Wine Glass Vortex: Boundary Layers in Rotating Fluids · AI Deep Generator #DEEP-C56AC0CA

Physics · mechanics

Problem

When a sommelier swirls wine in a glass, the liquid climbs up the walls before slowly descending in droplets. This phenomenon involves a delicate interplay between rotation, viscosity, and boundary effects. Consider a cylindrical glass of radius R=4.0R = 4.0 cm containing wine (density ρ=990\rho = 990 kg/m³, kinematic viscosity ν=1.5×10−6\nu = 1.5 \times 10^{-6} m²/s) initially at rest. The glass is suddenly set into rotation about its vertical axis with angular velocity ω0=3.0\omega_0 = 3.0 rad/s. (a) [8 pt] After a long time, the bulk of the fluid rotates as a solid body. However, near the vertical walls, there exists a boundary layer where the azimuthal velocity vθ(r)v_\theta(r) transitions from zero at the wall to ω0r\omega_0 r in the bulk. Using the steady-state Navier-Stokes equation in cylindrical coordinates, show that the boundary layer thickness δ\delta scales as δ∼ν/ω0\delta \sim \sqrt{\nu/\omega_0} and calculate its numerical value. (b) [10 pt] The reduced centripetal acceleration in the boundary layer creates a pressure imbalance. Show that this drives a radial secondary flow with velocity vr∼ω0δv_r \sim \omega_0 \delta near the bottom of the glass. Derive an expression for the mass flux per unit length along the bottom boundary. (c) [12 pt] This radial outflow must be balanced by upward flow along the walls. Using conservation of mass in a control volume near the wall-bottom junction, determine the vertical velocity vzv_z as a function of height zz above the bottom. Show that the flow forms a thin climbing layer of thickness ∼δ\sim \delta. (d) [10 pt] The climbing wine experiences both viscous drag and gravitational force. Find the maximum height hmaxh_{max} the wine can climb, expressing your answer in terms of the Ekman number E=ν/(ω0R2)E = \nu/(\omega_0 R^2) and the Froude number Fr=ω02R/gFr = \omega_0^2 R/g. (e) [10 pt] The wine tears that form are actually enhanced by alcohol evaporation (Marangoni effect), but the primary climbing mechanism is the boundary layer flow. Estimate the time scale for the establishment of the steady climbing pattern and compare it to the spin-up time of the bulk fluid. Which process is faster and why?

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