1 Introduction

For thermophysical investigations of hot, highly reactive metallic melts or melts in the undercooled liquid state below their melting temperature containerless handling and contactless measurement methods are beneficial or even mandatory. To measure contactlessly the surface tension and viscosity of a liquid metal the so-called “Oscillating Drop Method” is generally used [1]. For this, a liquid droplet, freely suspended without any mechanical contact by an electrostatic or a high frequency electromagnetic levitation field, is squeezed by a short-time pulse of the external levitation force. It is apparent, that the following damped oscillation of the droplet surface around its static equilibrium shape is governed by the surface tension of the liquid, which acts as restoring force, and by the viscous shear flow in the liquid, which acts as damping mechanism. To determine these two thermophysical properties, the movement of the surface has at first to be detected non-invasively by optical [1] or electrical inductive means [2] and evaluated with regard to its angular frequency \(\omega\) and damping time \(\tau\). Finally, these measurement quantities have to be associated via a reasonable physical theory to the surface tension coefficient \(\gamma\) and the viscosity coefficient \(\eta\). A revision of it is the main subject of the present work.

Under the condition of only marginal external forces acting on the oscillating but non-rotating droplet, which means that its equilibrium shape is spherical, and of only small oscillation amplitudes, the time dependence of the distance from the center of the equilibrium sphere of radius a to any point \(\mathbf{x }_{S}\) on its surface, as schematically shown in Fig. 1 left, can be described by [3]

$$\begin{aligned} r(t,\mathbf{x }_{S} )=a\,\left( {1+\sum \limits _{l=2}^\infty {\varepsilon _{l} (\mathbf{x }_{S} )\cdot \text {Re} \left\{ {\exp (-\tilde{{s}}_{l} \,t)} \right\} } } \right) . \end{aligned}$$
(1)

Here \(\varepsilon _{l} (\mathbf{x }_{S} )\) with \(\left| {\varepsilon _{l} (\mathbf{x }_{S} )} \right| \ll 1\) describes the surface shape of the oscillation mode l (\(l\in {\mathbb {N}}\), \(l\ge 2)\) expanded in spherical harmonics. The complex quantity

$$\begin{aligned} \tilde{{s}}_{l} :=i\omega _{l} +1/\tau _{l} \end{aligned}$$
(2)

is a combination of the real mode frequency \(\omega _{l}\) and the real damping time \(\tau _{l}\).

Fig. 1
figure 1

Left: Cut through a liquid droplet oscillating around its equilibrium spherical shape of radius a. Right: Example for the time dependent surface distance change \((r(t)-a)/a\) of a freely levitated, oscillating liquid metal droplet in weightlessness. The data (dots) were measured during a parabolic flight campaign by M. Mohr of the University Ulm, Germany. The points are plotted together with a fitted exponentially decaying sine curve (line), cf. Eq. 1. There are only \(N_{2} \approx 2\) oscillations until the amplitude dropped down to 1/e of its original value

Full size image

For undamped surface oscillations of inviscid, non-rotating droplets a relation between the mode frequency \(\omega _{l}\) and the surface tension coefficient \(\gamma\) has already been derived by Rayleigh [4] in 1879

$$\begin{aligned} \omega _{l}^{2} =\omega _{R,l}^{2} (\gamma ):=l(l+2)(l-1)\frac{4\pi }{3}\frac{\gamma }{M}, \end{aligned}$$
(3)

where M denotes the droplet mass. For very low damped surface oscillations a similar relation between the damping time \(\tau _{l}\) of the oscillation mode l and the viscosity coefficient \(\eta\) has little later been published by Lamb [5] (see also [3, §98 Eq. (291)])

$$\begin{aligned} \frac{1}{\tau _{l} }=\frac{1}{\tau _{L,l} (\eta )}:=(2l+1)(l-1)\frac{4\pi a}{3}\frac{\eta }{M}. \end{aligned}$$
(4)

According to Eq. 1, Eq. 4 also reveals that the oscillation modes with \(l\ge 3\) are considerably stronger damped than the \(l=2\) mode, so that only the latter is usually observed in real experiments. By this reason the standard oscillating drop experiments [1] use the formulas

$$\begin{aligned} \omega ^{2}=\omega _{R,2}^{2} (\gamma ):=\frac{32\pi }{3}\frac{\gamma }{M} \; \text{ and } \; \frac{1}{\tau }=\frac{1}{\tau _{L,2} (\eta )}:=\frac{20\pi a}{3}\frac{\eta }{M} \end{aligned}$$
(5)

to obtain the surface tension and viscosity coefficients \(\gamma\) and \(\eta\) of the liquid from the measured angular frequency \(\omega\) and damping time \(\tau\) of its surface oscillation.

The above mentioned conditions for the Eq. 5 immediately raise the question how well these formulas still work for higher viscous and thus stronger damped liquids like e.g., molten metallic glasses. Furthermore, it is puzzling that the Eqs. 3 and 4 are not interrelated, i.e., that (3) is independent of \(\tau _{l}\) and (4) independent of \(\omega _{l}\), although the opposite is true e.g., for the simple and from textbooks well known damped harmonic oscillator, the resonance frequency \(\omega ^{2}=\omega _{0}^{2} (k)-1 \big / {\tau ^{2}}\) of which depends also on the damping time \(\tau\). In this example \(\omega _{0}^{2} (k):=k \big / M\) is a function of the elastic constant k and can be compared with \(\omega _{R,2}^{2} (\gamma )\) of Eq. 5. Moreover, whether the equations (5) deliver reasonable values depends strongly also on the external experimental conditions and the impact on the liquid from the used levitation forces, which keep the droplet suspended against gravity without any mechanical contact. All this yields the motivation for a revision of the usually applied models of Rayleigh and Lamb.

A summary of all for the practical application important results derived in the present paper can be found in Sect. 5.

2 Formulation of the Problem

The following starts with a survey of a former work of Suryanarayana and Bayazitoglu [6] dealing with the same subject. Under the simplifying assumptions of small oscillation amplitudes and negligible external forces exerted on the liquid the flow field \(\mathbf{u }(\mathbf{x },t)\) inside of an incompressible liquid oscillating drop D (\(\mathbf{x }\in D)\) of constant mass density \(\rho\) can well be described by the mass conservation equation

$$\begin{aligned} \nabla \cdot \mathbf{u }(\mathbf{x },t)=0 \end{aligned}$$
(6)

and the linearized momentum conservation equation

$$\begin{aligned} \rho \,\partial _{t} \mathbf{u }(\mathbf{x },t)=-\nabla \cdot {\varvec{\Pi }}(\mathbf{x },t) \end{aligned}$$
(7)

with the Navier–Stokes stress tensor

$$\begin{aligned} {\varvec{\Pi }} (\mathbf{x },t):=p(\mathbf{x },t)\mathbf{U }-\eta \left( {\nabla \,\mathbf{u }(\mathbf{x },t)+\left[ {\nabla \,\mathbf{u }(\mathbf{x },t)} \right] ^{T}} \right) , \end{aligned}$$
(8)

where \(\mathbf{U }\) denotes the unit tensor. The terms on the right hand side of Eq. 8 describe the internal stress on a volume element in the liquid resulting from the pressure \(p(\mathbf{x },t)\) and the viscous shear flow with constant coefficient \(\eta\). The appropriate conditions at the boundary points \(\mathbf{x }_{S} (t)\in \partial D\) between the liquid droplet and a surrounding vacuum are determined on the one hand from the conservation of mass flow at the boundary surface \(\partial D\). This results in

$$\begin{aligned} \mathbf{n }(\mathbf{x }_{S} )\cdot \left( {\mathbf{u }(\mathbf{x },t)-\partial _{t} \mathbf{x }_{S} (t)} \right) =0 \; \text{ for } \; \mathbf{x }\rightarrow \mathbf{x }_{S} , \end{aligned}$$
(9)

meaning, that the fluid flow in normal direction \(\mathbf{n }(\mathbf{x }_{S} )\) on the droplet surface corresponds to the movement of the surface itself. On the other hand they are determined from the conservation of momentum flow through \(\partial D\). This results in

$$\begin{aligned} \mathbf{n }(\mathbf{x }_{S} )\cdot {\varvec{\Pi }}(\mathbf{x },t)-\,\gamma \,\nabla \cdot \mathbf{n }(\mathbf{x })\,\mathbf{n }(\mathbf{x }_{S} )=\mathbf{0 } \; \text{ for } \; \mathbf{x }\rightarrow \mathbf{x }_{S} , \end{aligned}$$
(10)

meaning amongst others, that the pressure and the stress from the viscous flow in normal direction at the droplet surface is balanced by the pressure from the surface tension \(-\gamma \,\nabla \cdot \mathbf{n }(\mathbf{x }_{S} )\), which is the product of the surface curvature \(\nabla \cdot \mathbf{n }(\mathbf{x }_{S} )\) and the constant surface tension coefficient \(\gamma\).

In the present form Eq. 10 does not consider any external tangential stress on the liquid which would e.g., provoke Marangoni convection. More important however is the fact, that the above set of equations neglects the non-linear convection term and also the influence of the levitation forces, which are under real experimental conditions necessary to suspend the droplet against gravity. The applicability of the Eqs. 6–10 even in this case will be discussed in a future work.

In consideration of the kinematic boundary condition (1) Reid [7, Eq. (19)] and Chandrasekhar [3, §98 Eq. (280)] solved the above set of equations and conditions and presented the resulting characteristic equations for the droplet surface oscillations of modes \(l\ge 2\) in the following form

$$\begin{aligned} \frac{2(l^{2}-1)}{\tilde{{x}}_{l}^{2} -2\tilde{{x}}_{l} Q_{l+1/2} (\tilde{{x}}_{l} )}-1+\frac{2l(l-1)}{\tilde{{x}}_{l}^{2} }\left[ {1-\frac{(l+1)Q_{l+1/2} (\tilde{{x}}_{l} )}{{\tilde{{x}}_{l} } \big / 2-Q_{l+1/2} (\tilde{{x}}_{l} )}} \right] =\frac{\omega _{R,l}^{2} (\gamma )}{\tilde{{s}}_{l}^{2} }. \end{aligned}$$
(11)

Equation (11) uses the definition

$$\begin{aligned} Q_{l+1/2} (\tilde{{x}}_{l} ):=\frac{J_{l+3/2} (\tilde{{x}}_{l} )}{J_{l+1/2} (\tilde{{x}}_{l} )}, \end{aligned}$$
(12)

where \(J_{l+1/2} (\tilde{{x}}_{l} )\) denotes the half-integer order Bessel function of the complex argument \(\tilde{{x}}_{l}\) which is the principal root of

$$\begin{aligned} \tilde{{x}}_{l}^{2} :=\frac{3}{4\pi a\,}\frac{M}{\eta }\tilde{{s}}_{l} =(2l+1)(l-1)\,\tau _{L,l} (\eta )\,\tilde{{s}}_{l} . \end{aligned}$$
(13)

The second term on the right hand side of (13) follows from the definition in Eq. 4. The cumbersome equations 11–13 have been used by [6] to determine numerically in a rather intricate way the surface tension \(\gamma\) and viscosity \(\eta\) of the oscillating liquid sphere from a measurement of its mode frequencies \(\omega _{l}\) and damping times \(\tau _{l}\).

3 Characteristic Equation Rearranged

This procedure can however considerably be simplified by a rearrangement of Eq. 11. The multiplication with \(\tilde{{s}}_{l}^{2}\) and the use of (13) transforms Eq. 11 to

$$\begin{aligned} \tilde{{s}}_{l}^{2} -\frac{2\tilde{{s}}_{l} }{\tau _{L,l} (\eta )}\left( {1-\varGamma _{l} (\tilde{{x}}_{l} )} \right) + \omega _{R,l}^{2} (\gamma )=0 \end{aligned}$$
(14)

with the definition

$$\begin{aligned} \varGamma _{l} (\tilde{{x}}_{l} ):= & {} 1-\frac{1}{2l+1}\left[ {\frac{l+1}{1-{Q_{l+1/2} (\tilde{{x}}_{l} )\,2} \big / {\tilde{{x}}_{l} }}+l\left( {1-\frac{(l+1){Q_{l+1/2} (\tilde{{x}}_{l} )\,2} \big / {\tilde{{x}}_{l} }}{1-{Q_{l+1/2} (\tilde{{x}}_{l} )\,2} \big / {\tilde{{x}}_{l} }}} \right) } \right] \nonumber \\\equiv & {} \frac{l^{2}-1}{2l+1}\;\frac{{Q_{l+1/2} (\tilde{{x}}_{l} )\,2} \big / {\tilde{{x}}_{l} }}{1-{Q_{l+1/2} (\tilde{{x}}_{l} )\,2} \big / {\tilde{{x}}_{l} }}, \end{aligned}$$
(15)

where the second term on the right hand side of Eq. 15 results from an identical rearrangement of the first one. It is remarkable that without the function \(\varGamma _{l} (\tilde{{x}}_{l} )\) the form of the characteristic equations (14) for the droplet coincides exactly with that of the simple damped harmonic oscillator already considered in the Introduction.

3.1 Calculation of \(\eta\)

Taking Eq. 2 into account, the imaginary part of Eq. 14 reads

$$\begin{aligned} \frac{\tau _{L,l} (\eta )}{\tau _{l} }=1-\text {Im}\left\{ {\tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} \end{aligned}$$
(16)

with the complex function

$$\begin{aligned} \tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) :=\left( {\frac{1}{\omega _{l} \tau _{l} }+i} \right) \,\varGamma _{l} \left( {\tilde{{x}}_{l} } \right) . \end{aligned}$$
(17)

In consideration of Eq. 2 the principal root of Eq. 13, i.e.,

$$\begin{aligned} \tilde{{x}}_{l} =\sqrt{(2l+1)(l-1)} \sqrt{\frac{\tau _{L,l} (\eta )}{\tau _{l} }} \sqrt{1+i\omega _{l} \tau _{l} } , \end{aligned}$$
(18)

and therefore also \(\tilde{{F}}_{l}\), are functions solely of the product between mode frequency \(\omega _{l}\) and damping time \(\tau _{l}\), which are assumed to be known by measurements, and of the a priory unknown quantity \(\tau _{L,l} (\eta )/\tau _{l}\), which contains, according to the definition in Eq. 4, the desired viscosity coefficient \(\eta\). Being the single unknown quantity in Eq. 16, the real implicit equation (16) can very easily be solved for \(\tau _{L,l} (\eta )/\tau _{l}\) by numerical iteration. This is a simple matter if modern mathematical calculation programs are used.

3.2 Calculation of \(\gamma\)

In consideration of Eq. 2 the real part of Eq. 14 reads

$$\begin{aligned} \frac{\omega _{R,l}^{2} (\gamma )}{\omega _{l}^{2} }=1+\frac{1}{\omega _{l}^{2} \tau _{l}^{2} }\left( {\frac{2 \, \tau _{l} }{\tau _{L,l} (\eta )}-1} \right) -\frac{2}{\omega _{l} \tau _{l} }\frac{\tau _{l} }{\tau _{L,l} (\eta )}\text {Re}\left\{ {\tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} . \end{aligned}$$
(19)

With the knowledge of the measured mode frequency \(\omega _{l}\) and damping time \(\tau _{l}\) and with the knowledge of the relation \(\tau _{L,l} (\eta )/\tau _{l}\) from a numerical solution of Eq. 16, the simple explicit equation (19) enables immediately the calculation of \(\omega _{R,l}^{2} (\gamma )\) and via its definition in Eq. 3 the determination of the surface tension coefficient \(\gamma\).

3.3 Comparison with the Results of Lamb and Rayleigh

Although the numerical calculation of the accurate viscosity and surface tension coefficients \(\eta\) and \(\gamma\) via the Eqs. 16 and 19 is a relatively simple task now, the question remains from a practical point of view whether it is really worth to do this work. The relative deviation between the accurate viscosity coefficient \(\eta\) calculated from the measurement values \(\omega _{l}\) and \(\tau _{l}\) via Eq. 16, i.e.,

$$\begin{aligned} \frac{1}{\eta }=\tau _{l} \,(2l+1)(l-1)\frac{4\pi a}{3M}\left[ {1-\text {Im}\left\{ {\tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} } \right] , \end{aligned}$$
(20)

and that one, denoted in the following by \(\eta _{L}\), which is calculated via the Lamb formula (4), i.e.,

$$\begin{aligned} \frac{1}{\eta _{L} }=\tau _{l} \,(2l+1)(l-1)\frac{4\pi a}{3M}, \end{aligned}$$

amounts to

$$\begin{aligned} \varDelta \eta _{L} :=\frac{\eta -\eta _{L} }{\eta }=\text {Im}\left\{ {\tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} . \end{aligned}$$
(21)

Similarly, the relative deviation between the accurate surface tension coefficient \(\gamma\) calculated from the measurement values \(\omega _{l}\) and \(\tau _{l}\) via Eq. 19 and that one, denoted in the following by \(\gamma _{R}\), which is calculated via the Rayleigh formula (3), reads

$$\begin{aligned} \varDelta \gamma _{R} :=\frac{\gamma -\gamma _{R} }{\gamma _{R} }=\frac{1}{\omega _{l}^{2} \tau _{l}^{2} }\left( {\frac{2 \, \tau _{l} }{\tau _{L,l} (\eta )}-1} \right) -\frac{2}{\omega _{l} \tau _{l} }\frac{\tau _{l} }{\tau _{L,l} (\eta )}\text {Re}\left\{ {\tilde{{F}}_{l} \left( {\frac{\tau _{L,l} (\eta )}{\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} . \end{aligned}$$
(22)

Both quantities are plotted as function of the measurement value

$$\begin{aligned} N_{l} :={\omega _{l} \tau _{l} } / {2\pi } \end{aligned}$$
(23)

in Fig. 2. According to the Eqs. 1 and 2, \(N_{l}\) denotes just the number of damped oscillations of mode l until the amplitude reached 1/e of its original value, cf. Fig. 1 right. For a reasonable evaluation of experimentally measured droplet oscillation data with regard to the frequency \(\omega _{2}\) and damping time \(\tau _{2}\) (usually only oscillations of the fundamental mode \(l=2\) are experimentally observed) at least \(N_{2} \ge 2\) oscillations should be detectable.

Fig. 2
figure 2

Plot of the relative deviations: \(\varDelta \eta _{L}\) of Eq. 21 (line), \(\varDelta \gamma _{R}\) of Eq. 22 (red long dashes) and \(\varDelta \eta _{A}\) of Eq. 32 (short dashes) for the fundamental oscillation mode \(l=2\) versus the number \(N_{2}\) of measured droplet oscillations

Full size image

The diagram in Fig. 2 shows the remarkable fact, that even for the case of highly damped oscillations, i.e., \(N_{2} \approx 2\), the evaluation of the droplet oscillations via the Rayleigh formula delivers a surface tension coefficient \(\gamma _{R}\) which differs by only \(\varDelta \gamma _{R} \approx 2\,\%\) from the accurate surface tension coefficient \(\gamma\), although the Rayleigh formula has originally been derived for undamped oscillations only. Thus, from a practical point of view, the simple Rayleigh formula \(\omega _{l}^{2} =\omega _{R,l}^{2} (\gamma )\) of Eq. 3 determines the surface tension coefficient \(\gamma\) from the measured oscillation frequency \(\omega _{l}\) for the whole range \(N_{2} \ge 2\) sufficiently accurate.

However, this is no longer true for the simple Lamb formula \(\tau _{l} =\tau _{L,l} (\eta )\) of Eq. 4 which is generally used for the determination of the droplet viscosity from the measured oscillation damping time \(\tau _{l}\). In this case the resulting viscosity coefficient \(\eta _{L}\) differs in the highly damped case (\(N_{2} \approx 2)\) by more than \(\varDelta \eta _{L} \approx 10\,\%\) from the accurate viscosity coefficient \(\eta\) determined via Eq. 16, and this deviation decreases only very slowly with increasing number of oscillations, see Fig. 2.

4 Asymptotic Solutions

Although the numerical calculation of \(\tau _{L,l} (\eta )/\tau _{l}\) from the implicit equation (16), which delivers the accurate viscosity coefficient \(\eta\), is no major task, it is more convenient for practical applications to have a formula which provides \(\tau _{L,l} (\eta )/\tau _{l}\) in a simple explicit presentation.

4.1 Asymptotic Solution for \(\tau _{L,l} (\eta )/\tau _{l}\)

In the following we look for an asymptotic solution of Eq. 16 which matches \(\tau _{L,l} (\eta )/\tau _{l}\) in the limit of \(\omega _{l} \tau _{l} =2\pi \,N_{l} \rightarrow \infty\). In the Appendix an asymptotic relation of the function \(\tilde{{F}}_{l} \left( {{\tau _{L,l} (\eta )} {\tau _{l} },\omega _{l} \tau _{l} } \right)\) has been derived, see Eq. 41. With this, Eq. 16 amounts for \(\omega _{l} \tau _{l} \rightarrow \infty\) to

$$\begin{aligned} \frac{\tau _{L,l} (\eta )}{\tau _{l} }\sim 1-\frac{\alpha _{l} }{\sqrt{\omega _{l} \tau _{l} } }\frac{1}{\sqrt{{\tau _{L,l} (\eta )} \big / {\tau _{l} }} } \end{aligned}$$
(24)

with

$$\begin{aligned} \alpha _{l} :=\sqrt{\frac{2(l+1)^{2}(l-1)}{(2l+1)^{3}}} . \end{aligned}$$
(25)

Hence, the lowest order asymptotic solution of Eq. 24 for \(\tau _{L,l} (\eta )/\tau _{l}\) yields

$$\begin{aligned} \frac{\tau _{L,l} (\eta )}{\tau _{l} }\sim 1, \end{aligned}$$
(26)

which just corresponds to the Lamb formula (4). To obtain a next higher order asymptotic solution, Eq. 26 is inserted in the right hand side of Eq. 24 providing in the limit of \(\omega _{l} \tau _{l} \rightarrow \infty\) a modified Lamb formula

$$\begin{aligned} \frac{\tau _{L,l} (\eta )}{\tau _{l} }\sim 1-\frac{\alpha _{l} }{\sqrt{\omega _{l} \tau _{l} } }. \end{aligned}$$
(27)

This procedure could be continued, but we limit ourselves to Eq. 27 and consider it as an approximate solution of Eq. 16 for the whole range of \(N_{l} ={\omega _{l} \tau _{l} } \big / {2\pi }\ge 1\). The resultant viscosity coefficient, denoted in the following by \(\eta _{A}\), follows finally from (27) via the definition of \(\tau _{L,l} (\eta )\) in Eq. 4

$$\begin{aligned} \frac{1}{\eta _{A} }=(2l+1)(l-1)\frac{4\pi a}{3M}\tau _{l} \left( {1-\frac{\alpha _{l} }{\sqrt{\omega _{l} \tau _{l} } }} \right) \end{aligned}$$
(28)

for arbitrary oscillation modes l. Especially for the experimentally mainly measured fundamental mode \(l=2\) it results in

$$\begin{aligned} \frac{1}{\eta _{A} }=\frac{20\pi a}{3M}\tau _{2} \left( {1-\sqrt{\frac{18}{125}} \frac{1}{\sqrt{\omega _{2} \tau _{2} } }} \right) . \end{aligned}$$
(29)

4.2 Asymptotic Solution for \(\omega _{R,l}^{2} (\gamma )/\omega _{l}^{2}\)

Although it is of minor interest only, because the Rayleigh formula is sufficiently accurate for practical purposes, see Sect. 3.3, we present for the sake of completeness in the following also a higher order asymptotic solution of \(\omega _{R,l}^{2} (\gamma )/\omega _{l}^{2}\) for \(\omega _{l} \tau _{l} =2\pi \,N_{l} \rightarrow \infty\). Since \(\tau _{L,l} (\eta )/\tau _{l}\) and the function \(\tilde{F}_{l} (\tau _{L,l} (\eta )/\tau _{l}, \omega _{l} \tau _{l})\) are bounded, the lowest order asymptotic solution of Eq. 19 results for \(\omega _{l} \tau _{l} \rightarrow \infty\) in

$$\begin{aligned} \frac{\omega _{R,l}^{2} (\gamma )}{\omega _{l}^{2} }\sim 1, \end{aligned}$$
(30)

which just corresponds to the Rayleigh formula in Eq. 3. To obtain a next higher order asymptotic solution, Eq. 26 is together with the asymptotic approximation of the function \(\tilde{F}_{l} (\tau _{L,l} (\eta )/\tau _{l}, \omega _{l} \tau _{l})\) from Eq. 41 inserted in the right hand side of Eq. 19. Taking only the dominant terms into account this yields in the limit of \(\omega _{l} \tau _{l} \rightarrow \infty\)

$$\begin{aligned} \frac{\omega _{R,l}^{2} (\gamma )}{\omega _{l}^{2} }\sim 1+\frac{2\alpha _{l} }{\left( {\omega _{l} \tau _{l} } \right) ^{3/2}}. \end{aligned}$$
(31)

This result can be used as improved version of the Rayleigh formula in the whole range of \(N_{l} = \omega _{l} \tau _{l} / 2\pi \ge 1\).

4.3 Comparison with the Accurate Results

The relative deviation between the accurate viscosity coefficient \(\eta\) calculated from the measurement values \(\omega _{l}\) and \(\tau _{l}\) via Eq. 20, where \({\tau _{L,l} (\eta )} / {\tau _{l} }\) results from the numerical solution of (16), and that one, denoted by \(\eta _{A}\), calculated via the improved Lamb formula (28), is given by

$$\begin{aligned} \varDelta \eta _{A} :=\frac{\eta -\eta _{A} }{\eta }=\frac{\text {Im}\left\{ {\tilde{{F}}_{l} \left( {{\tau _{L,l} (\eta )} \big / {\tau _{l} },\omega _{l} \tau _{l} } \right) } \right\} -{\alpha _{l} } \big / {\sqrt{\omega _{l} \tau _{l} } }}{1-{\alpha _{l} } \big / {\sqrt{\omega _{l} \tau _{l} } }}. \end{aligned}$$
(32)

A plot of this quantity for \(l=2\) versus the known measurement value \(N_{2} ={\omega _{2} \tau _{2} } \big / {2\pi }\) is shown in Fig. 2. It illustrates, that even for the case of high damped oscillations, i.e., \(N_{2} \approx 2\), the evaluation of the droplet oscillations via the modified Lamb formulas of Eqs. 28 or 29 delivers now a viscosity coefficient \(\eta _{A}\) which differs by only \(\varDelta \eta _{A} \approx 2\,\%\) from the accurate viscosity coefficient \(\eta\), and is thus for most practical purposes sufficiently accurate in the whole range \(N_{2} \ge 2\).

5 Summary

By a measurement of frequency \(\omega _{l}\) and damping time \(\tau _{l}\) of any surface oscillation mode l (\(l\ge 2)\), excited by a short pulse on a freely levitated liquid droplet, the surface tension coefficient \(\gamma\) and the viscosity coefficient \(\eta\) of the liquid can contactlessly be determined. Under the ideal assumptions of negligible external forces acting on the liquid and marginal oscillation amplitudes the commonly applied mathematical relations connecting the thermophysical material properties \(\gamma\) and \(\eta\) with the measurement quantities \(\omega _{l}\) and \(\tau _{l}\) are the Rayleigh (3) and the Lamb formula (4), respectively. However, the derivation of both equations supposes amongst others only undamped or very weakly damped oscillations.

In the preceding sections the characteristic equation of the linear Navier–Stokes equation has been derived in a simple form, see Eqs. 14 and 15. From its imaginary and real part, i.e., from Eqs. 16 and 19, the accurate values of \(\eta\) and \(\gamma\) for weakly as well as strongly damped oscillations can be calculated by means of simple numerically iteration. As shown in Fig. 2, the comparison of these values with those ones calculated via the Rayleigh and Lamb formula reveals that the Rayleigh formula (3) for the general oscillation mode l (\(l\ge 2)\)

$$\begin{aligned} \omega _{l}^{2} =l(l+2)(l-1)\frac{4\pi }{3M}\gamma , \end{aligned}$$

which reduces for the experimentally mainly observed oscillation mode \(l=2\) to

$$\begin{aligned} \gamma =\frac{3M}{32\pi }\omega _{2}^{2} , \end{aligned}$$

delivers even in the case of strongly damped oscillation and thus in the whole range of \(N_{l} ={\omega _{l} \tau _{l} } \big / {2\pi }\ge 1\) for most practical purposes a good approximation of the surface tension coefficient \(\gamma\). Here, \(N_{l}\) of Eq. 23 denotes the number of measured damped oscillations of mode l until the amplitude reached 1/e of its original value, cf. Fig. 1 right.

The viscosity coefficient calculated from the Lamb formula (4) deviates, however, in the strongly damped case \(N_{l} \approx 2\) by more than 10 % from the accurate value of \(\eta\). To avoid cumbersome numerics in calculating the latter from Eq. 16, an explicit approximate solution of Eq. 16 for \(\eta\) has been derived in Eq. 28. Denoted by \(\eta _{A}\), it reads for the general oscillation mode l (\(l\ge 2)\)

$$\begin{aligned} \frac{1}{\tau _{l} \,\left( {1-{\alpha _{l} } \big / {\sqrt{\omega _{l} \tau _{l} } }} \right) }=(2l+1)(l-1)\frac{4\pi a}{3M}\eta _{A} \end{aligned}$$

and reduces for the experimentally mainly observed oscillation mode \(l=2\) to Eq. 29

$$\begin{aligned} \frac{1}{\eta _{A} }=\frac{20\pi a}{3M}\tau _{2} \left( {1-\sqrt{\frac{18}{125}} \frac{1}{\sqrt{\omega _{2} \tau _{2} } }} \right) . \end{aligned}$$

Evidently these equations represent extensions of the Lamb formulas Eqs. 4 and 5 right. As shown in Fig. 2, the relative deviation between the viscosity coefficient, calculated form the Eqs. 28 or 29, and the accurate value, calculated numerically from Eq. 16, is in the whole range of \(N_{l}={\omega _{l} \tau _{l} } / {2\pi }\ge 1\) in the same small order of magnitude as that one of the Rayleigh formula and thus for most practical cases sufficiently exact.