两层流体界面上的孤立波
Solitary Waves at the interface of a Two-Layer Fluid
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摘要: 本文讨论两水平固壁间两层不可压无粘流体界面上的孤立波,计及界面上的表面张力效应.首先建立了适用于这种模型的基本方程组,并在弱色散近似下应用约化摄动法,导得了一阶界面升高所满足的Korteweg-de Vries方程,指出了按该方程系数α和μ的符号的异同,KdV孤立波可能凸向上或凸向下.然后详细讨论了原有近似下非线性效应与色散效应不能平衡的两种临界情形.在采用了适当的近似之后,对第一种临界情形(α=0)得到了修正的KdV方程,并指出,在所考虑的情形中,当μ>0时孤立波不存在,当μ<0时,孤立波仍可能存在,其形式与KdV孤立波不同;对第二种临界情形(μ=0),导得了推广的KdV方程,这时存在振荡型孤立波.文中还对近临界情形作了讨论.本文结果与一些经典结果完全一致,并把它们作了拓广.Abstract: In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account.First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries(KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion.lt is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients a and u of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind(a=0 and conclude that solitary waves cannot exist in the case considered as μ>0, but may still occur as μ<0, being in the form other than that of the KdV solitary wave. As for the critical case of second kind (μ=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.
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[1] Yih,C.S.,Stratifiea Flow.Chap.3,Academic Press,New York(1980),(1980)或易家训(李家春整理),分层流,《流体力学与应用数学讲座》之一,科学出版社(1982). [2] Miles,J.W.,Solitary waves,Ann.Rev.Fluid Mech.,12(1980),11-43. [3] Miles,J.W.,On internal solitary waves,Tellus,32(1979),456-462. [4] Keulegan,G.H.,Characteristics of internal solitary waves,J.Res.Natl.Bur.Stand.,51(1953),133-140. [5] Long,R.R.,Solitary waves in one-and two-fluid systems.Tellus,8(1956),460-471. [6] Benjamin,T.B.Internal waves of finite amplitude and permanent form.J.Fluid Mech.,25(1966),241-270. [7] Peters,A.S.and J.J.Stoker,Solitary waves in liquid having nonconstant density,Comm.Pure Appl,Math.,13(1960),115-164. [8] Kakutani,T.and N.Yamasaki,Solitary waves on a two-layer fluid,J.Phys.Soc.Japan,45(1978),674-679. [9] 戴世强,约化摄动法和非线性波远场分析,力学进展,12(1982). [10] Taniuti,T.,et al,Reductive Perturbation Method for Nonlinear wave PPopagation,Suppl.Progr.Theor.Phys.,No.55(1974),1-306. [11] 桥本英典,水の波——えの分散と集り,科学,40(1970),401-408. [12] Whitham,G.B.,Linear and Nonlinear Waves,Chap.13,Wiley-Interscience,New York(1974). [13] Mei,C.C.,The Applied Dynamics of Ocean Surface Waves.Chap.11,Wiley-Interscience,New York(1982),或梅强中(戴世强、周显初整理),水波动力学,第七章,《流体力学与应用数学讲座》之二,科学出版社(1982). [14] Jeffrey,A.and T.Kakutani,Weak nonlinear Weak nonlinear waves:a discussion centered arround the Korteweg-de Vries equation,SIAN Rev.14(1972),582-643. [15] Walker,L.R.,Interfacial solitary waves in a two-fluid medium.Phys.Fluids,16(1973),1794-1804. [16] Kakutani,T.and H,Ono.,Weak nonlinear hydromagnetic waves in a cold collision-free plasma,J.Phys.Soc.Japan,26(1969),1305-1318. [17] Kawahara,T.,Oscillatory solitary waves in dispersion media.J.Phys.Soc.Japan,33(1972),260-264. -
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