The Rayleigh-Plateau Instability

Introduction The Rayleigh-Plateau instability explains why a jet or cylinder of fluid breaks up into identical, evenly spaced droplets. We will be studying this instability for the case of a column of fluid on a string.

Experiment Fluids Strings Methods Canola oil Corn syrup Strings Fishing line 18 gauge wire Methods The wire was clamped, and an eye dropper was used to apply the fluid. The eye dropper was clamped, and the wire was pulled through it. Pictures were taken using a high speed camera operating at 60 fps.

Data Canola oil on fishing line applied using the first method.

Canola oil on fishing line applied using the second method.

Measurements (cm) Diameter of the fluid and cylinder Diameter of the droplets Distance is measured between peaks of droplets Theoretical distance is from Lord Rayleigh’s calculations CanolaFishpull CanolaFish CanolaWire CornFish CornWire diameter(fluid) 6.80E-02   8.80E-02 1.20E-01 1.60E-01 diameter(line) 5.30E-02 5.80E-02 9.60E-02 9.90E-02 distance(theory) 3.00E-01 3.90E-01 5.30E-01 7.10E-01 distance(average) 3.40E-01 3.60E-01 4.60E-01 6.40E-01 distance(12) 3.30E-01 3.50E-01 4.30E-01 4.20E-01 6.60E-01 distance(23) 3.70E-01 5.40E-01 7.70E-01 distance(34) 4.10E-01 4.80E-01 6.10E-01 distance(45) 4.00E-01 7.30E-01 distance(56) 4.50E-01 5.70E-01 distance(67) 2.90E-01 6.70E-01 5.10E-01 5.90E-01 distance(78) 4.90E-01 5.20E-01 diameter(1) 1.40E-01 1.30E-01 1.80E-01 diameter(2) 1.10E-01 2.30E-01 diameter(3) diameter(4) 2.80E-01 2.10E-01 diameter(5) diameter(6) 3.10E-01 1.50E-01 diameter(7) 2.00E-01 diameter(8) 1.90E-01

Theory Determine stability of a perturbed column of fluid Assume axial symmetry Use a Fourier expansion Derive Rayleigh’s criteria Investigate shape of a single drop Use Lagrange multiplier to minimize surface area Define conditions Nondimensionalize Determine the governing equations

Determine stability of a perturbed column of fluid Assume the perturbations are axially symmetric. Consider one mode of the Fourier expansion. Let the radius of the fluid be given by: We consider a column of length . The volume constraint becomes

Now we have the relation , or Now consider the surface area of the column.

Substituting ao into S we get If , then the perturbation decreases the surface area and is therefore unstable. Thus the perturbation is unstable for wavelengths satisfying

Investigating the shape of a single drop Consider a droplet on a section of string with length equal to the wavelength corresponding to the fastest growing mode. S - Radius of the string u(x) - Radius of the fluid q - Contact angle b - domain of x

Minimize droplet surface area constrained to a constant volume using: Using Lagrange multiplier l, we minimize surface energy with the volume constraint:

We now nondimensionalize our equation by introducing the dimensionless quantities: Now we are minimizing subject to

Since, when S – lV is optimized we have, by Euler-Lagrange equation we have or

After using some of our conditions and rearranging the terms, we arrive at the governing equations for the droplet profile: In order to get a numerical solution to this problem, one must know the contact angle for the fluid string pair, or measure it directly. The maximum height would also have to be measured. Having to take these measurements weakens the predictive value of this model. Therefore we did not pursue this any further.

Conclusions We were able to successfully derive Rayleigh’s criteria. We determined the governing equations for the shape of the drop. The measured wavelengths were reasonably close to the wavelength predicted by Rayleigh for the fastest growing mode. Future work Find a reasonable way to solve for the radius of the drop. Derive the wavelength corresponding to the fastest growing mode. Use more “high-tech” means to observe the instablility.

The Amazing Apparatus!