\partial \varPhi _{11} }{\partial z}\frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}\right) \right] \right| _{z=z_{2}} r\hbox {d}r\\&+\int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{21} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial z\partial r}\frac{\partial \varPhi _{01}}{\partial r}\right. \right. \\&\left. \left. \left. \left. +\frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{01}}{\partial z\partial r} \right) \right] \right| _{z=z_{2}} r\hbox {d}r \right\} ,\\&H_{12} =\frac{\pi }{2}\left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \frac{\partial \varPhi _{11}}{\partial z}\frac{\partial \varPhi _{21}}{\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&+\, 2\int _0^{r_{2}} {\left. {\left( {\frac{\varPhi _{11}^{2} \varPhi _{21}}{r}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&\left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11} \frac{\partial \varPhi _{11}}{\partial r}\frac{\partial \varPhi _{21}}{\partial r}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{13} =\frac{\pi }{2}\int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{01} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}\frac{\partial \varPhi _{21}}{\partial z}\right. \right. \\&\left. \left. \left. \quad +\frac{\partial \varPhi _{11}}{\partial z}\frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}\right) \right] \right| _{z=z_{2}} r\hbox {d}r\\&\qquad \!\!+\pi \int _0^{r_{2}} \left[ \frac{\varPhi _{11} \varPhi _{01}}{r}\left( \varPhi _{21} \frac{\partial \varPhi _{11}}{\partial z}\right. \right. \end{aligned}$$
$$\begin{aligned}&\left. \left. \left. \quad +\varPhi _{1} \frac{\partial \varPhi _{21}}{\partial z}\right) \right] \right| _{z=z_{2}} \hbox {d}r\\&\qquad \!\!+\frac{\pi }{2}\int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{01} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}\frac{\partial \varPhi _{21}}{\partial r}\right. \right. \\&\left. \left. \left. \quad +\frac{\partial \varPhi _{11}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}\right) \right] \right| _{z=z_{2}} r\hbox {d}r,\\&H_{14} =\frac{\pi }{2}\left\{ \int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{21} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial z^{2}}\frac{\partial \varPhi _{21}}{\partial z}\right. \right. \right. \\&\left. \left. \left. \quad +\frac{\partial \varPhi _{11} }{\partial z}\frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}\right) \right] \right| _{z=h} r\hbox {d}r\\&\quad \,\,+\int _0^{r_{2}} \left[ \varPhi _{11} \varPhi _{21} \left( \frac{\partial ^{2}\varPhi _{11}}{\partial r\partial z}\frac{\partial \varPhi _{21}}{\partial r}\right. \right. \\&\left. \left. \left. \left. \quad +\frac{\partial \varPhi _{11} }{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}\right) \right] \right| _{z=h} r\hbox {d}r \right\} , \end{aligned}$$
$$\begin{aligned}&H_{15} =\pi \int _0^{r_{2}} \left( \varPhi _{21}^{2} \varPhi _{11} \frac{\partial \varPhi _{11}}{r\partial z}\right. \\&\left. \left. +\, \varPhi _{11}^{2} \varPhi _{21} \frac{\partial \varPhi _{21}}{r\partial z}\right) \right| _{z=z_{2}} \hbox {d}r,\\&H_{16} =2\pi \left\{ \int _0^{r_{2}} \left. \left[ \varPhi _{01} \left( {\frac{\partial \varPhi _{01}}{\partial z}}\right) ^{2}\right] \right| _{z=h_{2}} r\hbox {d}r\right. \\&\left. +\int _0^{r_{2}} {\left. {\left[ {\varPhi _{01} \left( {\frac{\partial \varPhi _{01}}{\partial r}}\right) }\right] ^{2}} \right| _{z=h_{2}}} r\hbox {d}r \right\} ,\\&H_{17} =\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{01}}{\partial z}\frac{\partial ^{2}\varPhi _{01} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{01}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{18} =2\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{01} }{\partial r\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{01}}{\partial r\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{19} =\pi \left[ \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{01}}{\partial z}\frac{\partial ^{2}\varPhi _{01} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\right. \\&\left. +\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{01}}{\partial r\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\right] ,\\&H_{20} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21} \frac{\partial \varPhi _{01}}{\partial z}\frac{\partial \varPhi _{21}}{\partial z}}\right) } \right| _{z=z_{2}}} r\hbox {d}r\\&+\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial \varPhi _{21} }{\partial r}}\right) }\right| _{z=z_{2}}} r\hbox {d}r,\\&H_{21} =\frac{\pi }{4}\left[ \int _0^{r_{2}} \left( \varPhi _{11}^{2} \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}\frac{\partial \varPhi _{21}}{\partial z}\right. \right. \\&\left. \left. + \varPhi _{11}^{2} \frac{\partial \varPhi _{01}}{\partial z}\frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}\right) \right| _{z=z_{2} } r\hbox {d}r\\&+ \int _0^{r_{2}} \left( \varPhi _{11}^{2} \frac{\partial ^{2}\varPhi _{01}}{\partial r\partial z}\frac{\partial \varPhi _{21}}{\partial r}\right. \\&\left. \left. \left. +\varPhi _{11}^{2} \frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}\right) \right| _{z=z_{2}} r\hbox {d}r\right] ,\\&H_{22} =\pi \int _0^{r_{2}} \left[ \varPhi _{01} \varPhi _{21} \left( \frac{\partial ^{2}\varPhi _{01}}{\partial z^{2}}\frac{\partial \varPhi _{21} }{\partial z}\right. \right. \end{aligned}$$
$$\begin{aligned}&\left. \left. \left. +\frac{\partial \varPhi _{01}}{\partial z}\frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}\right) \right] \right| _{z=z_{2}} r\hbox {d}r+\pi \int _0^{r_{2}}\\&{\left. {\left[ {\varPhi _{01} \varPhi _{21} \left( {\frac{\partial ^{2}\varPhi _{01}}{\partial r\partial z}\frac{\partial \varPhi _{21}}{\partial r}+\frac{\partial \varPhi _{01}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}}\right) }\right] }\right| _{z=z_{2}} } r\hbox {d}r,\\&H_{23} =\pi \int _0^{r_{2}} {\left. {\left[ {\varPhi _{01} \left( {\frac{\partial \varPhi _{21}}{\partial z}}\right) ^{2}}\right] } \right| _{z=z_{2}}} r\hbox {dr}\\&+\, 4\pi \int _0^{r_{2}} {\left. {\left( {\frac{\varPhi _{01} \varPhi _{21}^{2}}{r}}\right) }\right| _{z=z_{2}}} \hbox {dr}\\&+\, \pi \int _0^{r_{2}} {\left. {\left[ {\varPhi _{01} \left( {\frac{\partial \varPhi _{21}}{\partial r}}\right) ^{2}}\right] } \right| _{z=z_{2}}} r\hbox {dr},\\&H_{24} =\frac{\pi }{2}\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{21}}{\partial z}\frac{\partial ^{2}\varPhi _{21} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\\&+ 2\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \varPhi _{21} \frac{\partial \varPhi _{21}}{\partial z}}\right) }\right| _{z=z_{2}}} \frac{\hbox {d}r}{r}\\&+ \frac{\pi }{2}\int _0^{r_{2}} {\left. {\left( {\varPhi _{11}^{2} \frac{\partial \varPhi _{21}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r,\\&H_{25} =\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{21}}{\partial z}\frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}} \right) }\right| _{z=z_{2}}} r\hbox {d}r\\&+ 4\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \varPhi _{21} \frac{\partial \varPhi _{21}}{r\partial z}} \right) }\right| _{z=z_{2}}} \hbox {d}r\\&+ \pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{01}^{2} \frac{\partial \varPhi _{21}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r,\\&H_{26} =\frac{3\pi }{4}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{21}}{\partial z}\frac{\partial ^{2}\varPhi _{21} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\\&+ \pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{3} \frac{\partial \varPhi _{21} }{r\partial z}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&+ \frac{3\pi }{4}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{21}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}} \right) }\right| _{z=z_{2}}} r\hbox {d}r,\\&H_{27} =\frac{\pi }{4}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{21}}{\partial z}\frac{\partial ^{2}\varPhi _{21} }{\partial z^{2}}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\\&+ 3\pi \int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{3} \frac{\partial \varPhi _{21} }{r\partial z}}\right) }\right| _{z=z_{2}}} \hbox {d}r\\&+ \frac{\pi }{4}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial \varPhi _{21}}{\partial r}\frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}} \right) }\right| _{z=z_{2}}} r\hbox {d}r, \end{aligned}$$
$$\begin{aligned}&H_{28} =\frac{\pi }{2}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial ^{2}\varPhi _{21}}{\partial z^{2}}\frac{\partial \varPhi _{21} }{\partial z}}\right) }\right| _{z=z_{2}}} r\hbox {d}r\\&- 2\pi \int _0^{r_{2}} \left. {\left( {\varPhi _{21}^{3} \frac{\partial \varPhi _{21}}{r\partial z}} \right) }\right| _{z=z_{2}} \hbox {d}r\\&+ \frac{\pi }{2}\int _0^{r_{2}} {\left. {\left( {\varPhi _{21}^{2} \frac{\partial ^{2}\varPhi _{21}}{\partial r\partial z}\frac{\partial \varPhi _{21}}{\partial r}}\right) } \right| _{z=z_{2}}} r\hbox {d}r. \end{aligned}$$

Appendix 3: Coefficients \(D_{i}\,(i=1-12)\) in Eq. (77)

$$\begin{aligned}&D_{1} =\frac{E_{11}}{H_{a}}, \quad D_{2} =\frac{E_{12} H_{a} -E_{11} H_{1}}{H_{a}^{2}}, \\&D_{3} =\frac{E_{12} H_{b} -E_{31} H_{9}}{H_{a} H_{b}}, \quad D_{4} =\frac{E_{13} H_{a} -2E_{11} H_{2}}{2H_{a}^{2}},\\&vD_{6} =\frac{4E_{11} H_{9}^{2} H_{c} -4E_{32} H_{\hbox {9}} H_{a} H_{c} \hbox {+8}E_{11} H_{\hbox {12}}^{2} H_{b} -4E_{42} H_{12} H_{a} H_{b} -8E_{11} H_{3} H_{b} H_{c} +3E_{14} H_{a} H_{b} H_{c}}{8H_{a}^{2} H_{b} H_{c}},\\&D_{5} =\frac{E_{13} H_{b} -2E_{41} H}{2H_{a} H_{c}}, \\&D_{7} =\frac{8E_{11} H_{12}^{2} -4E_{42} H_{12} H_{a} -8E_{11} H_{4} H_{c} +E_{14} H_{a} H_{c}}{8H_{a}^{2} H_{c}},\\&D_{8} =\frac{2E_{11} H_{9}^{2} -2E_{32} H_{9} H_{a} -4E_{11} H_{8} H_{b} +E_{14} H_{a} H_{b}}{4H_{a}^{2} H_{b}}, \\&D_{9} =\frac{E_{31}}{H_{b}},\\&D_{10} =\frac{E_{32} H_{a} -E_{11} H_{9}}{2H_{a} H_{b}}, \quad D_{11} =\frac{E_{41}}{H_{c}}, \\&D_{12} =\frac{E_{42} H_{a} -2E_{11} H_{12}}{2H_{a} H_{c}}. \end{aligned}$$

Appendix 4: Coefficients \(K_{i}\,(i =1-10)\) in Eq. (78)

$$\begin{aligned}&K_{1} =\frac{2H_{3} D_{1}^{2} +4E_{11} D_{6} +2E_{32} D_{10} +E_{42} D_{12} +2H_{9} D_{1} D_{9} +2H_{12} D_{1} D_{12}}{2E_{11} D_{1}},\\&K_{2} =\frac{E_{14}D_{1} +8E_{11}D_{7} +4E_{42}D_{12}}{8E_{11} D_{1}}, \\& K_{3} =\frac{E_{11}D_{3} +E_{32}D_{9}}{E_{11} D_{1}}, K_{4} =\frac{E_{13}D_{1} +2E_{11}D_{4}}{2E_{11} D_{1}},\\&K_{5} =\frac{E_{11}D_{2} +E_{12}D_{1}}{E_{11} D_{1}}, \\&K_{6} =\frac{E_{13}D_{1} +2E_{11}D_{4}}{2E_{11} D_{1}}, \quad K_{7} =-\frac{H_{2}D_{1}^{2} +2E_{41}D_{12}}{2E_{41} D_{11}},\\&K_{8} =-\frac{H_{1}D_{1}^{2} +4E_{31}D_{10}}{4E_{31} D_{9}},\\&K_{9} =-\frac{E_{13}D_{1} +2E_{41}D_{12}}{2E_{41} D_{11}}, \quad K_{10} =-\frac{E_{12}D_{1} +2E_{31}D_{10}}{2E_{31} D_{9}}. \end{aligned}$$

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Zhou, D., Wang, J.D. & Liu, W.Q. Nonlinear sloshing of liquid in rigid cylindrical container with a rigid annular baffle: free vibration. Nonlinear Dyn 78, 2557–2576 (2014). https://doi.org/10.1007/s11071-014-1610-z

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