Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth
Abstract
:1. Introduction
2. Dispersion Relation
2.1. Small Viscosity Case
2.2. Behaviour at Large Wavelengths
2.3. Numerical Methods for Finding Roots
2.4. Numerical Results for Frequencies
3. Parameter Dependence
3.1. Wave Modes
3.1.1. Modes with Real Q
3.1.2. Modes with Imaginary Q
3.1.3. Modes with Complex Q
3.2. Bifurcations
- I.
- Only imaginary solutions (infinitely many of them) are present;
- II.
- There is a single real solution and there are infinitely many imaginary solutions;
- III.
- There are two real solutions and there are infinitely many imaginary solutions;
- IV.
- There is a single complex solution and there are infinitely many imaginary solutions.
- 1.
- Imaginary to complex Q (border between regions I and IV, the red curve in Figure 6):For small K () we have
- 2.
- Complex to real Q (border between regions IV and III, the blue curve in Figure 6):For large K we haveThis implies (cf. Equations (21)–(24) and Equations (30) and (31)) that the transition to the over-damped mode happens at wavelengthIt is remarkable that this value does not depend on layer thickness h or gravity of Earth g, as it is a purely material constant. Numerically, we get for water, for glycerin and for mercury. In our opinion, this effect cannot be observed in water or mercury, but might be observed in glycerin. It is questionable if hydrodynamics are applicable at the scales obtained for water and mercury, albeit at least one study states that it is applicable down to the nanoscales [28]. It is obvious that in case of these short waves gravity does not play a role. Here we demonstrate, that our more general setup leads to the right results in this limiting case. There are other situations, however, when gravity becomes important. This happens, e.g., at the bifurcation at long wavelengths. Gravity and surface tension become equally important when the two bifurcations (at large and small wavelength) are close to each other, as in Figure 2a.
- 3.
- Imaginary to real Q (borders between regions I, II, and III, the black curve in Figure 6):For small K we haveAt parameter p diverges asFor a given material this implies (cf. Equation (35))
- 4.
- The common point of the bifurcation parameter curves (red, blue, and black lines in Figure 6 satisfiesThe solution of this equation yields for the coordinates of the common point
4. Minimal Layer Thickness Necessary for Wave Propagation
5. Particle Motion at Surface
6. Time Evolution of Surface Elevations
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Material | h | ||||
---|---|---|---|---|---|
water | 58.40 | ||||
glycerin | 0.45 1 | 0.085 | |||
mercury | 174.86 |
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Ghahraman, A.; Bene, G. Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth. Fluids 2023, 8, 173. https://doi.org/10.3390/fluids8060173
Ghahraman A, Bene G. Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth. Fluids. 2023; 8(6):173. https://doi.org/10.3390/fluids8060173
Chicago/Turabian StyleGhahraman, Arash, and Gyula Bene. 2023. "Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth" Fluids 8, no. 6: 173. https://doi.org/10.3390/fluids8060173