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Flow with a free surface
Published in Bernard S. Massey, John Ward-Smith, Mechanics of Fluids, 2018
Bernard S. Massey, John Ward-Smith
Waves whose characteristics are governed principally by surface tension are known as capillary waves. For waves of very short length, usually termed ripples, the term in eqn 10.54 involving g becomes negligible. For ripples on water with λ = 3 mm, say, gravity affects c by only 1.5%. When λ is small compared with h then tanh(2πh/λ) → 1 and the velocity of capillary waves → (2πγ/ϱλ)1/2. The frequency, that is, the number of wave crests passing a given fixed point divided by time interval, is c/λ, which, for λ = 3 mm on deep water, is 131 Hz. Thus capillary waves can be generated by a tuning fork held in a liquid – although they rapidly decay and cannot be seen for more than a few centimetres. Nevertheless, the measurement of the length of waves produced by a tuning fork of known frequency is the basis of one method of determining surface tension. (In practice, non-uniformity of the value of γ caused by contamination of the surface can affect the results somewhat for short waves.)
Polymeric Surfactants
Published in E. Desmond Goddard, James V. Gruber, Principles of Polymer Science and Technology in Cosmetics and Personal Care, 1999
E. Desmond Goddard, James V. Gruber
The tangential bulk-phase stress component evaluated at the interface combines an elastic (interfacial tension gradient) effect, ε', and an apparent viscous effect, ηsd+ηs+ε''/ω. One of the most convenient methods of measuring capillary waves is to use light scattering (29), which can yield information on both the tension and dilational modulus of the interface.
Non-simultaneous impact of multiple droplets on liquid film
Published in Numerical Heat Transfer, Part A: Applications, 2019
Gangtao Liang, Tianyu Zhang, Yang Chen, Liuzhu Chen, Shengqiang Shen
Many capillary waves with various wavelengths are created on the liquid sheet enclosing entrained bubble prior to its breakup, with high pressure beneath crests but low pressure beneath troughs, caused by collision of liquid moving from different directions within the bump, and loss of liquid in the neck, respectively. The criterion devised purposely for the liquid jet breakup can be used to estimate the breakup of enclosed liquid sheet.
Free-surface behaviour of shallow turbulent flows
Published in Journal of Hydraulic Research, 2021
Fabio Muraro, Giulio Dolcetti, Andrew Nichols, Simon J. Tait, Kirill V. Horoshenkov
Gravity-capillary wave theory describes the behaviour of the free surface of an incompressible fluid without viscosity, under the restoring effects of gravity and surface tension. A weakly deformed surface can be approximated as a linear combination of mutually independent sinusoids (waves). Each wave is characterized by a frequency, f, and by a wavenumber vector k with modulus , where λ is the wavelength. The wavenumber vector is pointing in the direction of propagation of the wave, while the wave celerity is the ratio between the frequency and wavenumber modulus, . The celerity of gravity-capillary waves is a function of the wavenumber, as well as of the flow depth, and mean flow velocity distribution. The relation between frequency and wavenumber is a dispersion relationship. Assuming a horizontally homogeneous flow, and no external forcing, the dispersion relation of gravity-capillary waves was obtained either analytically or numerically for a variety of average velocity vertical profiles, including a linear profile (Bièsel, 1950), power-function profile (Fenton, 1973), or more generic profiles (Ellingsen & Li, 2017). If the flow shear-rate is small (Shrira, 1993), the wave frequency can be approximated as: where U0 is the flow velocity vector at the surface, ci(k) is the intrinsic celerity and D is the water depth. The two solutions with opposite signs represent waves with opposite direction of propagation. The intrinsic celerity ci(k) in Eq. (2) indicates the speed of propagation of the waves relative to the mean flow. For water with depth larger than 2(T/ρfg)1/2 = 5.5 mm, ci(k) has a minimum of approximately 0.23 m s−1 when the effects of gravity and surface tension balance, at λ ≈ 1.7 cm (k ≈ (ρf g/T)1/2 = 367 m−1). Longer waves are governed by gravity and have a maximum celerity limit of (gD)1/2.
Wave turbulence: the case of capillary waves
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
With gravity waves, capillary waves constitute the most common surface waves encountered in nature. The latter have an advantage over the first in the sense that they are easier to treat analytically in the nonlinear regime. To introduce the physics of capillary waves, we will consider an incompressible fluid () (like water) subject to irrotational movements ( with φ the velocity potential). This condition is well justified when the air–water interface is disturbed by a wind blowing unidirectionally (a typical condition encountered in the sea). The nonlinear equations describing the dynamics of capillary waves are obtained by first noting that the deformation of the fluid at the air–water interface verifies the exact Lagrangian relation where is the deformation and the velocity potential (see figure 1 for an illustration). The Bernoulli equation (inviscid case) applied to the free surface of the liquid (at ) is where with γ the coefficient of surface tension (for the air–water interface N/m) and the mass density of water. Note that the mass density of the air is negligible compared to that of water. The surface tension term is obtained by assuming that the deformation is relatively weak, i.e. . This tension is responsible for a discontinuity between the fluid pressure at its free surface and the pressure of the atmosphere ; it is modelled by the relation with R the radius of curvature of the free surface (Guyon et al.2015). The hypothesis of a weak deformation (or weak curvature) makes it possible to simplify the modelling.