Liquid sloshing in a circular cylindrical container containing a two-layer fluid

Abstract

In this work, a semi-analytical approach is employed to investigate sloshing in an immiscible, incompressible and inviscid two-layer fluid. We evaluate natural sloshing frequencies of the two-layer fluid of different layer-wise densities $\rho _1 < \rho _2$ with an interface and a surface in a vertical circular cylinder. On the basis of the velocity potential formulation of the fluid motion inside the container, an infinite system of homogeneous linear equations is obtained. We demonstrate the effects of the parameters such as fluid heights and density ratio on the natural sloshing frequencies. The results are supported by relevant graphs.

Appendix

The detailed expressions for the non-zero coefficients of the matrices given by (31) and (32) are obtained as follows:

$$\begin{aligned} a^{11}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \lambda _{mn} \sinh \lambda _{mn}\beta _{1}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(33)

$$\begin{aligned} a^{12}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}(1-\delta ^{1}_{m})\times \frac{\lambda _{mn}}{\cosh \lambda _{mn}\beta _{1}}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(34)

$$\begin{aligned} a^{22}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(35)

$$\begin{aligned} a^{23}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \tanh \lambda _{mn}\beta _{2}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(36)

$$\begin{aligned} a^{32}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \rho \times \lambda _{mn} \times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(37)

$$\begin{aligned} a^{33}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \lambda _{mn}\tanh \lambda _{mn}\beta _{2}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(38)

$$\begin{aligned} {\bar{a}}^{11}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \cosh \lambda _{mn}\beta _{1}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(39)

$$\begin{aligned} {\bar{a}}^{31}_{mn{\bar{n}}}= & {} \delta _{n{\bar{n}}}\times \rho \times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(40)

$$\begin{aligned} {\bar{a}}^{32}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \rho \times \tanh \lambda _{mn}\beta _{1}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi , \end{aligned}$$

(41)

$$\begin{aligned} {\bar{a}}^{33}_{mn{\bar{n}}}= & {} -\delta _{n{\bar{n}}}\times \int ^{1}_{0}\xi J^{2}_{m}(\lambda _{mn}\xi )d\xi . \end{aligned}$$

(42)