The statistical mechanics of meandering R. Bruinsma To cite this version: R. Bruinsma. The statistical mechanics of meandering. Journal de Physique, 1990, 51 (9), pp.829-845. 10.1051/jphys:01990005109082900. jpa-00212412 HAL Id: jpa-00212412 https://hal.archives-ouvertes.fr/jpa-00212412 Submitted on 1 Jan 1990 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Phys. France 51 (1990) 829-845 1er MAI 1990, 829 Classification Physics Abstracts 05.40 - 46.30 - 68.42 The statistical mechanics of meandering R. Bruinsma Physics Department & Solid State Science Center, University of California, Los Angeles, Los Angeles, CA, 90024, U.S.A. (Reçu le 17 juillet 1989, accepté sous forme définitive le 1 b novembre 1989) Résumé. 2014 On présente un modèle simple décrivant la statistique du serpentement d’écoulements étroits sur un substrat propre et lisse. Ce modèle présente trois régimes différents suivant le débit: (i) pour des faibles débits, le trajet est une marche aléatoire stationnaire ; (ii) pour des débits intermédiaires, des méandres stationnaires apparaissent. L’apparition de ces méandres est très similaire à la physique des transitions de phase continues ; (iii) pour des débits élevés, les méandres commencent à glisser. Le problème du décrochage des méandres est équivalent au problème du décrochage des parois dans les systèmes magnétiques désordonnés. Cette correspondance permet de calculer le débit critique où apparaît le glissement des méandres. Abstract. 2014 We present a simple mathematical model to describe the statistical properties of the meandering of narrow streams on clean and smooth substrates. The model is shown to contain three different regimes, depending on the flow rate : (i) at low rates, the stream path is a time- independent random walk ; (ii) for intermediate flow rates, static meanders appear. The onset of meandering is found to be closely analogous to the physics of continuous phase transitions ; (iii) at high flow rates, the meanders start to slide downhill. The problem of the depinning of meanders can be mapped onto the problem of domain-wall depinning in disordered magnets. Using this correspondence, we can compute the critical flow rate for the onset of meander sliding. 1. Introduction. Under non-equilibrium conditions, the surfaces and interphases encountered in solid-state physics can exhibit fascinating patterns. Well known examples are dendrites, diffusion-limited aggregation and ballistic deposition. Fluid interfaces can also show pattern formation as demonstrated by viscous fingering. We will discuss in this paper a familiar hydrodynamic instability - stream meandering [1] - which shows pattern formation. This instability exhibits a number of features reminiscent of critical phenomena in condensed matter physics - a similarity which we will exploit later on. A well known example of the instability is the meandering of rivers. River meandering has been a long-standing fascinating problem with a considerable literature [2] to which even Einstein contributed. It involves a complex interplay between soil erosion and hydrodynamics. In the present paper, we will consider a closely related but simpler problem namely the question of finding the morphology of the stream-path of a narrow stream flowing down a Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005109082900 830 (rigid) inclined plane. The fluid is assumed to be non-wetting. There are a number of simplifications in this case : (i) for sufficiently narrow streams and sufficiently low flow rates, one may use the Poiseuille approximation ; (ii) erosion plays no role for a rigid substrate ; (iii) for narrow streams, surface tension provides an important stabilizing action which simplifies the analysis. The problem was investigated experimentally by Nakagawa and Scott [3] and by Walker [4]. We will briefly review their results for different values of the volume flow rate I. (i) For large inclinations of the plane (> 30° ), the stream forms stable meanders. The meanders consist of relatively straight diagonals connected by sharp bends. At low I, the meanders are less prominent while the stream-path is strongly correlated with the path taken by the stream when the flow was turned on. It also depends on height irregularities and chemical contamination. The shape of the meanders of narrow streams differs from that of the meanders of rivers (which are « sine-generated » curves [2]) but for convenience we will retain the name. (ii) With increasing I, the stream-path is reorganized and meandering becomes stronger. The appearance of the meanders appears to be triggered by turbulence and/or deformation of the stream cross-section. For larger I, it may take a long time before the stream-path stabilizes into a static pattern. (iii) Above a critical flow rate, 1,,2, the stream-path is unstable. Meanders constantly break up, reform, and slide downwards. Streams also may bifurcate. (iv) For low inclinations of the plane ( 30° ), there is a second critical current, ici. For I less then Ici the stream breaks up into droplets sliding individually down the plane. The physical origin of the destabilization of straight stream profiles is the centrifugal force f k exerted by a flowing fluid on a curved boundary surface (see Fig. 1). If the boundary forces a narrow stream of fluid to flow along a curve, then the change in momentum of the fluid Fig. 1. - Forces exerted on a curve in the stream. The centrifugal force f k tries to increase curvature while the surface tension force tries to reduce curvature. The normal of the fS component f g gravitational force tries to slide the stream down the plane. 831 elements, as they move through the curve, must be absorded by the boundary. The resulting force tries to increase the curvature of the stream profile and to deform the cross-section of the stream. The stream becomes longer as a result. The ratio S (L ) of the stream length and geometrical distance L between the initial and final points of the stream is called the sinusuosity. Mandelbrot [5] noted that for rivers, S (L ) has a power-law dependence on L. The instability may be triggered by small initial deformations in the stream cross-section, as was emphasized by Nakagawa and Scott. The increase in length of the stream path is opposed by the surface tension force f which tries to minimize the surface area of the fluid (Fig. 1). At low flow rates, surface tension wins so the stream should be relatively straight. With increased flow rates, surface tension is overcome by the centrifugal force and meandering starts. The third important force is gravity. The component of the gravitational force parallel to the stream-path is reduced in the diagonal sections as compared with that of a stream-path flowing in the direction of steepest descent. This means that the flow velocity also is reduced. Under steady state conditions, the volume flow rate I should be a fixed quantity so the stream cross-section of the diagonal sections must have increased. This indeed is seen experimentally. The diagonal sections become unstable due to the component f g of the gravitational force in the direction normal to the stream-path (Fig.1). This is also the force which is responsible for the sliding. If we rotate a diagonal section towards the horizontal, then the parallel force decreases in magnitude while the normal force increases. The reduction of the parallel force leads, as we saw, to a reduced flow velocity and a concomitant increase in the mass per unit length. The normal force then tries to slide the heavier sections downhill. The aim of this paper is to construct a mathematical model for narrow-stream meandering which is sufficiently simple so that it can be treated either analytically or numerically. Eventhough we are dealing with a simpler problem then river meandering, our model still involves a number of simplifications. The model certainly does not attempt to provide an exact description of the hydrodynamics of the stream. It has however been the experience in growth problems that the large distance geometrical properties of a surface or interface are relatively insensitive to details as long as the basic physical mechanisms are properly included. The hope is thus that the model presented in section 2 is useful for computing large scale properties. Whether this is indeed the case would of course would have to be confirmed experimentally. Some of the limitations of the model are discussed in the final section. In section 3, we first develop an analogy with the theory of continuous phase transitions and discuss the associated « phase-diagram ». In section 4 we compute the relationship between the threshold flow rate 1 c for the onset of meandering and the treshold flow rate Ic2 for the onset of meander sliding. In section 5, we compare with experiment, briefly discuss the dynamical aspects of the problem, such as the sliding velocity, and we finish by reexamining the « Landau » description of section 2. 2. The meander model. 2.1 FORMULATION. - We start by defining the inclined plane down which the stream is flowing. Let r = (x, y ) be a horizontal surface with x e [0, L ] and let y be unbounded. The inclined plane is assumed to have an average height ho (r ) = - ax, with a the tangent of the angle between the horizontal plane and the inclined surface.