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Interfacial Phenomena and Heat Transfer

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ISSN Print: 2169-2785

ISSN Online: 2167-857X

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NONLINEAR WAVE DYNAMICS OF LIQUID-SATURATED POROUS MEDIA

Volume 11, Issue 3, 2023, pp. 43-59
DOI: 10.1615/InterfacPhenomHeatTransfer.2023045651
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ABSTRACT

The propagation of plane longitudinal waves in a liquid-saturated porous media with cavities is considered in a linear elastic approximation. We show that, as distinct from the classical porous medium (Biot's medium) where two longitudinal nondispersive waves (a fast wave and a slow wave) can propagate, in a cavity-porous medium three dispersive longitudinal waves propagate, one of which is solely induced by available cavities in the material. The study of wave behavior is carried out by obtaining and analyzing the dispersion equation, a phase velocity and a group velocity. The frequency spectrum density is considered, which enables us to determine the degree of dispersion. At specific values of the system parameters with some regions of strong and weak dispersion, regions of normal and anomalous dispersion are identified. The propagation of nonlinear periodic (cnoidal) plane longitudinal waves and solitons in the liquid-saturated porous medium with cavities is studied on the assumption that the wave energy dissipation in the medium may be neglected. The effect of the nonlinear periodic wave velocity on the wave amplitude and wavenumber is shown. The influence of the system parametersminus;in particular, the size of spherical cavities on the main parameters (the amplitude and width) of localized waves−is identified. The obtained results are compared to classic soliton behavior.

Figures

  • FIG. 1: Dynamics of the dispersion curve ω(k) (solid line) upon changing parameter b2: (a) b b 2 2
= (1); (b) b b 2 2
= (2); (c) b b 2 2
= (3); (d)
b b 2 2
= (4); b b b b 2
1
2
2
2
3
2
 ( )  ( )  ( )  (4) 
  • FIG. 2: Dynamics of the dispersion curve ω(k) (solid line) upon changing parameter b1: (a) b b 1 1
= (1); (b) b b 1 1
= (2); (c) b b 1 1
= (3); (d)
b b 1 1
= (4); (e) b b 1 1
= (5); (f) b b 1 1
= (6); (g) b b 1 1
= (7); b b b b b b b 1
1
1
2
1
3
1
4
1
5
1
6
1
 ( )  ( )  ( )  ( )  ( )  ( )  (7) 
  • FIG. 3: Dependences (a), ω(k) (solid line); (b), vph(k) (long dash line) vgr(k) (dash-and-dot line) at b b 3 3
= (1); 1, 2, 3 are branches of
the dispersion curve
  • FIG. 4: Dependences ρ(ω) (solid line) at different values of parameter b3: (a) b b 3 3
= (1); (b) b b 3 3
= (2); b b 3
1
3
 ( )  (2) ; 1, 2, 3 are branches
of the dispersion curve
  • FIG. 5: Dependences (a) ω(k) (solid line); (b) vph(k) (long dashed line); vgr(k) (dash-dot line); (b) at b b 3 3
= (2)
  • FIG. 6: Phase portrait (V, Vξ) of an anharmonic oscillator with quadratic nonlinearity, (a) m1 < 0; (b) m1 > 0
  • FIG. 7: Wave profiles V+(ξ) corresponding to different trajectories of the phase portrait (m1 > 0): s → 0 (dashed line), s < 1 (dashdot
line), s → 1 (solid line)
  • FIG. 8: Dependence k(A) at fixed parameter values
  • FIG. 9: Dependences Δ(Ac) at fixed value of parameter r2 and parameter c change in the range: (a) 0  
 c c and r r 2 2
= (1); (b)
0  
 c c and r r 2 2
= (2); (с) c A c 

 
  and r r 2 2
= (3); (d) c  c
 and r r 2 2
= (4); r r 2
1
2
( )  ( )
 , r2
2 1 ( ) >> , 0 2
3
2   
r( ) r  ( ) , r2
4 0
  • FIG. 10: Dependences Ac(r2) (solid line) and Δ(r2) (long dashed line) at fixed velocity value c: (a) 0 < c < c* or c c c 

 
  ; (b)
c c c
  

or c
c

 
; 1 and 2 are the first and second types of solitons
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