3.1. Modes of droplet aerobreakup

Figure 3 shows global observation of the temporal evolution of liquid droplet with $c/c^*=0.70$ subjected to $M_s$ of 1.18, 1.34 and 1.71 with respective $We$ of 320, 949 and 7642. Here, $t^*=t/t_I$ is the non-dimensionalized time such that $t_I=({D_0}/{U_g})\sqrt {{\rho _l}/{\rho _g}}$ is the inertial time scale generally used in describing the results of liquid droplet aerobreakup (Nicholls & Ranger Reference Nicholls and Ranger1969; Theofanous et al. Reference Theofanous, Mitkin and Ng2013). Time $t$ is counted from the moment of droplet–shock wave interaction. The three cases shown in figure 3 illustrate the three different breakup modes observed in the present study (supplementary movies 1, 2 and 3 available at https://doi.org/10.1017/jfm.2023.377). At low $We$ (<700), the droplet suffers large deformation to form a flattened sheet ($t^*=3.08$ in figure 3a). Finally, the surface tension and the elastic forces (if present) overwhelm the aerodynamic forces leading to rebound and oscillation of the liquid mass. This is identified as the vibrational mode of breakup (figure 3a). In this regime, the liquid mass generally remains as a single integral structure, and even if it breaks up, only a few daughter droplets are formed. Liquid rebound is the main characteristic of the vibrational mode. At moderate $We$ ($700< We<2800$), in addition to droplet deformation, KH waves also appear on the droplet surface between the front stagnation point and the droplet equator. These waves grow with time and travel towards the equator region, finally leading to the ejection and liquid mass entrainment in the airflow near the equator region ($t^*=1.14$ in figure 3b). This is identified as the SIE mode of breakup. In this regime, the liquid mass drawn in the airflow goes beyond the rebound limit of restoring forces, and this feature separates it from the vibrational mode observed at low $We$. At high $We$ (>2800), in addition to droplet deformation and KH waves, RT waves also contribute to the breakup process. The RT waves appear as surface corrugations on the flattened frontal region of the droplet ($t^*=1.27$ in figure 3c), leading to a widespread erratic breakup of liquid mass which is identified as the catastrophic breakup mode (figure 3c).

Figure 3. Global observation of temporal evolution of polymeric droplet with $c/c^*=0.70$ subjected to $M_s$ of (a) 1.18, (b) 1.34 and (c) 1.71. Respective Weber numbers are 320, 949 and 7642. The three examples shown here represent the three modes of droplet breakup: (a) vibrational, (b) SIE and (c) catastrophic modes.

It is observed that the range of $We$ corresponding to different breakup modes discussed above is independent of the polymer concentration, and even the water droplets follow the same trend. But this $We$ range is different from the range of $We$ for the aerobreakup of Newtonian droplets reported in conventional shock tube flow or continuous air jet flow-based studies. Further, two approaches are employed in existing literature for classifying the breakup modes. First is the classical approach in which five different breakup modes (vibrational, bag, multi-bag, shear-stripping, catastrophic) are identified based on the morphology of the atomizing liquid mass (Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009; Jain et al. Reference Jain, Prakash, Tomar and Ravikrishna2015; Sharma et al. Reference Sharma, Chandra, Basu and Kumar2022). Second is the modern approach where breakup modes (RT piercing and SIE) are classified based on the underlying hydrodynamic instabilities (Theofanous & Li Reference Theofanous and Li2008; Theofanous Reference Theofanous2011; Sharma et al. Reference Sharma, Singh, Rao, Kumar and Basu2021c). We adopted the modern way of classification, but the present range of $We$ corresponding to a particular breakup mode differs from that of the existing studies. For instance, the maximum $We$ for the vibrational breakup mode is $\sim$700 in the present work, whereas it is reported as $\sim$11 in the existing literature (Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009). This difference arises because $We$ reported in the present work is based on the gas flow properties ($\rho _g$, $U_g$) at the instant of droplet–shock wave interaction, which decays rapidly with time in the present experimental set-up (exploding-wire-based shock tube). This is in contrast to most of the existing studies where a conventional shock tube or continuous air jet set-up has been employed, resulting in constant gas flow properties during droplet breakup. Further details of the transient characteristics associated with our experimental set-up can be found in Appendix A. The present shock tube provides a new approach for investigating droplet aerobreakup and its usage is justified because it has several advantages, like ease of operation, the small size of the test facility, precise control and high repeatability compared with the conventional shock tube set-up. In fact, the blast wave obtained in an exploding-wire-based shock tube set-up is more relevant to many practical scenarios compared with the ideal conditions obtained in the conventional shock tube set-up (Igra et al. Reference Igra, Falcovitz, Houas and Jourdan2013; Pontalier et al. Reference Pontalier, Loiseau, Goroshin and Frost2018). Further, the primary objective of the present work is to study the role of liquid elasticity in the aerobreakup of polymeric droplets. This remains unaltered by choice of the present experimental set-up because conclusions are drawn by testing all the liquid samples in the same experimental conditions.

3.2. Stages of droplet aerobreakup

On the basis of the temporal evolution of the liquid mass, we describe each mode of droplet aerobreakup as a three-stage process (supplementary movie 4). Figure 4 illustrates the three stages with representative experimental images showing the state of liquid mass in each stage. Stage I is the droplet deformation that occurs immediately after the passage of the shock wave at the droplet location. In this stage the cross-stream diameter of the droplet increases, and the shape of the droplet changes from spherical to a cupcake geometry. Stage II marks the appearance and the growth of different hydrodynamic instabilities. State of the liquid mass obtained in stage II acts as a precursor for stage III, where breakup modes are decided, and morphological evolution of liquid mass is observed. In stage II and stage III, three different regimes are observed based on $We$. At low $We$ (<700), a deformed droplet undergoes further deformation to form a flattened sheet. Sometimes small ligaments are also observed emanating from the periphery of the flattened sheet. Finally, the liquid mass is pulled back, leading to the vibrational mode of breakup. At moderate $We$ ($700< We<2800$), the droplet undergoes KH wave-assisted SIE mode. At high $We$ (>2800), RT wave-assisted catastrophic mode of breakup is observed. A clear-cut demarcation between the three stages is not always possible, and the overlap of two consecutive stages also happens. At high $We$, due to a higher growth rate and quick appearance of hydrodynamic instabilities, it appears that stage I and stage II proceed simultaneously. Capturing them separately is beyond the scope of the present temporal resolution. It is important to provide a stage-wise description of the aerobreakup process because the effect of liquid elasticity is insignificant in the first two stages. The dominant role of elasticity appears only in stage III in terms of the morphology of the liquid mass. In this stage, depending upon liquid elasticity, different liquid morphologies like sheet, ligaments and beads-on-a-string (BOAS) can be observed. Detailed discussion of each stage is provided in the following subsections.

Figure 4. The three stages in droplet aerobreakup based on the temporal evolution of the liquid mass. Scale bar in each image represents 1 mm.

3.3. Stage I: droplet deformation

At early times of droplet–airflow interaction, the air stream almost achieves stagnation pressure at the windward and the leeward side of the droplet. Under the action of this pressure, the droplet deforms to change its shape from a sphere to a cupcake geometry, often approximated to an oblate spheroid for modelling purposes (Sor & García-Magariño Reference Sor and García-Magariño2015; Sharma, Singh & Basu Reference Sharma, Singh and Basu2021b). Deformation is quantified in terms of aspect ratio $D/D_0$, where $D$ is the maximum cross-stream diameter of the deformed droplet. Figures 5(a) and 5(b) show the temporal evolution of the aspect ratio for water and polymeric droplet with $c/c^*=16.67$ at comparable $We$. Deformation data for all concentrations are collated in 5(c). Here, $t' =tU_g/R_o$ is the non-dimensional time and $R_0$ is the initial radius of the droplet. Time $t$ is counted from the instant of droplet–shock wave interaction and until $D/D_0 \sim 1.7$. It is clear from figure 5(a–c) that variation of polymer concentration and hence $El$ have no significant effect on the deformation dynamics of the polymeric droplets.

Figure 5. Effect of liquid elasticity on droplet deformation dynamics. (a,b) Experimental images showing aspect ratio evolution for water and polymeric droplet with $c/c^*=16.67$. (c) Temporal variation of droplet deformation for water and polymeric droplets with different $c/c^*$ at comparable $We$. (d) Schematic of a spherical droplet deforming into the shape of an oblate spheroid. Comparison of experimental deformation with the modified DRD model for $c/c^*$ values of (e) 0 and ( f) 16.67.

A deforming droplet is assumed to undergo purely extensional flow, changing its shape from spherical to an oblate spheroid with semi-major axis length $b$ as shown in the schematic diagram figure 5(d). Parameter $y$ is the distance between the half-droplet centre of mass and the centre of the spheroid such that $b=\frac {8}{3}y$. To predict the deformation, force balance is performed in terms of motion of the half-droplet centre of mass. Sor & García-Magariño (Reference Sor and García-Magariño2015) proposed a droplet ratio deformation (DRD) model to predict the deformation of Newtonian droplets. Here we present a modified DRD model for polymeric droplets by adding the viscoelastic force term. The final force balance is given by

(3.1)\begin{equation} m a_c=F_v+F_s+F_p+F_{ve} .\end{equation}

Here, $m$ is the mass of the half-droplet and $a_c$ is the deformational acceleration, i.e. acceleration in the cross-stream direction, of the half-droplet centre of mass. Terms on the right-hand side of (3.1) represent the forces on the half-droplet due to viscous$(F_v)$, surface tension$(F_s)$, pressure$(F_p)$ and viscoelastic $(F_{ve})$ effects. It should be noted that $F_v$ is the viscous force only due to contribution from solvent (water in the present case), whereas $F_{ve}$ is the viscoelastic force due to polymer contribution. Except for $F_{ve}$, all the other terms in (3.1) can be evaluated in the same manner as done in the literature (Sor & García-Magariño Reference Sor and García-Magariño2015). To estimate $F_{ve}$, we have used the upper convected Maxwell model for viscoelastic fluid subjected to a two-dimensional incompressible and purely elongational flow. The expression for $F_{ve}$ is provided in Appendix B. After substituting the expression for all the terms in (3.1) and performing suitable non-dimensionalization, the equation which governs the deformation is given by

(3.2)\begin{equation} \frac{{\rm d}^2y'}{{\rm d} t'^2} ={-}\frac{8N}{KRe} \left(\frac{1}{y'^2}\frac{{{\rm d}y}'}{{\rm d} t'}\right) - \frac{4}{KWe} \left(\frac{{\rm d} A_s'}{{\rm d} b'}\right) + \frac{3C_p}{4K} - \frac{2M}{KReDe} \left(\frac{\xi}{y'}\right).\end{equation}

Here, quantities with primes represent non-dimensional terms. Parameters $R_0$ and ${R_0}/{U_g}$ have been used as the length scale and time scale for non-dimensionalization, $C_p$ is the coefficient of pressure and $A_s'={A_s}/{{\rm \pi} R_0^2}$ such that $A_s$ is the surface area of a half-spheroid. Non-dimensional terms appearing in (3.2) are density ratio ($K$), viscosity ratio ($N$ and $M$), Reynolds number ($Re$) and Deborah number ($De$) defined as

(3.3a–e)\begin{equation} K=\frac{\rho_l}{\rho_g};\quad N=\frac{\mu_s}{\mu_g};\quad M=\frac{\mu_p}{\mu_g};\quad Re=\frac{\rho_g U_g D_0}{\mu_g};\quad De=\frac{\lambda U_g}{R_0}. \end{equation}

Here, $\mu _p$ is the polymer contribution to the zero-shear viscosity of the polymeric solution and $\mu _g$ and $\mu _s$ are the dynamic viscosities of the gas phase and the solvent (water in the present case). Since deformation occurs at early times of droplet–shock wave interaction, a constant value of gas-phase velocity obtained at the initial time instant can be used in (3.2) to get an estimation of deformation. Comparison of deformation predicted from (3.2) with experimental data is shown in figures 5(e) and 5( f) for two extreme values of $c/c^*$ studied in the present work. Here, $C_p$ is used as a fitting parameter, and from the different experimental data on deformation rate, its suitable value is obtained between 0.3 and 0.4. For the present range of parameters, it can be checked from (3.2) that contributions of viscous, viscoelastic and surface tension forces are at least an order of magnitude less than that of the pressure term. This suggests that droplet deformation is mainly governed by the balance between aerodynamic pressure and the inertia of the deforming liquid. Since liquid properties play an insignificant role in the deformation dynamics, this explains that the temporal evolution of the aspect ratio is unaffected by the variation in $c/c^*$ of the test liquid (figure 5c). In non-dimensional form, equation (3.2), neglecting viscous, viscoelastic and surface tension terms, leads to a constant deformational acceleration of the half-droplet. This means that the net deformation should be quadratic in time which is indeed observed experimentally as well as predicted theoretically (figure 5).

3.4. Stage II: KH instability

Stage II represents the appearance and growth of the hydrodynamic instabilities. As shown in figure 4, three different regimes based on $We$ can be observed in stage II. With increasing $We$, the first transition ($We\sim 700$) is observed when instead of only deformation, KH waves also appear on the droplet surface. To probe this further, KH instability wavelength $\lambda _{KH}$ is measured for different cases. To measure $\lambda _{KH}$, high-speed imaging (75 000 fps) at high resolution (4.5 $\mathrm {\mu }$m px$^{-1}$) is performed as shown in figures 6(a) and 6(b). These experimental images show the formation and evolution of KH waves for the case when a normal shock wave with $M_s=1.34$ interacts with a water and polymeric droplet with $c/c^*=16.67$, which is the maximum concentration considered in the present study. Only a quarter portion of the initially spherical droplet is captured to keep a high spatio-temporal resolution. Levitating droplet makes it convenient to keep the desired portion of the droplet in the camera field of view. The use of levitating droplets proves to be superior than the pendant droplet (Jackiw & Ashgriz Reference Jackiw and Ashgriz2021) and the falling droplet (Arcoumanis et al. Reference Arcoumanis, Khezzar, Whitelaw and Warren1994; Theofanous et al. Reference Theofanous, Mitkin and Ng2013) methods used in previous studies. Pendant droplets are not symmetric due to the presence of needle contact on one side, whereas in the case of a falling droplet, it is difficult to synchronize everything to capture images at high speed and high resolution in a small field of view. Moreover, falling polymeric droplets may exhibit asymmetry due to the presence of a long liquid tail formed while detaching from the source needle. All these challenges are overcome by using a levitating droplet.

Figure 6. Effect of liquid elasticity on the dynamics of KH instability. Zoomed-in images showing KH waves on the surface of droplets with $c/c^*$ values of (a) 0 and (b) 16.67. (c) Experimental value of KH wavelength for different concentrations at different shock Mach number. (d) Dispersion plot of KH instability for different concentrations at $M_s=1.34$. Inset shows the dispersion plot near peak growth rate values. (e) Dispersion plot of KH instability for $c/c^*=16.67$ at different shock Mach numbers. Horizontal line corresponds to the experimental wavenumbers with standard error as the span.

To understand the effect of liquid elasticity, $\lambda _{KH}$ is measured for water and polymeric droplets with three different concentrations subjected to four different $M_s$. The result is shown in figure 6(c). Insignificant effect of $c/c^*$ variation on $\lambda _{KH}$ is observed for a given $M_s$. However, a monotonic decrease in $\lambda _{KH}$ with increasing $M_s$ is observed for all concentrations. Considering an inviscid and incompressible airflow on the droplet surface, neglecting gravity effects and assuming a finite vorticity layer in the gas phase, one can apply the linear perturbation analysis to obtain a dispersion relation for KH instability as done for Newtonian droplets (Marmottant & Villermaux Reference Marmottant and Villermaux2004; Padrino & Joseph Reference Padrino and Joseph2006). We extended this approach for viscoelastic liquids by using the Oldroyd-B constitutive equation for the liquid phase. The details are provided in Appendix C. Finally, The dispersion relation for KH instability in an Oldroyd-B fluid can be written as

(3.4)\begin{equation} {\rm e}^{{-}2 \eta}=[1+(\varOmega-\eta)] \left[\frac{\varPhi+(\varOmega+\eta)\left\{2 \hat{\rho}-(1+\hat{\rho}) (\varOmega+\eta)-(1+\hat{\mu}) \beta \eta^{2}\right\}}{\varPhi+(\varOmega+\eta) \left\{2 \hat{\rho}-(1-\hat{\rho})(\varOmega+\eta)-(1-\hat{\mu}) \beta \eta^{2}\right\}}\right].\end{equation}

Equation (3.4) is the same as (C.4) and meanings of different symbols are provided in Appendix C. This equation is solved numerically to get the dispersion plot of KH instability. Since the growth of these instability waves happens on a very short time scale (${\sim }10^1$ $\mathrm {\mu }$s), the dispersion relation can be solved by neglecting the transient decay and considering a constant value of $U_g$. Figure 6(d) shows the dispersion plot for water and polymeric droplets with different concentrations subjected to $M_s=1.34$. The dispersion curve for all the concentrations overlaps for a given $M_s$. Near the peak point, the growth rate for water is slightly lower compared with that for the polymeric droplet, as shown in the inset of figure 6(d). This is because of the small difference in surface tension of water as compared with the polymeric solution. Figure 6(e) shows the KH instability dispersion curve for a polymeric droplet with $c/c^*=16.67$ subjected to different $M_s$, and the horizontal line on the plot indicates the experimental value of wavenumbers with standard error as the span. It should be noted that the growth rate is not measured experimentally, and hence the horizontal lines are accurate only to the wavenumbers. Although the linear stability analysis only accounts for the first-order effects of perturbation, it is a powerful tool for deciphering the essential physics. This is evident from figure 6(e), where the predicted wavenumber $k_{KH_{max}}$ corresponding to the maximum growth rate of KH instability is in reasonable agreement with the experimental values. It is clear from the experimental observations and theoretical predictions (figure 6) that the dynamics of KH instability is governed by $M_s$ (and hence $We$), but the elasticity of the liquid phase plays an insignificant role.

3.5. Stage II: RT instability

As $We$ is increased, the second transition ($We\sim 2800$) in stage II is marked by the appearance of RT waves, in addition to the deformation and the KH waves (figure 4). Again, the onset of critical $We$ for this transition is independent of $El$, similar to the first transition due to KH waves, as discussed in the previous section. The RT waves appear as surface corrugations on the flattened front surface of the droplet as shown in figures 7(a) ($t^*=1.63$), 7(b) ($t^*=1.59$) and 7(c). Figures 7(a) and 7(b) present the development of RT waves on the surface of a water droplet and high-concentration-polymer droplet ($c/c^*=16.67$) both subjected to a shock wave with $M_s=1.57$. For a given aerodynamic condition, the wavelength of front-surface corrugations has a similar magnitude irrespective of elastic properties. Joseph et al. (Reference Joseph, Beavers and Funada2002) provided a theoretical analysis to relate these surface corrugations with the RT waves. Later, Theofanous & Li (Reference Theofanous and Li2008) and Theofanous et al. (Reference Theofanous, Mitkin and Ng2013) performed laser-induced fluorescence imaging at an oblique angle of $30^{\circ }$ from the direction of side-view imaging and showed that the windward surface of the droplet remains smooth even at high $We$. They claimed that the appearance of surface corrugation at high $We$ is an artefact due to side-view shadowgraphic imaging. Despite this ambiguity in existing literature, there is no denying a few facts. First, even with the shadowgraphic images, the transition from smooth to the corrugated front surface of the droplet is observed when $We$ is increased from moderate to high value. Second, the RT waves do not appear as quickly as the KH waves due to large differences in their growth rate (Sharma et al. Reference Sharma, Singh, Rao, Kumar and Basu2021c). Before the appearance of RT waves, the droplet has already suffered some deformation and growth of KH waves, resulting in the development of a significant cross-sectional area normal to the airflow direction. This liquid–air interface, perpendicular to the airflow direction, provides a suitable platform for the RT instability to occur. Further, the cross-stream dimension of the flattened front surface is observed to be at least 1.3 times $D_0$ at the time of appearance of corrugations, whereas Theofanous & Li (Reference Theofanous and Li2008) reported the span of the front smooth surface to be only 0.65 times $D_0$. This means that even if the RT waves do not form in the central portion of the front surface, there is enough normal surface area away from the central portion for the RT waves to form. This also suggests the possibility of hybrid KH–RT instability where RT waves are modulated on the crest of KH waves (Zhao et al. Reference Zhao, Hou, Liu, Tian, Xu, Li, Liu, Wu, Zhang and Lin2014; Mitkin & Theofanous Reference Mitkin and Theofanous2017; Sharma et al. Reference Sharma, Singh, Rao, Kumar and Basu2021c) and appear as front-surface corrugations. It is worth noting that the ‘hybrid KH–RT’ mechanism referred to here is different from the ‘combined KH–RT’ mechanism studied by Vadivukkarasan & Panchagnula (Reference Vadivukkarasan and Panchagnula2016, Reference Vadivukkarasan and Panchagnula2017) in which the KH and the RT instabilities occur simultaneously as equal contributors to the destabilization process. Whereas, in the hybrid KH–RT mechanism, the KH waves appear first, leading to an increase in the area of the air–liquid interface normal to the airflow direction, which provides a suitable platform for the subsequent occurrence of RT instability. The combined KH–RT mechanism could provide an explanation for the periodicity of ligaments in the azimuthal direction as shown in figure 4 (stage III of the SIE regime) and figure 3(b), but studying this is beyond the scope of the present work. However, as far as the corrugations in the frontal surface of the liquid mass are concerned, the hybrid KH–RT mechanism is more suited. In a recent study, Mansoor & George (Reference Mansoor and George2023) used a digital holography imaging technique and provided evidence for the presence of RT instability on the droplet front surface in the catastrophic regime of shock-induced aerobreakup.

Figure 7. Effect of liquid elasticity on the dynamics of RT instability. Development of RT waves as surface corrugations on the flattened front surface of (a) water droplet and (b) polymeric droplet with $c/c^*=16.67$. (c) Image of RT waves on the front surface of polymeric droplet with $c/c^*=1.2\times 10^{-3}$ at $M_s=1.57$, $We=5062$. Red lines indicate the wavelength of the surface corrugations, i.e. the experimental value of RT wavelength, $\lambda _{RT_e}$. Two horizontal blue lines show the RT wavelength $\lambda _{RT_m}$ predicted by the theoretical model. (d) Dispersion plot of the RT instability for different concentrations at $M_s=1.57$. Inset shows the dispersion plot near peak growth rate values. Horizontal line corresponds to the experimental wavenumbers with standard error as the span. Scale bar for all the experimental images corresponds to 1 mm.

In the previous section, we have already shown that the viscoelastic properties do not play a significant role in the KH instability for the considered range of parameters. Now we show that similar results are obtained for the RT instability. We have used the dispersion relation for RT instability at high $We$ in an Oldroyd-B fluid derived by Joseph et al. (Reference Joseph, Beavers and Funada2002). For all practical purposes, $\rho _g<<\rho _l$ holds true. Therefore, neglecting $\rho _g/(\rho _l+\rho _g)$ and assuming $\rho _l/(\rho _l+\rho _g)=1$, the dispersion relation for RT instability with growth rate ${\rm i}\omega _{RT}$ and wavenumber $k_{RT}$ can be written as

(3.5)\begin{align} & -1+ \frac{1}{\omega_{RT}^2} \left({-}ak_{RT}+\frac{\gamma k_{RT}^3}{\rho_l}\right) + \frac{4k_{RT}^2}{{\rm i}\omega_{RT}} \left( \frac{\mu_g-\mu_{eff}}{\rho_l}\right) \nonumber\\ &\quad -\frac{4k_{RT}^3}{\omega_{RT}^2} \left( \frac{\mu_g-\mu_{eff}}{\rho_l}\right)^2 \left( q_{2}-k_{RT}\right)=0, \end{align}

where

(3.6)\begin{equation} q_2 = \sqrt{ k_{RT}^2 + \frac{{\rm i}\omega_{RT}\rho_l}{\mu_{eff}} }.\end{equation}

Here, $\mu _{eff}$ can be obtained from (C.3) by replacing $\omega _{KH}$ with $\omega _{RT}$. Acceleration $a$ of the deformed droplet in the streamwise direction can be estimated from the following relation (Zhao et al. Reference Zhao, Wu, Li, Xu and Liu2018):

(3.7)\begin{equation} a = \frac{3\rho_g U_g^2 C_d D_m^2}{4 D_0^3 \rho_l}, \end{equation}

where $C_d$ is the drag coefficient and $D_m$ is the maximum cross-stream diameter of the liquid mass. The RT instability depends strongly on the acceleration of the air–liquid interface. For an exact estimation of acceleration, in (3.7), $U_g$ should be replaced by relative velocity between the air and the droplet, which changes continuously with time. Coefficient $C_d$ depends on $D_m$, which in turn is a transient quantity. This makes it difficult to obtain the exact value of acceleration. However, an approximate estimate can be obtained using (3.7) and taking $C_d=1.2$ and $D_m=2.15D_0$ (Zhao et al. Reference Zhao, Wu, Li, Xu and Liu2018). Now, (3.5) can be solved numerically to plot the dispersion curve for RT instability. In the original formulation, Joseph et al. used retardation time as a fitting parameter to match the theory with experiments. However, retardation time is not an independent parameter for an Oldroyd-B fluid. It depends on the relaxation time, zero-shear viscosity of the solution and the solvent viscosity such that $\lambda _r=(\mu _s/\mu _0)\lambda$ (Larson Reference Larson2013). Using the correct value of $\lambda _r$, (3.5) is solved to get the RT dispersion plot for all the test liquid droplets subjected to $M_s=1.57$ equivalent to $We \approx 4900$ for polymeric droplet with $D_0=1.8$ mm. The results are shown in figure 7(d). The inset shows the dispersion plot near peak growth rate values, and horizontal lines represent wavenumbers measured from experimental images with standard error as the span. Growth rates are not measured experimentally; therefore, the location of the horizontal lines in figure 7(d) corresponds only to the wavenumbers but not the growth rates. It can be observed that the dispersion curve overlaps for all the polymeric liquids. Water shows a slight deviation from the polymeric liquids, mainly because of the small difference in the surface tension values. Wavelength of RT instability $\lambda _{RT_{m}}$ (blue) predicted from the model is shown in figure 7(c) for comparison with the experimental data, $\lambda _{RT_{e}}$ (red). Due to various approximations in theory, especially in estimating the acceleration $a$, this model is good only for an order-of-magnitude estimation of RT wavelengths. Despite its drawbacks, this model brings out the essential physics that the RT instability is insensitive to the liquid elasticity for a wide range of values studied in the present work.

3.6. Stage III: breakup morphology

From the perspective of viscoelastic droplet aerobreakup, stage III is the most important stage where the effect of the liquid elasticity enters into play. This stage corresponds to the evolution of liquid mass morphology pertaining to different modes of aerobreakup. Three distinct modes are identified in the present study as shown in stage III of figure 4. At low $We$, the droplet undergoes large deformations and subsequent rebound of the liquid mass, leading to the vibrational mode of breakup. At moderate $We$, the droplet undergoes KH wave-assisted SIE mode and, at high $We$, the droplet experiences RT wave-assisted catastrophic mode. These breakup modes are governed by the underlying hydrodynamic instabilities during stage II. We have already shown in the previous subsections that the liquid elasticity plays an insignificant role in the first two stages; therefore, the critical $We$ for the onset of different breakup modes is also independent of $El$. However, the effect of $El$ is seen during stage III in terms of the liquid mass morphology obtained within a particular mode of breakup (figure 8). An increase in $El$ caused a higher degree of rebound in the vibrational regime and a larger size of fragmented chunks in the catastrophic regime. The effect of elasticity is even more prominent in stage III of the SIE mode. In this regime, based on $El$, a hierarchy of well-defined morphological structures is observed. At high $El$, the liquid mass entrained in the airflow forms a stable sheet denoted as SS in figure 8(b). As $El$ is lowered, sheet structure is not stable, they break and long stable ligament structures are observed, denoted as SB–SL. In the case of water, ideally $El=0$, neither sheet nor ligament structure is stable, and they break to form fine daughter droplets denoted as SB–LB. Schematic representations of these different morphologies observed during stage III of the SIE mode are shown below their corresponding experimental images in figure 8(b). Another interesting observation is the formation of a BOAS structure, reported widely in extensional flows of dilute polymeric solutions (Clasen et al. Reference Clasen, Eggers, Fontelos, Li and McKinley2006; Bhat et al. Reference Bhat, Appathurai, Harris, Pasquali, McKinley and Basaran2010; Malkin, Arinstein & Kulichikhin Reference Malkin, Arinstein and Kulichikhin2014; Scharfman et al. Reference Scharfman, Techet, Bush and Bourouiba2016). The BOAS structure consists of small beads of liquid mass connected by threadlike ligaments, as shown in figure 8(b). The BOAS structure is observed for a moderate $We$ (SIE regime) and a lower range of $El$. Below this $El$, ligaments are not stable and form fine droplets through the mechanism of Rayleigh–Plateau instability. Above this $El$, stable ligaments are formed. The BOAS is the intermediate morphology, representing the transition from the stable ligament to the fine daughter droplet formation.

Figure 8. Effect of $El$ on morphology of liquid mass obtained in stage III of (a) vibrational mode, (b) SIE mode and (c) catastrophic breakup mode. Well-defined morphologies obtained in stage III of the SIE mode are denoted as: SB–LB, sheet breakup–ligament breakup; BOAS, beads on a string; SB–SL, sheet breakup–stable ligament; SS, stable sheet. Schematic representations of these morphologies are shown below their corresponding experimental images in (b).

All these morphologies observed in stage III show that the liquid elasticity enters as a stabilizing agent or resistance against the aerobreakup process. It is a challenging task to theoretically model these complex phenomena, especially the fragment size at high $We$. There are theoretical studies related to the breakup of viscoelastic sheets and ligaments, but the results are ambiguous in the sense that some literature reported elasticity as a destabilizing agent (Liu, Brenn & Durst Reference Liu, Brenn and Durst1998; Wang et al. Reference Wang, Yang, Xie and Chen2015; Dasgupta et al. Reference Dasgupta, Sharma, Nath and Bhanja2021), whereas others reported it as a stabilizing agent (Goren & Gottlieb Reference Goren and Gottlieb1982; Bousfield et al. Reference Bousfield, Keunings, Marrucci and Denn1986; Ruo et al. Reference Ruo, Chen, Chung and Chang2011; Yang, Tong & Fu Reference Yang, Tong and Fu2013; Xie et al. Reference Xie, Yang, Fu and Qin2016). The resolution for this discrepancy comes in terms of unrelaxed tension (Xie et al. Reference Xie, Yang, Fu and Qin2016) in the liquid phase. The role of elasticity changes from a destabilizing to a stabilizing agent if the unrelaxed tension is accounted for in the relevant stability analysis. Unrelaxed tension develops in strong extensional flows of polymeric solution, where flow time scale is small compared with $\lambda$, such that it does not allow enough time for relaxation of polymer molecules (Bousfield et al. Reference Bousfield, Keunings, Marrucci and Denn1986). The condition for the presence of unrelaxed tension can be given in terms of the Weissenberg number, $Wi=\lambda \dot {\varepsilon }>1$, where $\dot {\varepsilon }$ is the strain rate of extensional flow. Various literature mentioned above provides stability analyses for viscoelastic sheets and ligaments in the context of flow coming out from a spray nozzle. The same physics can be extended to the stability of morphologies obtained in the SIE regime of the present study. In this regime, the liquid mass entrained in the airflow suffers strong extensional flow as it becomes drawn into sheets and ligaments (figure 9). An order-of-magnitude estimation of $\dot {\varepsilon }=({1}/{l})({{\rm d} l}/{{\rm d} t})$ can be obtained from the images as shown in figure 9, where $l$ is the instantaneous length of liquid mass entrained in the airflow. Average values of $\dot {\varepsilon }$ pertaining to the SIE regime are in the range of ${\sim }10^4\,{\rm s}^{-1}$. The value of $\lambda$ for the polymeric solutions used in the present study is in the range of ${\sim }10^{-3}\unicode{x2013}10^{-1}\,{\rm s}$. Here, $Wi \gg 1$ suggests the presence of unrelaxed tension in the liquid phase and, hence, the stabilizing role of elasticity. This explains the increased stability of liquid morphologies with increasing $El$ as observed in stage III of the present study. It is interesting to note that the stabilizing role of elasticity is not observed either in theory or from experiments during RT and KH instability analysis, as discussed in the two previous subsections. This can be attributed to insignificant unrelaxed tension, which was not accounted for in those analyses. For the existence of unrelaxed tension, a significant amount of strain accumulation is also required, in addition to a high strain rate (Bousfield et al. Reference Bousfield, Keunings, Marrucci and Denn1986). While this condition is fulfilled for stage III, where large deformations are present, it is not true for RT and KH wave formation in stage II. The RT and KH wavelengths are measured at their early stage of inception, where the deformation suffered by a liquid element near the instability wave is small and insufficient for unrelaxed tension in the liquid phase. Therefore, the stabilizing effect of elasticity is not observed in stage II, and a simple linear stability analysis with zero stress in the base state of the liquid phase can capture the physics.

Figure 9. Strong extensional flow near the equator region of a polymeric droplet with $c/c^*=1.67$, $El=4\times 10^{-3}$ undergoing SIE mode of breakup at $We=987$.

3.7. Regime map of droplet breakup

Traditionally, aerobreakup modes of Newtonian droplets are presented in a $We$–$Oh$ number space (Guildenbecher et al. Reference Guildenbecher, López-Rivera and Sojka2009). A similar plot, by replacing $Oh$ with an effective Ohnesorge number, $Oh_{eff}$, has been used for non-Newtonian droplets (Theofanous et al. Reference Theofanous, Mitkin and Ng2013; Mitkin & Theofanous Reference Mitkin and Theofanous2017). The drawback of $Oh_{eff}$ is that it only accounts for the shear thinning/thickening effect through liquid viscosity but not the elastic effects. Parameter $We$ represents the ratio of aerodynamic force to the surface tension forces, and it is equally important for the aerobreakup of viscous and viscoelastic droplets. Whereas $Oh$ brings in the effect of liquid viscosity, which is relevant only for viscous droplets. In the case of viscoelastic liquids, apart from viscosity and surface tension, elasticity also plays an important role. The elasticity of any liquid appears in terms of a non-zero finite relaxation time, $\lambda$. It is obvious that a non-dimensional number like $El$ that accounts for both viscous and elastic effects must be chosen in the case of viscoelastic droplet breakup, which has not been done before. The elasticity number represents the relative strength of the viscoelastic force to the inertial force in the liquid phase, and it can be estimated from (1.2). Parameter $El$ has been shown to be relevant for understanding the primary breakup of viscoelastic liquid sheets and ligaments (Thompson & Rothstein Reference Thompson and Rothstein2007; Wang et al. Reference Wang, Yang, Xie and Chen2015). Such structures are also observed in the present study (figure 8b) in a way that the morphological hierarchy is in coherence with $El$. This motivates the usage of $El$ as a crucial governing parameter in the present work.

Results obtained from all the experimental runs are plotted in the form of a three-dimensional phase plot in $El$–$We$–time coordinates as shown in figure 10. Representative experimental images are shown as insets in the plot. Droplet deformation (stage I) happens for the complete range of $We$ and $El$ considered in the present study. At high $We$ (>2800), stage I and stage II seem to proceed simultaneously; therefore, these data points are not shown in stage I. Stage II and stage III are divided into three distinct regimes. The orange region corresponds to the low-$We$ (<700) regime where an unstable flattened sheet formed in stage II leads to the vibrational mode of breakup in stage III. The green region indicates the moderate-$We$ ($700< We<2800$) regime, where KH instability in stage II leads to the SIE breakup mode in stage III. The blue region corresponds to the high-$We$ (>2800) regime where RT instability in stage II causes catastrophic breakup in stage III. The critical $We$ for the transition from one regime to the next regime is independent of $El$, and this is supported by the experimental evidence and mathematical analysis provided in the previous subsections. Inset images provided in stage III show the stabilizing effect of $El$ in the aerobreakup process. Well-defined liquid morphologies obtained in the SIE regime are shown with different symbols.

Figure 10. Three-dimensional regime map showing different stages and modes of viscoelastic droplet breakup on $We$–$El$–time axes. Symbols with colour partition indicate transition between two consecutive modes.