An official website of the United States government
Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.
more »
« less
Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons
We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.
more »
« less
- Award ID(s):
- 2029540
- PAR ID:
- 10309980
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 477
- Issue:
- 2256
- ISSN:
- 1364-5021
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.more » « less
-
We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L 2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $$\lambda > 0$$ , there is a travelling wave solution to fKdV and fNLS $$\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $$ , which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in H α/2 [ − T , T ] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.more » « less
-
The existing concepts of non-reciprocity in propagation of acoustic or elastic waves are based either on nonlinear effects, or on local circulation of linear elastic fluid that leads to red or blue Doppler shift, depending on the direction of sound wave. The same concepts exist for electromagnetic non-reciprocity, where external magnetic field may produce the effect similar to local rotation of the medium. These two concepts originate from two known methods of breaking a time-reversal symmetry (T-symmetry), that is necessary for observation of nonreciprocal wave propagation. Both concepts require additional electrical or mechanical devices to be installed with their own power sources. Here we propose to explore viscosity of fluid as a natural factor of the T-symmetry breaking through energy dissipation. We report experimental observation of the nonreciprocal transmission of ultrasound through a water-submerged phononic crystals consisting of several layers of aluminum rods arranged in a square lattice. While viscous losses break the T-symmetry, making the wave propagation thermodynamically irreversible, the transmission remains reciprocal if the scatterers are symmetrical. To generate different energy losses for opposite directions of propagation, the P-symmetry of the crystal is broken by using asymmetric scatterers. Due to asymmetry, two sound waves propagating in the opposite directions produce different distributions of velocity and pressure that leads to different local absorption. Dissipation of acoustic energy occurs mostly near the surface of the scatterers and it strongly depends on surface roughnesses. Using two phononic crystal with smooth and rough aluminum rods we demonstrate low (2-5 dB) and high (10-15 dB) level of non-reciprocity within a wide range of frequencies, 300-600 kHz. Experimental results are in agreement with numerical simulations based on the Navier-Stokes equation. This nonreciprocal linear device is very cheap, robust and does not require energy source.more » « less
-
null (Ed.)In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. ( J. Fluid Mech. , vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas ( J. Fluid Mech. , vol. 689, 2011, pp. 129–148) and Haut & Ablowitz ( J. Fluid Mech. , vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2021.0550
