String Cheese - Water Stream Dynamics · USAPhO 2020 #B1

Physics · fluid-mechanics

Problem

a. When a faucet is turned on, a stream of water flows down with initial speed v0v_0 at the spout. For this problem, we define yy to be the vertical coordinate with its positive direction pointing up. Assuming the water speed is only affected by gravity as the water falls, find the speed of water v(y)v(y) at height yy. Define the zero of yy such that the equation for v2v^2 has only one term and find y0y_0, the height of the spout. b. Assume that the stream of water falling from the faucet is cylindrically symmetric about a vertical axis through the center of the stream. Also assume that the volume of water per unit time exiting the spout is constant, and that the shape of the stream of water is constant over time. In this case, the radius rr of the stream of water is a function of vertical position yy. Let the radius at the faucet be r0r_0. Using your result from part (a), find r(y)r(y). If r(y)r(y) is not constant, it implies that the water has some radial velocity during its fall, in contradiction to our assumptions in part (a) that the motion is purely vertical. You may assume throughout the problem that any such radial velocity is negligibly small. c. The water-air interface has some surface tension, σ\sigma. The effect of surface tension is to change the pressure in the stream according to the Young-Laplace equation, ΔP=σ(1r+1R),\Delta P = \sigma \left( \frac{1}{r} + \frac{1}{R} \right), where ΔP\Delta P is the difference in pressure between the stream and the atmosphere and RR is the radius of curvature of the vertical profile of the stream, visualized below. (R<0R < 0 for the stream of water; the radius of curvature would be positive only if the stream profile curved inwards.) [Diagram showing |R|] For this part of the problem, we assume that ∣R∣≫∣r∣|R| \gg |r|, so that the curvature of the vertical profile of the stream can be ignored. Also assume that water is incompressible. Accounting for the pressure in the stream, find a new equation relating for r(y)r(y) in terms of σ,r0,v0\sigma, r_0, v_0, and ρ\rho, the density of water. You do not need to solve the equation for rr. d. After falling for some distance, the water stream usually breaks into smaller droplets. This occurs because small random perturbations to the shape of the stream grow over time, eventually breaking the stream into apart. For the rest of this problem we ignore the change in the radius of the stream due to changing speed of the water, as considered earlier. Instead, we examine small random variations in the radius of the stream. Random variations can be broken down into a sum of sinusoidal variations in stream radius, each with a different wavenumber kk. We can analyze these different sinusoidal variations independently. Consider a stream of water whose radius obeys r(y)=r0+Acos⁡(ky),r(y) = r_0 + A\cos(ky), where A≪r0A \ll r_0 is the perturbation amplitude. To analyze such a stream, it is sufficient to consider only the thickest and thinnest parts of the stream. Accounting for both sources of curvature, find a condition on r0r_0 and kk such that the size of perturbations increases with time. [Diagram showing r(y) and |R(y)|]

Related theory

Browse the problem library