Two separate situations

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Fluid statics and the continuum picture

Fluid Dynamics, Ch Interfaces: capillary statics and the stress conditions at a surface

Problem

Two separate situations, both worked from the general interfacial stress balance (σij(2)−σij(1))nj=γ(∇⋅n^)ni−(∇sγ)i(\sigma^{(2)}_{ij}-\sigma^{(1)}_{ij})n_j=\gamma(\nabla\cdot\hat{\mathbf n})n_i-(\nabla_s\gamma)_i. The drop. A spherical drop of liquid (fluid 1, radius a=3.0 mma=3.0\,\mathrm{mm}) is suspended at rest in a large volume of gas (fluid 2) whose viscosity is negligible, so the gas exerts no shear on the drop's surface. The drop's shape is unaffected by gravity. A steady temperature difference of ΔT=6.0 K\Delta T=6.0\,\mathrm{K} is maintained between its two poles, and the surface tension of the liquid falls with temperature at the standard rate dγ/dT=−1.5×10−4 N m−1 K−1\mathrm{d}\gamma/\mathrm{d}T=-1.5\times10^{-4}\,\mathrm{N\,m^{-1}\,K^{-1}}, taken uniform and varying linearly with the polar angle θ\theta (measured from the colder pole, 0≤θ≤π0\le\theta\le\pi) between the two poles. The unit normal n^\hat{\mathbf n} points from the liquid into the gas. The flow this drives is slow enough that the normal viscous stress in the liquid may be neglected next to the pressure (a quasi-static approximation). The coated fibre. A liquid film of uniform unperturbed outer radius a=0.50 mma=0.50\,\mathrm{mm} completely wets a rigid, fixed cylindrical fibre of radius b=0.20 mmb=0.20\,\mathrm{mm}. There is no evaporation or condensation, so the volume of liquid per unit length of fibre is fixed, and the effect of gravity on the film's shape is neglected. The outer (free) surface is given a small sinusoidal ripple r=a0+ϵcos⁡kzr=a_0+\epsilon\cos kz with ϵ≪a0\epsilon\ll a_0, where zz runs along the fibre's axis and a0a_0 is fixed by volume conservation. ((a)) For the drop, starting from the general stress balance, obtain separate expressions for the normal and tangential components of the stress that the liquid must supply at its own surface, in terms of γ(θ)\gamma(\theta), aa and ∇sγ\nabla_s\gamma. ((b)) For the drop, evaluate the magnitude of the tangential (shear) stress the liquid surface carries, and state towards which pole it acts. ((c)) For the coated fibre, by repeating the fixed-volume, fixed-area argument used for a free liquid cylinder, but now for the annulus of liquid between r=br=b and the free surface, show that the relation between a0a_0 and the nominal radius aa, and the resulting change in free-surface area per unit length to order ϵ2\epsilon^2, does not involve bb. ((d)) For the coated fibre, state the threshold wavelength below which the coating is stable, evaluate it numerically, and say whether a ripple of wavelength λ=4.0 mm\lambda=4.0\,\mathrm{mm} grows or decays.

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