Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.

A two-layer long-wave approximation of the homogeneous Euler equations for a free-surface flow evolving over mild slopes is derived. The upper layer is turbulent and is described by depth-averaged equations for the layer thickness, average fluid velocity and fluid turbulent energy. The lower layer is almost potential and can be described by Serre–Su–Gardner–Green–Naghdi equations (a second-order shallow water approximation with respect to the parameter $H/L$, where $H$ is a characteristic water depth and $L$ is a characteristic wavelength). A simple model for vertical turbulent mixing is proposed governing the interaction between these layers. Stationary supercritical solutions to this model are first constructed, containing, in particular, a local turbulent subcritical zone at the forward slope of the wave. The non-stationary model was then numerically solved and compared with experimental data for the following two problems. The first one is the study of surface waves resulting from the interaction of a uniform free-surface flow with an immobile wall (the water hammer problem with a free surface). These waves are sometimes called ‘Favre waves’ in homage to Henry Favre and his contribution to the study of this phenomenon. When the Froude number is between 1 and approximately 1.3, an undular bore appears. The characteristics of the leading wave in an undular bore are in good agreement with experimental data by Favre (Ondes de Translation dans les Canaux Découverts, 1935, Dunod) and Treske (J. Hydraul Res., vol. 32 (3), 1994, pp. 355–370). When the Froude number is between 1.3 and 1.4, the transition from an undular bore to a breaking (monotone) bore occurs. The shoaling and breaking of a solitary wave propagating in a long channel (300 m) of mild slope (1/60) was then studied. Good agreement with experimental data by Hsiao et al. (Coast. Engng, vol. 55, 2008, pp. 975–988) for the wave profile evolution was found.

Abstract. Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related with the classical shallow water model with the change of variables. The derived symmetries are used to generate new exact solutions of the rotating shallow equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

In this paper we consider a system of integrodifferential equations describing, in a long wave approximation, plane-parallel rotational motions of an ideal incompressible liquid with a free surface. Using special identities, we calculate integral Riemann invariants which are conserved along the trajectories and characteristics. A new class of solutions for this system is found. The description of these special solutions simplifies essentially since in this case the integrodifferential system reduces to a hyperbolic system of two differential equations admitting formulation in the Riemann invariants. The solutions describing propagation of simple waves are obtained in an analytical form. Numerical simulations of rotational flows are performed using both the system describing the special class of the solutions and shallow water equations for rotational flows. In order to describe discontinuous rotational flows, the equations of motion are written in a special conservation form and jump conditions are derived. A non-oscillatory, second-order central scheme is used in the calculations. Some illustrative solutions are presented for smooth and discontinuous rotational flows.

This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Twodimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained.Introduction. Approximate models of shallow-water theory are used to model wave processes in fluids and to describe large-scale motions in the atmosphere and ocean. A mathematical foundation for the classical (depth-averaged) shallow-water approximation was given by Ovsyannikov [1]. The long-wave model taking into account velocity shear along the depth, especially in the two-dimensional case has been studied to a lesser extent. The nonlinear equations of rotational shallow water for plane-parallel motions were studied in [2-7], etc., where infinite series of conservation laws were found, classes of exact solutions were constructed, and conditions for the generalized hyperbolicity and well-posedness of the Cauchy problem were formulated. Teshukov [8,9] studied the shallow-water equations for two-dimensional shear flow, established the existence of simple waves, constructed an extension of Prandtl-Meyer waves, and formulated conditions for the generalized hyperbolicity of the steady-state equations.In the present work, a theoretical group analysis of the two-dimensional shallow-water equations for shear flows was performed. The 9-dimensional group of the admitted transformations was found. It was established that the Lie algebra of operators L 9 corresponding to these transformations is isomorphic to the Lie algebra of the admitted operators for the equations of two-dimensional isentropic motion of a polytropic gas with an adiabatic exponent γ = 2, for which the optimal system subalgebras [10] is known. New classes of exact solutions were constructed using an optimal system of subalgebras that allows a classification of submodels. Steady-state rotationally symmetric solutions describing motion with zones of return flow were obtained. Stable unsteady shear solutions describing the spread (collapse) of a parabolic cavity were found.1. Mathematical Model and Admitted Transformations. The solutions of the system of differential equations

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A new type of efficient chiral catalyst has been elaborated for asymmetric C-alkylation of CH acids under PTC conditions. Sodium alkoxides formed from chiral derivatives of tartaric acid and aminophenols (TADDOL's 2a-e and NOBIN's 3a-h) can be used as chiral catalysts in the enantioselective alkylation, as exemplified by the reaction of Schiff's bases 1a-e derived from alanine esters and benzaldehydes with active alkyl halides. Acid-catalyzed hydrolysis of the products formed in the reaction afforded (R)-alpha-methylphenylalanine, (R)-alpha-naphthylmethylalanine, and (R)-alpha-allylalanine in 61-93% yields and with ee 69-93%. The procedure could be successfully scaled up to 6 g of substrate 1b. When (S,S)-TADDOL or (R)-NOBIN are used, the (S)-amino acids are formed. A mechanism rationalizing the observed features of the reaction has been suggested.

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