4.1 Formulation of the axisymmetric exit problem and its solution

The problem of water exit is formulated in cylindrical coordinates $(r,z)$ , where $r$ is the radial coordinate and $z$ is the vertical one, which coincides with the symmetry axis of the plate. The plane $z=0$ corresponds to the initial position of both the lower surface of the disc and the liquid surface, with the liquid occupying the lower half-space, $z<0$ . The disc touches the water surface with zero draft. Thus the initial position of the lower surface of the disc is $z=0$ , $r<R$ . Then the disc is moved suddenly upwards by an external force applied at the disc centre. The position of the lower surface of the moving disc is described by the equation $z=w(r,t)$ , where $w(r,t)$ accounts for both the rigid and elastic components of the disc displacement. During the early stage, when displacements are small compared to the disc radius $R$ , the flow equations and the boundary conditions can be linearised. In particular, this means that the boundary conditions can be imposed on the initial position of the liquid boundary.

Under the assumptions of the early stage, the flow caused by lifting an elastic disc from the water surface is described by a velocity potential, $\unicode[STIX]{x1D711}(r,z,t)$ , which satisfies Laplace equation,

(4.1) $$\begin{eqnarray}\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}r}\right)+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z^{2}}=0\quad (z<0),\end{eqnarray}$$

in the initial flow region, the linearised dynamic condition on the free surface,

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D711}=0\quad (z=0,~r>R),\end{eqnarray}$$

and the body boundary condition on the disc,

(4.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}z}=\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}t}\quad (z=0,~r<R).\end{eqnarray}$$

Furthermore, this potential decays at infinity, as $r^{2}+z^{2}\rightarrow \infty$ .

The disc displacement, $w(r,t)$ , is governed by the Bernoulli–Euler equation of thin elastic plates,

(4.4) $$\begin{eqnarray}m\frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}t^{2}}+D\unicode[STIX]{x1D6FB}^{4}w=p_{ext}(r,t)+p_{h}(r,0,t)\quad (r<R,~t>0),\end{eqnarray}$$

where $m=\unicode[STIX]{x1D70C}_{p}h_{p}$ is the mass of the disc per unit area, and $D=Eh_{p}^{3}/[12(1-\unicode[STIX]{x1D708}^{2})]$ is the rigidity coefficient for an elastic disc of constant thickness (see table 1 for the elastic constants of the disc material). The hydrodynamic pressure, $p_{h}(r,0,t)$ , acting on the disc–water interface, $z=0$ , is given by the linearised Bernoulli equation,

(4.5) $$\begin{eqnarray}p(r,0,t)=-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D711}_{t}(r,0,t),\end{eqnarray}$$

whereas $p_{ext}(r,t)$ is the external load caused by the driving mechanism (see figure 1) and acting on the disc through the load cell and the connector. The external load is assumed to be uniformly distributed over a small circular area of radius $r_{c}$ , $p_{ext}(r,t)=P(t)$ , where $r<r_{c}$ , with $F_{ext}(t)=\unicode[STIX]{x03C0}r_{c}^{2}P(t)$ being the total force acting on the disc. Later, we will simplify the analysis by letting $r_{c}\rightarrow 0$ . The displacement $w(r,t)$ is positive where the disc moves upwards. Note that different parts of the disc can move in different directions. For example, the centre of the disc may move upwards but the edge of the disc moves downwards at the same time. The boundary-value problem (4.1)–(4.5) is similar to that studied by Scolan (Reference Scolan2004) for water entry of an elastic conical shell.

The connector is not axisymmetric in experiments. Therefore, the external load and the disc deflection should depend on the azimuthal coordinate, strictly speaking. These azimuthal modes are expected to introduce a high-frequency jitter in the measured acceleration curves, although its amplitude must be small. Indeed, to trigger the non-axisymmetric modes in an efficient way, the force should be applied far from the disc centre, which is not the case here.

In the experiments, the weight of the connector, $M_{c}$ , which is placed in between the load cell and the disc, is comparable with the weight, $M_{p}=\unicode[STIX]{x03C0}R^{2}m$ , of the disc. To account for this extra weight, Newton’s second law for the connector,

(4.6) $$\begin{eqnarray}M_{c}\frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}t^{2}}(0,t)=F_{exp}(t)-F_{ext}(t),\end{eqnarray}$$

is used, where $F_{exp}(t)$ is the force measured by the load cell, and $-F_{ext}(t)$ is the force acting on the connector from the disc. We assume that the connector is rigidly connected to the centre of the disc, and thus the displacement of the connector is approximately $w(0,t)$ . Then the external load can be approximated by

(4.7) $$\begin{eqnarray}p_{ext}(r,t)=\left(F_{exp}(t)-M_{c}\frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}t^{2}}(0,t)\right)\frac{\unicode[STIX]{x1D6FF}(r)}{2\unicode[STIX]{x03C0}r},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}(r)$ is the Dirac delta function. In the present model, the force $F_{exp}(t)$ is assumed known from experimental measurements.

The disc edge, $r=R$ , is free of stresses and shear forces. The radial bending moment and the Kelvin–Kirchhoff edge reaction are zero at the edge. For axisymmetric deflections of a circular plate, these two conditions read

(4.8a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}r^{2}}+\frac{\unicode[STIX]{x1D708}}{r}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}r}=0,\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}r}\right)\right)=0\quad (r=R).\end{eqnarray}$$

Initially, $t=0$ , both the disc and the water are at rest,

(4.9a-c ) $$\begin{eqnarray}w(r,0)=0,\quad w_{t}(r,0)=0,\quad \unicode[STIX]{x1D711}(r,z,0)=0.\end{eqnarray}$$

The problem formulated by (4.1)–(4.9) is coupled: the hydrodynamic loads and the disc displacement should be determined at the same time. The problem is solved by the normal-mode method (Khabakhpasheva & Korobkin Reference Khabakhpasheva and Korobkin1998; Korobkin Reference Korobkin2000; Scolan Reference Scolan2004; Khabakhpasheva Reference Khabakhpasheva2006; Khabakhpasheva, Korobkin & Malenica Reference Khabakhpasheva, Korobkin and Malenica2013). Within this method the disc displacement is sought in the form

(4.10) $$\begin{eqnarray}w(r,t)=h(t)+\mathop{\sum }_{n=1}^{\infty }a_{n}(t)W_{n}(\tilde{r}),\end{eqnarray}$$

where $\tilde{r}=r/R$ is the non-dimensional radial coordinate, $\tilde{r}<1$ , $a_{n}(t)$ are the principal coordinates of the elastic deflections of the disc, which are to be determined, and $h(t)$ is the rigid-body displacement of the disc. The functions $W_{n}(\tilde{r})$ are the non-trivial bounded solutions to the homogeneous boundary-value problem

(4.11) $$\begin{eqnarray}\displaystyle & \tilde{\unicode[STIX]{x1D6FB}}^{4}W_{n}=k_{n}^{4}W_{n}\quad (\tilde{r}<1), & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & W_{n}^{\prime \prime }+\unicode[STIX]{x1D708}W_{n}^{\prime }=0,\quad (\tilde{\unicode[STIX]{x1D6FB}}^{2}W_{n})^{\prime }=0\quad (\tilde{r}=1), & \displaystyle\end{eqnarray}$$

where

(4.13) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FB}}^{2}W_{n}=\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{r}}\left(\tilde{r}\frac{\unicode[STIX]{x2202}W_{n}}{\unicode[STIX]{x2202}\tilde{r}}\right),\end{eqnarray}$$

a prime stands for the derivative in $\tilde{r}$ , and $k_{n}$ are the corresponding eigenvalues. The functions $W_{n}(\tilde{r})$ , known as normal modes of the free–free circular elastic disc, describe the axisymmetric shapes of free vibrations of a circular disc with its edge being free of forces and bending stresses, and with frequencies proportional to $k_{n}^{2}$ (Leissa Reference Leissa1969). The normal modes are orthogonal and read

(4.14) $$\begin{eqnarray}W_{n}(\tilde{r})=A_{n}\left(\text{J}_{0}(k_{n}\tilde{r})-\frac{\text{J}_{1}(k_{n})}{\text{I}_{1}(k_{n})}\text{I}_{0}(k_{n}\tilde{r})\right),\end{eqnarray}$$

where $\text{J}_{n}$ is the Bessel function and $\text{I}_{n}$ the modified Bessel function, both of the first kind. Moreover, $k_{n}$ , $n\geqslant 1$ , are the positive solutions of the equation

(4.15) $$\begin{eqnarray}\frac{\text{J}_{1}(k_{n})}{\text{J}_{0}(k_{n})}+\frac{\text{I}_{1}(k_{n})}{\text{I}_{0}(k_{n})}=\frac{2(1-\unicode[STIX]{x1D708})}{k_{n}},\end{eqnarray}$$

and the coefficients $A_{n}$ are determined by the normalisation condition,

(4.16) $$\begin{eqnarray}\int _{0}^{1}W_{n}(\tilde{r})W_{m}(\tilde{r})\tilde{r}\,\text{d}\tilde{r}=\unicode[STIX]{x1D6FF}_{nm},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FF}_{nm}=0$ for $n\neq m$ and $\unicode[STIX]{x1D6FF}_{nn}=1$ . Equations (4.14)–(4.16) give

(4.17) $$\begin{eqnarray}A_{n}=\left(\text{J}_{0}^{2}(k_{n})-\frac{2\unicode[STIX]{x1D708}(1-\unicode[STIX]{x1D708})}{k_{n}^{2}}\text{J}_{1}^{2}(k_{n})-\frac{2(1-\unicode[STIX]{x1D708})}{k_{n}}\text{J}_{0}(k_{n})\text{J}_{1}(k_{n})\right)^{-1/2}.\end{eqnarray}$$

Note that

(4.18) $$\begin{eqnarray}\int _{0}^{1}W_{n}(\tilde{r})\tilde{r}\,\text{d}\tilde{r}=0,\end{eqnarray}$$

which means that the elastic modes with $n\geqslant 1$ are orthogonal to the rigid mode, $W_{0}(\tilde{r})=\sqrt{2}$ , which corresponds to $k_{0}=0$ .

The solution of the hydrodynamic problem (4.1)–(4.3) is given by

(4.19a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D711}(R\tilde{r},0,t)=\int _{\tilde{r}}^{1}\frac{\unicode[STIX]{x1D712}(\unicode[STIX]{x1D707},t)\,\text{d}\unicode[STIX]{x1D707}}{\sqrt{\unicode[STIX]{x1D707}^{2}-\tilde{r}^{2}}},\quad \unicode[STIX]{x1D712}(\unicode[STIX]{x1D707},t)=\frac{2R}{\unicode[STIX]{x03C0}}\int _{0}^{\unicode[STIX]{x1D707}}\frac{w_{t}(R\unicode[STIX]{x1D70E},t)\unicode[STIX]{x1D70E}\,\text{d}\unicode[STIX]{x1D70E}}{\sqrt{\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D70E}^{2}}}\end{eqnarray}$$

(see appendix A in Korobkin & Scolan (Reference Korobkin and Scolan2006)). It is convenient to introduce the functions $Q_{n}(x)$ and the coefficients $W_{nk}$ by

(4.20a,b ) $$\begin{eqnarray}Q_{n}(x)=\frac{1}{x}\int _{0}^{x}\frac{W_{n}(\unicode[STIX]{x1D70E})\unicode[STIX]{x1D70E}\,\text{d}\unicode[STIX]{x1D70E}}{\sqrt{x^{2}-\unicode[STIX]{x1D70E}^{2}}},\quad W_{nk}=\frac{2}{\unicode[STIX]{x03C0}}\int _{0}^{1}x^{2}Q_{n}(x)Q_{k}(x)\,\text{d}x\end{eqnarray}$$

(see Scolan Reference Scolan2004; Pegg, Purvis & Korobkin Reference Pegg, Purvis and Korobkin2018). In particular, $Q_{0}(x)=\sqrt{2}$ . The functions $Q_{n}(x)$ and the coefficients $W_{nk}$ are expressed through the Bessel, trigonometric and hyperbolic functions similar to those in appendices B and C in Pegg et al. (Reference Pegg, Purvis and Korobkin2018), where the corresponding integrals were evaluated for a simply supported circular elastic disc.

Substituting (4.10) in (4.19) and using (4.20) we find the velocity potential on the disc surface,

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}(\unicode[STIX]{x1D707},t)=\frac{2R}{\unicode[STIX]{x03C0}}\unicode[STIX]{x1D707}\left(h^{\prime }(t)+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime }(t)Q_{n}(\unicode[STIX]{x1D707})\right), & \displaystyle\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D711}(R\tilde{r},0,t)=\frac{2R}{\unicode[STIX]{x03C0}}\left(h^{\prime }(t)\sqrt{1-\tilde{r}^{2}}+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime }(t)\int _{\tilde{r}}^{1}\frac{\unicode[STIX]{x1D707}Q_{n}(\unicode[STIX]{x1D707})\text{d}\unicode[STIX]{x1D707}}{\sqrt{\unicode[STIX]{x1D707}^{2}-\tilde{r}^{2}}}\right). & \displaystyle\end{eqnarray}$$

The latter equation provides the asymptotic behaviour of the radial flow velocity near the edge of the disc,

(4.23a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}r}(r,0,t)\sim -\frac{2RV(t)}{\sqrt{R^{2}-r^{2}}},\quad V(t)=h^{\prime }(t)+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime }(t)Q_{n}(1).\end{eqnarray}$$

Multiplying both sides of the plate equation (4.4) by $\tilde{r}$ and integrating in $\tilde{r}$ from 0 to 1 using (4.10), (4.7) and (4.18), we arrive at the following equation for the rigid displacement of the disc:

(4.24) $$\begin{eqnarray}h^{\prime \prime }(t)=\frac{F_{exp}(t)}{M+m_{a}}-\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime \prime }(t)\left\{\frac{M_{c}}{M+m_{a}}W_{n}(0)+\frac{3\unicode[STIX]{x03C0}}{2\sqrt{2}}\frac{m_{a}}{M+m_{a}}W_{n0}\right\}.\end{eqnarray}$$

Here $M=M_{c}+M_{p}$ is the mass of the equipped disc, and $m_{a}$ is the added mass of the disc (3.2). Equation (4.24) provides the acceleration at the disc centre, $a(t)$ , which is measured in the experiments:

(4.25) $$\begin{eqnarray}\displaystyle a(t) & = & \displaystyle \frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}t^{2}}(0,t)=h^{\prime \prime }(t)+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime \prime }(t)W_{n}(0)\nonumber\\ \displaystyle & = & \displaystyle \frac{F_{exp}(t)}{M+m_{a}}+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime \prime }(t)\left\{\frac{M_{p}+m_{a}}{M+m_{a}}W_{n}(0)-\frac{3\unicode[STIX]{x03C0}}{2\sqrt{2}}\frac{m_{a}}{M+m_{a}}W_{n0}\right\}.\end{eqnarray}$$

Therefore, the elastic deflection of the disc may significantly affect the acceleration of its centre. Formula (4.25) is reduced to (3.3) when the elastic accelerations are negligible, i.e.  $a_{k}(t)\approx 0$ .

The equations for elastic deflections of the disc follow from the plate equation (4.4). Multiplying both sides of the plate equation (4.4) by $\tilde{r}W_{k}(\tilde{r})$ and integrating in $\tilde{r}$ from 0 to 1 using (4.10), (4.7), (4.11) and (4.16), we arrive at the following equations for the principal coordinates $a_{k}(t)$ , $k\geqslant 1$ :

(4.26) $$\begin{eqnarray}a_{k}^{\prime \prime }(t)+\unicode[STIX]{x1D714}_{k}^{2}a_{k}=F_{exp}(t)f_{k}+\mathop{\sum }_{n=1}^{\infty }a_{n}^{\prime \prime }(t)S_{kn},\end{eqnarray}$$

where

(4.27a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{n}^{2}=\frac{Dk_{n}^{4}}{mR^{4}},\quad \unicode[STIX]{x1D6FE}=\frac{3\unicode[STIX]{x03C0}}{2\sqrt{2}}\frac{m_{a}}{M+m_{a}},\quad f_{k}=\frac{M_{p}+m_{a}}{2M_{p}(M+m_{a})}\left(W_{k}(0)-\unicode[STIX]{x1D6FE}W_{k0}\right),\end{eqnarray}$$

(4.28) $$\begin{eqnarray}\displaystyle S_{kn}=S_{nk} & = & \displaystyle \frac{3\unicode[STIX]{x03C0}}{4\sqrt{2}}\frac{m_{a}}{M_{p}}\unicode[STIX]{x1D6FE}W_{k0}W_{n0}-\frac{M_{c}}{2M_{p}}\frac{M_{p}+m_{a}}{M+m_{a}}W_{k}(0)W_{n}(0)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FE}M_{c}}{2M_{p}}\left(W_{k}(0)W_{n0}+W_{n}(0)W_{k0}\right)-\frac{3\unicode[STIX]{x03C0}}{4}\frac{m_{a}}{M_{p}}W_{nk}.\end{eqnarray}$$

The system (4.26) is integrated in time numerically subject to the initial conditions

(4.29a,b ) $$\begin{eqnarray}a_{k}(0)=0,\quad a_{k}^{\prime }(0)=0\quad (n\geqslant 1)\end{eqnarray}$$

and for the forcing function $F_{exp}(t)$ measured in the experiments. Then the acceleration at the centre of the disc is calculated using (4.25) and compared with the results gathered by the accelerometer.

4.2 Comparison between theoretical and experimental results

The computations are performed for the one-mode approximation with $a_{k}(t)\equiv 0$ for $n\geqslant 2$ . This is because the conditions of the experiments are axisymmetric only approximately. Therefore, we do not expect that including more axisymmetric terms in the series (4.10) would improve the theoretical predictions of the disc acceleration compared to the experimental ones.

Figure 7.

Comparison of experimental and theoretical results for session 3, repetition 1 (see figure 2). (a) Dashed line is for rigid wet-body acceleration $F_{exp}(t)/(M+m_{a})$ , solid line is for theoretical acceleration at the disc centre provided by (4.33), and the dotted line is for the measured acceleration. (b) Rigid wet-body acceleration and theoretical acceleration measured at the centre of the disc are the same as in (a); dash-dotted line is for the rigid-body acceleration $h^{\prime \prime }(t)$ given by (4.32), dotted line is for the elastic acceleration at the plate centre, $a_{1}^{\prime \prime }W_{1}(0)$ , and the solid line with dots markers are for the acceleration of the disc edge, $w_{tt}(R,t)$ .

For the conditions of the experiments with the plate thickness $h_{p}=1~\text{cm}$ (see table 1), we calculate $k_{1}=3.011$ . The frequency and the period of the first dry mode are $\unicode[STIX]{x1D714}_{1}=3853~\text{s}^{-1}$ and $T_{1}=1.63~\text{ms}$ (compare with the corresponding values for the second dry elastic mode, $k_{2}=6.2$ , $\unicode[STIX]{x1D714}_{2}=16\,363~\text{s}^{-1}$ , $T_{2}=0.38~\text{ms}$ ); also $\unicode[STIX]{x1D6FE}=2.423$ , $W_{10}=0.143$ , $W_{1}(0)=2.848$ , $W_{11}=0.218$ , $f_{1}=2.645~\text{kg}^{-1}$ and $S_{11}=-2.921$ . The resulting equation for the principal coordinate of the first elastic mode reads

(4.30) $$\begin{eqnarray}a_{1}^{\prime \prime }(t)+\unicode[STIX]{x1D6FA}_{1}^{2}a_{1}=\unicode[STIX]{x1D6FC}_{1}F_{exp}(t),\end{eqnarray}$$

where the frequency of the first wet mode, $\unicode[STIX]{x1D6FA}_{1}$ , and the forcing factor $\unicode[STIX]{x1D6FC}_{1}$ are given by

(4.31a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{1}=\frac{\unicode[STIX]{x1D714}_{1}}{\sqrt{1-S_{11}}},\quad \unicode[STIX]{x1D6FC}_{1}=\frac{f_{1}}{1-S_{11}}.\end{eqnarray}$$

In the conditions of these experiments, $\unicode[STIX]{x1D6FA}_{1}=1946~\text{s}^{-1}$ and $\unicode[STIX]{x1D6FC}_{1}=0.675~\text{kg}^{-1}$ . The corresponding period of the first wet elastic mode is equal to $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}_{1}=3.223~\text{ms}$ , which is twice as great as the period of the dry mode, and is comparable with the peak time of the measured acceleration (see figures 7 and 10). Equation (4.30) is integrated numerically for the function $F_{exp}(t)$ measured by the load cell and the initial conditions (4.29). Then the acceleration of the rigid-body motion of the disc is calculated by (4.24), which reads within the one-mode approximation,

(4.32a,b ) $$\begin{eqnarray}h^{\prime \prime }(t)=\frac{F_{exp}(t)}{M+m_{a}}-\unicode[STIX]{x1D6FC}_{2}\,a_{1}^{\prime \prime }(t),\quad \unicode[STIX]{x1D6FC}_{2}=\frac{M_{c}}{M+m_{a}}W_{1}(0)+\unicode[STIX]{x1D6FE}W_{10}.\end{eqnarray}$$

The theoretical acceleration of the disc centre, $a(t)$ , is calculated by (4.25):

(4.33a,b ) $$\begin{eqnarray}a(t)=\frac{F_{exp}(t)}{M+m_{a}}+\unicode[STIX]{x1D6FC}_{3}\,a_{1}^{\prime \prime }(t),\quad \unicode[STIX]{x1D6FC}_{3}=\frac{M_{p}+m_{a}}{M+m_{a}}W_{1}(0)-\unicode[STIX]{x1D6FE}W_{10}.\end{eqnarray}$$

For the conditions of our experiments, $\unicode[STIX]{x1D6FC}_{2}=0.59$ and $\unicode[STIX]{x1D6FC}_{3}=2.61$ . Therefore, the effect of the elastic deflection of the disc on the rigid motion of the disc is approximately four times smaller than on the acceleration of the disc centre.

Figure 8.

Sketch of the expected motion during the initial stages of the elastic disc lifting from the water surface.

Figure 7(a) compares the measured acceleration of the disc centre (dotted line) with the theoretical predictions (4.33) (solid line) for the first run of session 3. The first term in (4.33) is shown by the solid thin line. Therefore we can claim that accounting for the disc elasticity explains the relation between the measured force and acceleration. More details about the disc motion are shown in figure 7(b). It can be observed that the rigid-solid component of the acceleration $h^{\prime \prime }(t)$ (dash-dotted line) depends weakly on the elasticity of the disc and is close to the acceleration predicted by the rigid disc model (3.2) (dashed line). The total acceleration of the disc centre (solid line), $a(t)$ , is the sum of two components within the one-mode approximation, $a(t)=h^{\prime \prime }(t)+a_{1}^{\prime \prime }(t)W_{1}(0)$ ; see (4.25). The elastic acceleration, $a_{1}^{\prime \prime }(t)W_{1}(0)$ , is shown by the dotted line. It is larger than the rigid acceleration, $h^{\prime \prime }(t)$ , for $0<t<4~\text{ms}$ . The acceleration of the disc edge, $w_{tt}(R,t)=h^{\prime \prime }(t)+a_{1}^{\prime \prime }(t)W_{1}(1)$ , is shown by the solid line with dots markers. It is seen that this acceleration is negative for $0<t<3.8~\text{ms}$ . Therefore, initially the edge of the disc goes down but the disc centre goes up. This situation is sketched in figure 8. The edge of the plate moves down when $0<t<t^{\ast }$ , where $t^{\ast }$ is determined by the equation $w_{t}(R,t^{\ast })=0$ . The calculations provide $t^{\ast }=4.5~\text{ms}$ for the case of figure 7. Initially the edge of the disc penetrates the water even though the main part of the disc exits the water. The radial velocity of the flow near the disc edge is given by (4.23). Within the one-mode approximation, the vertical velocity of the disc edge,

(4.34) $$\begin{eqnarray}w_{t}(R,t)=h^{\prime }(t)+a_{1}^{\prime }(t)W_{1}(1),\end{eqnarray}$$

the coefficient $V(t)$ in (4.23),

(4.35) $$\begin{eqnarray}V(t)=h^{\prime }(t)+a_{1}^{\prime }(t)Q_{1}(1),\end{eqnarray}$$

and the coefficient

(4.36) $$\begin{eqnarray}U(t)=-h^{\prime \prime }(t)-a_{1}^{\prime \prime }(t)Q_{1}(1)\end{eqnarray}$$

in the asymptotic formula for the pressure,

(4.37) $$\begin{eqnarray}P(r,0,t)\sim \frac{2\unicode[STIX]{x1D70C}}{\unicode[STIX]{x03C0}}U(t)\sqrt{R^{2}-r^{2}}\quad (r\rightarrow R-0),\end{eqnarray}$$

near the contact line before it starts to move are depicted in figure 9. Physically speaking the contact line cannot move if the pressure under the disc edge is greater than atmospheric pressure, when $U(t)$ given by (4.36) is positive; see equation (4.37). The calculations provide that $U(t)$ is very close to zero for $0<t<4~\text{ms}$ and quickly decreases after this time. The contact line does not move also if the radial velocity of the flow at the disc edge is positive, $V(t)$ given by (4.35) is negative; see (4.23). The calculations show that $V(t)$ is positive but very small for $0<t<3~\text{ms}$ . The disc edge moves down for $0<t<4.6~\text{ms}$ ; see line 1 in figure 9(b). Therefore, figure 9 shows that the generalised exit theory predicts that the contact line is unlikely to move before $t=3.5{-}4~\text{ms}$ , which is in good agreement with the experiments (see figure 5 a).

Figure 9.

(a) Pressure coefficients $U(t)$ for two repetitions of session 3 (1 and 2). (b) Plot of $w_{t}(R,t)$ (line 1) and $V(t)$ (line 2) for session 3, repetition 1.

The theoretical accelerations $a(t)$ calculated for different forcing functions $F_{exp}(t)$ in different experimental runs are compared with the measured accelerations in figure 10. These results confirm that taking into account the elastic deflection of the disc explains the non-monotonic relation between the measured forces and measured accelerations. The discrepancies between the peak accelerations derived theoretically and measured experimentally shown in figure 10 are 13.47 % for (a), 17.72 % for (b) and 6.59 % for (c). The repetition shown in figure 10(d) corresponds to a drop that fell from a lower height, compared to the other runs. Thus, the force applied to pull up the disc was smaller and so was the acceleration. Nonetheless the agreement between the theory and experiments is still good even in these different conditions (the discrepancy in this case is 11.35 %). Even though the specific details of the measured forces and accelerations are rather sensitive to the precise initial conditions of each run, the theory predicts well the maximum accelerations and their duration.

Figure 10.

Comparison of the theoretical accelerations at the disc centre with the experimental ones for four runs (session 3, repetitions 2, 3, 4 and 5). Dashed lines represent the rigid wet-body acceleration, $F_{exp}(t)/(M+m_{a})$ , solid lines are the theoretical accelerations at the disc centre provided by (4.25), and the dotted lines are for the measured accelerations.

To prove that elastic effects are negligible in the absence of water, we show in figure 11 the acceleration $a_{exp}(t)$ measured by the accelerometer, the acceleration computed from the load-cell measurements under the rigid-body assumption, $F_{exp}(t)/M$ , and the acceleration predicted by the present elastic model, $a_{th}(t)$ , with $m_{a}=0$ . It is seen that all these three accelerations are close to each other, with the maximum difference between them being 5.5 %.

Figure 11.

Comparison of the measured acceleration, $a_{exp}$ , the acceleration computed using the load-cell measurements under the rigid-body assumption, $F_{exp}/M$ , and the theoretical acceleration, $a_{th}$ .

To conclude this subsection, a few comments can be made about the distribution of stresses in the disc. The radial $\unicode[STIX]{x1D70E}_{r}$ and tangential $\unicode[STIX]{x1D70E}_{t}$ stresses on the disc surface are given by

(4.38) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{r}(R\tilde{r},t)=\frac{Eh_{p}}{2R^{2}(1-\unicode[STIX]{x1D708}^{2})}\left(W_{1}^{\prime \prime }(\tilde{r})+\frac{\unicode[STIX]{x1D708}}{\tilde{r}}W_{1}^{\prime }(\tilde{r})\right)a_{1}(t), & \displaystyle\end{eqnarray}$$

(4.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{t}(R\tilde{r},t)=\frac{Eh_{p}}{2R^{2}(1-\unicode[STIX]{x1D708}^{2})}\left(\unicode[STIX]{x1D708}W_{1}^{\prime \prime }(\tilde{r})+\frac{1}{\tilde{r}}W_{1}^{\prime }(\tilde{r})\right)a_{1}(t),\quad (\tilde{r}<1), & \displaystyle\end{eqnarray}$$

in the one-mode approximation. For the conditions of figure 7, the theoretical amplitude of the elastic mode $a_{1}(t)$ increases monotonically from zero to $4\times 10^{-4}~\text{mm}$ at the end of the measurements. The functions $\tilde{\unicode[STIX]{x1D70E}}_{r}(\tilde{r})=W_{1}^{\prime \prime }(\tilde{r})+\unicode[STIX]{x1D708}W_{1}^{\prime }(\tilde{r})/\tilde{r}$ and $\tilde{\unicode[STIX]{x1D70E}}_{t}(\tilde{r})=\unicode[STIX]{x1D708}W_{1}^{\prime \prime }(\tilde{r})+W_{1}^{\prime }(\tilde{r})/\tilde{r}$ are shown in figure 12. Their maximum values are achieved at $\tilde{r}=0$ and are equal to each other.

Figure 12.

Distributions of radial and tangential normalised stresses along the disc radius; curve (1) corresponds to $\tilde{\unicode[STIX]{x1D70E}}_{r}(\tilde{r})$ and (2) to $\tilde{\unicode[STIX]{x1D70E}}_{t}(\tilde{r})$ .

4.3 Effect of plate thickness

To further support the conclusion that elastic effects are essential during the initial stages of the plate exit, we provide here additional experiments with two discs of thicknesses $h_{p}=0.5~\text{cm}$ and $h_{p}=2~\text{cm}$ (see table 3). As expected, the elastic vibrations of the thicker disc have smaller amplitude and higher frequency than those of the thinner one, as depicted in figure 13. In general, the period of these oscillations is reasonably well captured by the theory. It should be stressed that the results presented here have been obtained using the measured properties of the plates and that, once these parameters are fixed, the theory has no adjustable parameters and its only input is the excitation force measured experimentally.

Table 3.

Properties of the three different discs used in experiments.

Figure 13.

Comparison of (a) the experimental accelerations at the disc centre and (b) the measured rigid wet-body accelerations, $F_{exp}(t)/(M+m_{a})$ , for three different plate thicknesses, 0.5, 1 and 2 cm. (c,d) Comparison between the experimental accelerations (dotted lines) and the rigid wet-body accelerations $F_{exp}(t)/(M+m_{a})$ (dashed lines) for the plate thicknesses $h_{p}=0.5~\text{cm}$ (c) and $h_{p}=2~\text{cm}$ (d).

Figure 14.

Comparison of theoretical accelerations at the disc centre with the experimental ones (session 5) in four runs for disc thickness, $h_{p}=0.5~\text{cm}$ . Dashed lines are for the rigid wet-body acceleration, $F_{exp}(t)/(M+m_{a})$ , solid lines are for the theoretical accelerations at the disc centre provided by (4.25) and dotted lines are for the measured acceleration.

For the thinner plate, the first peak of the acceleration is very well described by the theory, as illustrated in figure 14, albeit the second one exhibits an amplitude substantially smaller than the calculated one. We hypothesise that this disagreement is caused by several effects. First, it should be recalled that, in the theory, we assume that the force is applied at the centre, whereas in reality it is applied on a non-axisymmetric region. Second, nonlinearity leads to the appearance of higher-order modes, including non-axisymmetric ones, which are excited more easily in a thinner plate, since their – longer – natural period is closer to the duration of the experiment. A third explanation may be sought in the fact that our theory does not account for any source of dissipation. Owing to the smaller elastic energy stored in a thin plate, the effects of dissipation are expected to be more important in relative terms. Finally, because of its smaller mass, a thin plate leaves the water surface at a slightly larger acceleration than a heavier one; thus the assumption that the virtual mass is that evaluated at $t=0$ becomes questionable. In figure 14, the maximum difference between the theoretical and experimental accelerations is 19.5 %, but the difference between the measured and the rigid wet-body accelerations is 400 %, which shows that the rigid-disc model of water exit is not applicable to the case of this thin disc.

Regarding the results for the thick plate, $h_{p}=2~\text{cm}$ (see figure 15), the theory predicts accelerations that oscillate around the experimental ones, albeit with a larger amplitude, which we attribute again to the lack of dissipation in the model. The maximum difference between the theoretical and the experimental acceleration is 33 % in figure 15(c). For the other cases these differences are smaller than 13 %. Despite this discrepancy, the period of these oscillations is well predicted by the elastic exit theory. Note that these oscillations are not present in the rigid wet-body accelerations shown by dashed lines in figure 15. Thus, we find it fair to state that the elastic theory predicts better the experimental behaviour in this case. Indeed, for a thick plate we expect the arguments used in the previous paragraph for a thin plate to reverse, which explains the better agreement between theory and experiments.

In view of these results, we find it reasonable to claim that the generalised linear theory of water exit presented here can be successfully used to describe the motions of an elastic body that leaves the water surface at a large acceleration, such that its elastic vibrational modes are excited.

Figure 15.

Comparison of theoretical accelerations at the disc centre with the experimental ones (session 4) in four other runs for disc thickness, $h_{p}=2~\text{cm}$ . See figure 14 for an explanation of the different curves.