2 Dimensional analysis

If the minor effect of surface tension is neglected, the coiling frequency $\unicode[STIX]{x1D6FA}$ depends on $H$ , $Q$ , $\unicode[STIX]{x1D708}$ , $d$ and the gravitational acceleration $g$ . Of these six parameters, two have independent dimensions. Buckingham’s $\unicode[STIX]{x1D6F1}$ -theorem then states that there are four independent dimensionless groups, which I take to be

(2.1a-d ) $$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}=\unicode[STIX]{x1D6FA}\left(\frac{\unicode[STIX]{x1D708}}{g^{2}}\right)^{1/3},\quad \unicode[STIX]{x1D6F1}_{H}=H\left(\frac{g}{\unicode[STIX]{x1D708}^{2}}\right)^{1/3},\quad \unicode[STIX]{x1D6F1}_{Q}=Q\left(\frac{g}{\unicode[STIX]{x1D708}^{5}}\right)^{1/3},\quad \unicode[STIX]{x1D6F1}_{d}=d\left(\frac{g}{\unicode[STIX]{x1D708}Q}\right)^{1/4}.\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D6F1}_{d}$ is the ratio of the nozzle diameter $d$ to the length scale

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\left(\frac{\unicode[STIX]{x1D708}Q}{g}\right)^{1/4},\end{eqnarray}$$

which is of the order of the arcwise extent of the bending boundary layer in gravitational coiling (Ribe et al. Reference Ribe, Huppert, Hallworth, Habibi and Bonn2006b ). The dimensionless frequency obeys a law of the form

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}=\text{fct}\left(\unicode[STIX]{x1D6F1}_{H},\unicode[STIX]{x1D6F1}_{Q},\unicode[STIX]{x1D6F1}_{d}\right).\end{eqnarray}$$

The above relationship can be represented by contour plots of $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ as a function of $\unicode[STIX]{x1D6F1}_{H}$ and $\unicode[STIX]{x1D6F1}_{Q}$ for several values of $\unicode[STIX]{x1D6F1}_{d}$ . The ensemble of these plots will be the basis for the regime diagram that is sought.

3 Onset of coiling

The path to a regime diagram naturally starts with a consideration of the onset of coiling. The traditional way to determine the coiling onset criterion is to perform a linear stability analysis of a steady axisymmetric stagnation flow of a fluid jet falling from a nozzle onto a plate some distance below (Tchavdarov et al. Reference Tchavdarov, Yarin and Radev1993). However, the stagnation flow can only be steady if the plate is permeable, and if it is not then the jet must be infinitely wide at the bottom. This serious drawback of the linear stability approach means that we need to seek an alternative criterion for the onset of coiling.

A simple geometrical criterion arises from the fact that there are two distinct length scales in the coil: the radius $R$ of the coil and the radius $a_{1}$ of the rope itself. Now, coiling can only occur if $R\geqslant a_{1}$ ; otherwise, the rope would intersect itself in the centre of the coil. Accordingly, Habibi et al. (Reference Habibi, Hosseini, Khatami and Ribe2014) proposed that $a_{1}=R$ is the critical condition for the onset of coiling. Since the numerical continuation method of Ribe (Reference Ribe2004) predicts both $R$ and $a_{1}$ , it can be used to determine the critical surface as a function of $\unicode[STIX]{x1D6F1}_{H}$ , $\unicode[STIX]{x1D6F1}_{Q}$ and $\unicode[STIX]{x1D6F1}_{d}$ . The procedure is to start with a finite-amplitude coiling solution with $a_{1}<R$ , and then to continue it numerically until $a_{1}$ just equals $R$ .

Figure 2.

Existence of steady coiling solutions, determined numerically using the method of Ribe (Reference Ribe2004). Each curve is a locus of points where $a_{1}=R$ for a given value of $\unicode[STIX]{x1D6F1}_{d}=d(g/\unicode[STIX]{x1D708}Q)^{1/4}\equiv d/\unicode[STIX]{x1D6FF}$ . Each curve divides the diagram into regions where steady coiling solutions with $a_{1}<R$ exist (above and to the left) and do not exist (below and to the right). Black inverted triangles indicate values of $\unicode[STIX]{x1D6F1}_{H}$ above which critical solutions with $a_{1}=R$ do not exist. Inequalities give the criteria for the onset of coiling corresponding to different parts of the diagram.

The result of this procedure is shown in figure 2, which displays critical curves $a_{1}=R$ for seven different values of $\unicode[STIX]{x1D6F1}_{d}\in [0.01,10]$ . These curves are in fact cross-sections of a critical surface in the three-dimensional parameter space $(\unicode[STIX]{x1D6F1}_{H},\unicode[STIX]{x1D6F1}_{Q},\unicode[STIX]{x1D6F1}_{d})$ . The logarithms of the values of $\unicode[STIX]{x1D6F1}_{d}$ are evenly spaced to assist in identifying intermediate asymptotic limits, of which four are evident. The first is represented by the line with slope $-2$ at upper right, where the four curves for $\unicode[STIX]{x1D6F1}_{d}\geqslant 0.316$ coincide. Coiling in this limit occurs when $gQ^{2}H/\unicode[STIX]{x1D708}^{4}<2.21$ , and corresponds to one variety of the inertial (I) mode, as will be explained below. The second limit occurs for low values of $\unicode[STIX]{x1D6F1}_{d}\leqslant 0.1$ for which the upper parts of the critical curves are vertical. Here, coiling occurs when $Q/(\unicode[STIX]{x1D708}d)<0.85$ . The inverted triangles in figure 2 indicate where the critical solutions cease to exist, as evidenced by failure of the numerical algorithm. This limit, which corresponds to a second variety of inertial coiling, has not previously been reported to the best of my knowledge. The third and fourth limits are in the left portion of the diagram where inertia is negligible. At lower left, the critical curve with slope $1/4$ corresponds to the V mode, for which coiling occurs when $H/d>3.68$ . At centre left, the critical curve is that of the G mode, which occurs when $(H/\unicode[STIX]{x1D6FF})[\ln (H/\unicode[STIX]{x1D6FF})]^{1/2}>8.0$ . The logarithmic factor in the expressions for the radius $R$ and the frequency $\unicode[STIX]{x1D6FA}$ of gravitational coiling was first found by Blount (Reference Blount2010).

A natural question at this point is whether the critical curves of figure 2 agree with measurements of coiling onset in laboratory experiments. Figure 3 shows the critical curve for the V and G modes, obtained from the leftmost portion of figure 2. The V mode corresponds roughly to $\unicode[STIX]{x1D6F1}_{d}<0.5$ , while the G mode obtains for $\unicode[STIX]{x1D6F1}_{d}>2$ . Also shown in figure 3 are experimental measurements of the critical fall height for the onset of coiling for small values of the inverse Bond number $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}gd^{2}$ , where $\unicode[STIX]{x1D6FE}$ is the coefficient of surface tension (squares and circles; Cruickshank Reference Cruickshank1980). Finally, the dashed line is the prediction of the critical height for $\unicode[STIX]{x1D6FE}=0$ from the linear stability analysis of Tchavdarov et al. (Reference Tchavdarov, Yarin and Radev1993). While the experimental measurements do not agree perfectly with the critical curve $a_{1}=R$ , they are significantly closer to that curve than to the prediction of the linear stability analysis. The most likely reason for the remaining discrepancy is that the rope is fairly thick in the parameter range of the experiments, so that the slender-rope approximation upon which the numerical code is based becomes inaccurate. Nevertheless, it seems reasonable to conclude from figure 3 that $a_{1}=R$ is an appropriate criterion for the onset of coiling observed in laboratory experiments. In what follows, I shall study numerically how the coiling frequency depends on the dimensionless parameters $\unicode[STIX]{x1D6F1}_{H}$ , $\unicode[STIX]{x1D6F1}_{Q}$ and $\unicode[STIX]{x1D6F1}_{d}$ .

Figure 3.

Critical curve $a_{1}=R$ (solid line) in the inertia-free (Stokes) limit, separating regions where coiling occurs (above) and does not occur (below). The symbols are experimentally measured critical heights (Cruickshank Reference Cruickshank1980) for silicone oil with $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}gd^{2}=0.0063$ (circles) and 0.014 (squares), corresponding to the limit of small surface tension. The dashed line is the critical curve predicted by the linear stability analysis of Tchavdarov et al. (Reference Tchavdarov, Yarin and Radev1993) in the absence of surface tension. The asymptotic expressions for the portions of the critical curve corresponding to V and G onset are shown.

Figure 4.

Contour plots of the dimensionless coiling frequency $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ as a function of $\unicode[STIX]{x1D6F1}_{Q}$ and $\unicode[STIX]{x1D6F1}_{H}$ for four values of $\unicode[STIX]{x1D6F1}_{d}$ , determined numerically using the method of Ribe (Reference Ribe2004). The labels on the contours are $\log _{10}\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ . Black circles indicate where convergence of the numerical algorithm failed, and black squares show where the calculation was stopped deliberately to avoid entering the multivalued regions (grey). The heavy black lines are the critical curves for coiling onset from figure 2. The labels V, G, IG, I1, I2 and I3 indicate roughly where the corresponding coiling modes reside in the regime diagram.

4 Coiling frequency

The frequency of steady coiling can be predicted using the numerical method of Ribe (Reference Ribe2004) for any values of $\unicode[STIX]{x1D6F1}_{H}$ and $\unicode[STIX]{x1D6F1}_{Q}$ that lie above and to the left of the critical curves of figure 2. Figure 4 shows contour plots of the dimensionless coiling frequency $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ for four values of $\unicode[STIX]{x1D6F1}_{d}$ . The heavy line on each plot is the critical curve for coiling onset from figure 2. To make the connection with figure 1(b), note that the curve in that figure (for which $\unicode[STIX]{x1D6F1}_{d}=0.272$ ) is very similar to a vertical slice at $\unicode[STIX]{x1D6F1}_{Q}=3.78\times 10^{-7}$ through figure 4(c) (for which $\unicode[STIX]{x1D6F1}_{d}=0.316$ ). In figures 4(c) and 4(d), the contours at lower left loop back on themselves, indicating that the frequency passes through a minimum as the fall height $H$ is increased. This behaviour, which is also evident in figure 1(b), corresponds to the transition from V coiling (for which $\text{d}\unicode[STIX]{x1D6FA}/\text{d}H<0$ ) to G coiling (for which $\text{d}\unicode[STIX]{x1D6FA}/\text{d}H>0$ ). The grey regions in figure 4 correspond to the multivalued IG mode. A typical variation of $\unicode[STIX]{x1D6FA}$ as a function of $H$ for the IG mode is the portion $\unicode[STIX]{x1D6F1}_{H}\in [0.7,1.6]$ of the curve in figure 1(b).

I begin by determining the coiling onset frequency for the four intermediate asymptotic limits of the critical curves indicated in figure 2. As a preliminary, I note that the frequency of inertial coiling has three distinct scaling laws, depending on the magnitude of the gravitational stretching of the tail. All are based on the fundamental scaling law for the frequency of inertial coiling, which is (Mahadevan et al. Reference Mahadevan, Ryu and Samuel2000)

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{I}\sim \left(\frac{Q^{4}}{\unicode[STIX]{x1D708}a_{1}^{10}}\right)^{1/3}.\end{eqnarray}$$

In the first inertial mode, which I call I1, the tail is in free fall and the viscous resistance to stretching is negligible. In this case, the radius $a_{1}\sim (Q^{2}/gH)^{1/4}$ , whence (4.1) becomes

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{I}^{(1)}\sim \left(\frac{g^{5}}{\unicode[STIX]{x1D708}^{2}Q^{2}}\right)^{1/6}H^{5/6}.\end{eqnarray}$$

A second inertial mode (I2) has negligible stretching in the tail, so that $a_{1}\approx d/2$ . The scaling law for this mode is therefore

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{I}^{(2)}\sim \left(\frac{Q^{4}}{\unicode[STIX]{x1D708}d^{10}}\right)^{1/3}.\end{eqnarray}$$

Finally, in the third inertial mode (I3), the tail is strongly stretched by gravity, but the latter is balanced by the viscous resistance to stretching rather than by inertia. In this case, $a_{1}\sim (\unicode[STIX]{x1D708}Q/gH^{2})^{1/2}$ , and (4.1) becomes

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{I}^{(3)}\sim \left(\frac{g^{5}}{\unicode[STIX]{x1D708}^{6}Q}\right)^{1/3}H^{10/3}.\end{eqnarray}$$

As we shall see, these three inertial modes blend smoothly into one another on the regime diagram.

Figure 5 shows the dimensionless coiling frequency $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ on the critical surface as a function of relevant dimensionless parameters for the four onset modes V (a), G (b), I1 (c) and I2 (d). The dimensional equation for the best-fitting regression line is shown on each panel. As an illustration of how the onset frequency may be written in a more familiar form, consider the mode I1, for which the best-fitting regression line is (in dimensional form)

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=1.21Hg/\unicode[STIX]{x1D708}.\end{eqnarray}$$

By using the definition $gQ^{2}H/\unicode[STIX]{x1D708}^{4}=2.21$ of the critical surface for this mode, one can rewrite the right-hand side of (4.5) by multiplying it by the unitary quantity $(2.21\unicode[STIX]{x1D708}^{4}/gQ^{2}H)^{1/6}$ , whence

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=1.21\frac{Hg}{\unicode[STIX]{x1D708}}\left(\frac{2.21\unicode[STIX]{x1D708}^{4}}{gQ^{2}H}\right)^{1/6}=1.38\left(\frac{g^{5}}{\unicode[STIX]{x1D708}^{2}Q^{2}}\right)^{1/6}H^{5/6}.\end{eqnarray}$$

Comparison of (4.6) with (4.2) shows that the onset of coiling along the portion of the critical curve defined by $gQ^{2}H/\unicode[STIX]{x1D708}^{4}=2.21$ does indeed correspond to the I1 mode of inertial coiling.

Figure 5.

Dimensionless coiling frequency $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FA}}$ on the critical surface as a function of relevant dimensionless parameters for the four onset modes (a) V, (b) G, (c) I1 and (d) I2. The dimensional form of the best-fitting regression line is indicated on each panel.

A similar analysis applied to the remaining three intermediate asymptotic limits in figure 2 yields scaling laws for the onset frequency for the three additional modes V, G and I2. The results for all four modes are summarized in table 1. The auxiliary conditions in the third column specify where in the parameter space $\unicode[STIX]{x1D6F1}_{H}$ – $\unicode[STIX]{x1D6F1}_{Q}$ – $\unicode[STIX]{x1D6F1}_{d}$ each of the onset criteria applies.

Table 1.

Onset of coiling: asymptotic limits.

With the above results in hand, it is now possible to construct a complete regime diagram for liquid rope coiling. This diagram is just the set of four contour plots of figure 4, with the coiling modes corresponding to the different regions now indicated. The desired synoptic portrait of the ‘whole elephant’ is thus complete.