Keywords: roll wave, Bingham fluid, shallow water.

Flows that process under accentuated steep slopes may develop instabilities. However, the chaotic aspect present in the nature of these instable phenomena, seem to tend, at the end of a finite time, to a stationary flow, more stable, with the appearance of periodic long waves as a hydraulic jump or "bore waves". Such perturbations are called "roll waves". If, on one hand, these waves are rare in natural flows, on the other, they are frequent in artificial canals and in water flowing from spillways of dams.

The purpose of this article is to analyze the criteria for the occurrence of roll wave phenomenon in the supercritical and turbulent Newtonian and non-Newtonian flows from the engineering point of view. Rewriting the shallow water equations and taking into account the Bingham fluid behavior and fluid viscosity, first the conditions for the development of roll waves using the stability linear technique are presented. Second, it is presented a new mathematical model based on shallow water equations in which the numerical simulation was performed by a finite volume technique using Godunov-VanLeer schemes. Imposing a constant discharge at the upstream of the canal and superposing a small perturbation, it was observed that roll waves can be developed more easily for small wave numbers and for high cohesions. Finally, from the mathematical model used, it was demonstrated that the numerical viscosity is 10 times the physical viscosity.

Another study (mathematical approach ) developed by us is to investigate in detail the Van Der Pol problem adaptability to explain roll-waves phenomena. In this context, a linear instability analysis for a uniform flow has been undertaken. From this analysis we obtained two conditions:

1. with : Froude's Number, : wavenumber, : function of Froude, : steepness, : slope of the canal; : Reynold's Number, : Bingham's parameter (cohesion of the fluid) and the coefficient presented into the momentum equation that takes into account the diferent velocity profiles.
2. where is the "roll wave" celerity.

These last results are not presented in this paper, but they are shown in detail in other references.

1

B.S. Venant, "Théorie du mouvement non permanent des eaux", Institute de France, Acad�mie des sciences, Comptes rendus, vol 73, 147-237, 1871.

2

R.F. Dressler, "Mathematical solution of the problem of roll waves in inclined open channels", Communs pure appl. Math., vol 2, 149-194, 1949.

3

S.K. Gudunov, "A difference method for numeric calculation of discontinuous equations of hydrodinamics", Math. Sb, 47(89), 271-300, 1959.

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T. Takahashi, "Debris Flow" Monograph, IAHR, Balkema, Rotterdam, 1990.

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B. Vanller, "Toward the ultimate conservative difference sheme", Journal of Comp. Phys. 32, 101-136, 1981. doi:10.1016/0021-9991(79)90145-1

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G. Freitas Maciel, "Impulse waves generated by solid mass falling into the reservoirs". SBPN Scientific Journal, São Paulo,v. 4, n. 1, p. 54-66, 2000.

7

G. Freitas Maciel, "Analogia de roll waves ao problema de Van Der Pol. RBRH Revista Brasileira de Recursos H�dricos, Porto Alegre, v. 4, n. 4, p. 17-23, 1999.

8

G. Freitas Maciel, "Critérios de formação de instabilidades em canais de forte declividade para uma reologia não-newtoniana". RBRH Revista Brasileira de Recursos Hídricos, Porto Alegre, v. 4, n. 4, p. 25-39, 1999.

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