A fluid of uniform density moves with no viscosity under gravity alone

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Vorticity, circulation and vortex motion

Problem

A fluid of uniform density ρ\rho moves with no viscosity under gravity alone, so that Euler's equation holds in the form DuDt=−1ρ∇p−gz^,\frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho}\nabla p - g\hat{\mathbf{z}}, with zz measured upward from a fixed datum. Let C(t)C(t) be a material loop, that is a closed curve made always of the same fluid particles, and let Γ(t)=∮C(t)u⋅dℓ\Gamma(t) = \oint_{C(t)}\mathbf{u}\cdot\mathrm{d}\boldsymbol{\ell} be its circulation, taken anticlockwise. Label the particles of the loop by a parameter ss running from 00 to 11 round it, so that the loop is the curve x(s,t)\mathbf{x}(s,t) with x(0,t)=x(1,t)\mathbf{x}(0,t) = \mathbf{x}(1,t), and ∂x/∂t=u\partial\mathbf{x}/\partial t = \mathbf{u} at fixed ss. (a) Write Γ(t)\Gamma(t) as an integral over ss and differentiate it following the fluid, showing that two terms appear. (b) Show that the term coming from the stretching of the line element contributes nothing to DΓ/Dt\mathrm{D}\Gamma/\mathrm{D}t. (c) Substitute Euler's equation in the remaining term and hence prove Kelvin's circulation theorem, DΓ/Dt=0\mathrm{D}\Gamma/\mathrm{D}t = 0. State at each place which of the assumptions (no viscosity, density a function of pressure alone, conservative body force) you have used.

Browse the problem library