Fluid thread breakup

While linear theory is useful in considering the growth of small disturbances on the free surface, when the disturbances grow to have a significant amplitude, non-linear effects begin to dominate breakup behavior. The non-linear behavior of the thread governs its final breakup and ultimately determines the final shape and number of the resulting fluid masses.

Nonlinearity is captured through the use of self-similarity. Self-similarity assumes that the behavior of the fluid thread as the radius approaches zero is the same as the behavior of the fluid thread when it has some finite radius. Detailed understanding of non-linear thread behavior requires the use of asymptotic expansions to generate the appropriate scaling behavior. Numerous solutions have been found for the non-linear behavior of fluid threads based on the forces that are relevant in particular circumstances.[15][16][17]

How a fluid thread or jet undergoes breakup is governed by several parameters among which are the Reynolds number, the Weber number, Ohnesorge number, and the disturbance wavelength. While these numbers are common in fluid mechanics, the parameters selected as scales must be appropriate to thread breakup. The length scale most often selected is the radius of the fluid thread, while the velocity is most often taken to be the velocity of the bulk fluid motion. However, these scales can change based on the characteristics of the considered problem.

The Reynolds number is the ratio between inertia and viscous effects within the thread. For large Reynolds numbers, the effects of motion of the thread are much greater than viscous dissipation. Viscosity only has a minimal damping effect on the thread. For small Reynolds numbers, viscous dissipation is large and any disturbances are rapidly damped from the thread.

The Weber number is the ratio between inertia and surface tension effects within the thread. When the Weber number is large, the inertia of the thread is large which resists the tendency of surface tension to flatten bent surfaces. For small Weber numbers, the changes in the capillary pressure due to the surface disturbances is large and surface tension dominates thread behavior.

The Ohnesorge number is the ratio between viscous and surface tension effects within the thread. As it eliminates the effects of inertia and the need for a velocity scale, it is oftentimes more convenient to express scaling relationships in terms of the Ohnesorge number rather than the Reynolds and Weber number individually.

The perturbation wavelength is the characteristic length of the disturbance on the surface of the jet, assuming that any arbitrary disturbance can be decomposed via a Fourier transform into its constitutive components. The wavelength of the perturbation is critical in determining if a particular disturbance will grow or decay in time.

The linear stability of low viscosity liquids was first derived by Plateau in 1873.[14] However, his solution has become known as the Rayleigh-Plateau instability due to the extension of the theory by Lord Rayleigh to include fluids with viscosity. Rayleigh-Plateau instability is often used as an introductory case to hydrodynamic stability as well as perturbation analysis.

Plateau considered the stability of a thread of fluid when only inertial and surface tension effects were present. By decomposing an arbitrary disturbance on the free surface into its constitutive harmonics/wavelengths, he was able to derive the a condition for the stability of the jet in terms of the perturbation:

${\displaystyle \omega ^{2}={\frac {\sigma k}{\rho a^{2}}}{\frac {I_{1}(ka)}{I_{0}(ka)}}\left(1-k^{2}a^{2}\right),}$

where ω is the growth rate of the perturbation, σ is the surface tension of the fluids, k is the wavenumber of perturbation, ρ is the fluid density, a is the initial radius of the unperturbed fluid, and I is the modified Bessel function of the first kind. By computing the growth rate as a function of wavenumber, one can determine that the fastest growing disturbance wavelength occurs at:

${\displaystyle \lambda _{\text{max}}\approx 9.02a.}$

The wavelength of maximum instability increases as the radius of the fluid thread increases. As importantly, unstable modes are only possible when:

${\displaystyle ka<1.}$

Reynolds and later Tomotika extended Plateau's work to consider the linear stability of viscous threads. Rayleigh solved for the stability of a viscous thread of viscosity ${\displaystyle \mu _{A}}$ without the presence of an external fluid.[18] Tomokita solved for the stability of a fluid thread in the presence of an external fluid with its own viscosity ${\displaystyle \mu _{B}}$.[19] He considered three cases where the viscosity of the fluid thread was much greater than the external environment, the viscosity of the external environment was much greater than the fluid thread, and the general case where the liquids are of arbitrary viscosity.

For the limiting case where the fluid thread is much more viscous than the external environment, the viscosity of the external environment falls from the growth rate completely. The growth rate thus becomes only a function of the initial radius of thread, the perturbation wavelength, the surface tension of the thread, and the thread viscosity.

${\displaystyle \omega ={\frac {\sigma \left(k^{2}a^{2}-1\right)}{2a\mu _{A}}}{\frac {1}{k^{2}a^{2}+1-k^{2}a^{2}I_{0}^{2}(ka)/I_{1}^{2}(ka)}}}$

Plotting this, one finds that the longest wavelengths are the most unstable. As importantly, one can note that the viscosity of the fluid thread does not influence which wavelengths will be stable. Viscosity only acts to decrease how rapidly a given perturbation will grow or decay with time.

Examples of when this case would apply are when almost any liquid undergoes thread/jet breakup in an air environment.

For the limiting case where the external environment of the fluid thread is much more viscous than the thread itself, the viscosity of the fluid thread falls from the perturbation growth rate completely. The growth rate thus becomes only a function of the initial radius of the thread, the perturbation wavelength, the surface tension of the thread, the viscosity of the external environment, and the second order Bessel functions of the second kind.

${\displaystyle \omega ={\frac {\sigma \left(1-k^{2}a^{2}\right)}{2a\mu _{B}}}{\frac {1}{k^{2}a^{2}+1-k^{2}a^{2}K_{0}^{2}(ka)/K_{1}^{2}(ka)}}}$

If one were to plot the growth rate as a function of the perturbation wavelength, one would find that the most unstable wavelengths again occur at the longest wavelengths and that the viscosity of the external environment would only act to decrease how rapidly a perturbation would grow or decay in time.

Examples of when this case would apply are when gas bubbles enter a liquid or when water falls into honey.

The general case for two viscous fluids is much more difficult to solve directly. Tomotika expressed his solution as:

${\displaystyle \omega ={\frac {\sigma \left(1-k^{2}a^{2}\right)}{2a\mu _{B}}}\Phi (ka)}$

where ${\displaystyle \Phi }$ was defined as:

${\displaystyle {\begin{aligned}\Phi (ka)={}&{\frac {N(ka)}{D(ka)}}\\N(ka)={}&I_{1}(ka)\Delta _{1}-\{kaI_{0}(ka)-I_{1}(ka)\}\Delta _{2}\\D(ka)={}&{\frac {\mu _{A}}{\mu _{B}}}\{kaI_{0}(ka)-I_{1}(ka)\}\Delta _{1}-{\frac {\mu _{A}}{\mu _{B}}}\left\{\left(k^{2}a^{2}+1\right)I_{1}(ka)-kaI_{0}(ka)\right\}\Delta _{2}\\&-\{kaK_{0}(ka)+K_{1}(ka)\}\Delta _{3}-\left\{\left(k^{2}a^{2}+1\right)K_{1}(ka)+kaK_{0}(ka)\right\}\Delta _{4}\end{aligned}}}$

The ${\displaystyle \Delta }$ coefficients are most easily expressed as the determinants of the following matrices:

${\displaystyle {\begin{aligned}\Delta _{1}&={\begin{vmatrix}kaI_{0}(ka)-I_{1}(ka)&K_{1}(ka)&-kaK_{0}(ka)-K_{1}(ka)\\I_{0}(ka)+kaI_{1}(ka)&-K_{0}(ka)&-K_{0}(ka)+kaK_{1}(ka)\\{\frac {\mu _{A}}{\mu _{B}}}kaI_{0}(ka)&K_{1}(ka)&-kaK_{0}(ka)\end{vmatrix}}\\[3pt]\Delta _{2}&={\begin{vmatrix}I_{1}(ka)&K_{1}(ka)&-kaK_{0}(ka)-K_{1}(ka)\\I_{0}(ka)&-K_{0}(ka)&-K_{0}(ka)+kaK_{1}(ka)\\{\frac {\mu _{A}}{\mu _{B}}}I_{1}(ka)&K_{1}(ka)&-kaK_{0}(ka)\end{vmatrix}}\\[3pt]\Delta _{3}&={\begin{vmatrix}I_{1}(ka)&kaI_{0}(ka)-I_{1}(ka)&-kaK_{0}(ka)-K_{1}(ka)\\I_{0}(ka)&I_{0}(ka)+kaI_{1}(ka)&-K_{0}(ka)+kaK_{1}(ka)\\{\frac {\mu _{A}}{\mu _{B}}}I_{1}(ka)&{\frac {\mu _{A}}{\mu _{B}}}kaI_{0}(ka)&-kaK_{0}(ka)\end{vmatrix}}\\[3pt]\Delta _{4}&={\begin{vmatrix}I_{1}(ka)&kaI_{0}(ka)-I_{1}(ka)&K_{1}(ka)\\I_{0}(ka)&I_{0}(ka)+kaI_{1}(ka)&-K_{0}(ka)\\{\frac {\mu _{A}}{\mu _{B}}}I_{1}(ka)&{\frac {\mu _{a}}{\mu _{B}}}kaI_{0}(ka)&K_{1}(ka)\end{vmatrix}}\end{aligned}}}$

The resulting solution remains a function of both the thread and external environment viscosities as well as the perturbation wavelength. The most unstable combination of viscosities and perturbation occurs when ${\displaystyle \mu _{A}/\mu _{B}\approx 0.28}$ with ${\displaystyle \lambda \approx 10.66a}$.

For most applications, use of the general case is unnecessary as the two fluids in question have significantly different viscosities which permits the use of one of the limiting cases. However, some instances such as the mixing of oils or oils and water may require the use of the general case.

Dripping water is an everyday occurrence. As water leaves the faucet, the filament attached to the faucet begins to neck down, eventually to the point that the main droplet detaches from the surface. The filament cannot retract sufficiently rapidly to the faucet to prevent breakup and thus disintegrates into several small satellite drops.

Air bubbles are another common breakup phenomenon. As air enters a tank of liquid, like a fish tank, the thread again necks at the base to produce a bubble. Blowing bubbles from a straw into a glass behaves in much the same manner.

The pitch drop experiment is a famous fluid breakup experiment using high viscous tar pitch. The rate of breakup is slowed to such a degree that only 11 drops have fallen since 1927.

Honey is sufficiently viscous that the surface perturbations that lead to breakup are almost fully damped from honey threads. This results in the production of long filaments of honey rather than individual droplets.