Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and general properties

A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 24–36, March–April, 2009.     

A Semi-Analytical Solution for Large-Scale Injection-Induced Pressure Perturbation and Leakage in a Laterally Bounded Aquifer–Aquitard System

A number of (semi-)analytical solutions are available to drawdown analysis and leakage estimation of shallow aquifer–aquitard systems. These solutions assume that the systems are laterally infinite. When a large-scale pumping from (or injection into) an aquifer–aquitard system of lower specific storativity occurs, induced pressure perturbation (or hydraulic head drawdown/rise) may reach the lateral boundary of the aquifer. We developed semi-analytical solutions to address the induced pressure perturbation and vertical leakage in a “laterally bounded” system consisting of an aquifer and an overlying/underlying aquitard. A one-dimensional radial flow equation for the aquifer was coupled with a one-dimensional vertical flow equation for the aquitard, with a no-flow condition imposed on the outer radial boundary. Analytical solutions were obtained for (1) the Laplace-transform hydraulic head drawdown/rise in the aquifer and in the aquitard, (2) the Laplace-transform rate and volume of leakage through the aquifer–aquitard interface integrated up to an arbitrary radial distance, (3) the transformed total leakage rate and volume for the entire interface, and (4) the transformed horizontal flux at any radius. The total leakage rate and volume depend only on the hydrogeologic properties and thicknesses of the aquifer and aquitard, as well as the duration of pumping or injection. It was proven that the total leakage rate and volume are independent of the aquifer’s radial extent and wellbore radius. The derived analytical solutions for bounded systems are the generalized solutions of infinite systems. Laplace-transform solutions were numerically inverted to obtain the hydraulic head drawdown/rise, leakage rate, leakage volume, and horizontal flux for given hydrogeologic and geometric conditions of the aquifer–aquitard system, as well as injection/pumping scenarios. Application to a large-scale injection-and-storage problem in a bounded system was demonstrated.     

Linear viscoelastic master curves of neat and laponite-filled poly(ethylene oxide)–water solutions

Aqueous solutions composed of dispersed nanoparticles and entangled polymers are shown to exhibit common viscoelasticity over a range of particle and polymer concentrations. Time–temperature superposition and time–concentration superposition are applied to generate rheological master curves for neat and laponite-filled aqueous solutions of poly(ethylene oxide). The shift factors were correlated in terms of temperature and concentration and are found to differ from previous reports for ideal polymer solutions, which can be rationalized with a molecular interpretation of the structure of the laponite–polymer solutions. Laponite addition to the concentrated polymer solution is observed to increase the relaxation time but decrease the elastic modulus, which is a consequence of polymer adsorption and bridging. The addition of small amounts of laponite to stable PEO–water solutions also leads to ageing on the time scale of days.     

Analytical Solutions to Multiphase First-Contact Miscible Models with Viscous Fingering

In this paper, we analyze an empirical model of viscous fingering for three-component, two-phase, first-contact miscible flows. We present the complete range of analytical solutions to secondary and tertiary water-alternating-gas (WAG) floods. An important ingredient in the construction of analytical solutions is the presence of detached (nonlocal) branches of the Hugoniot locus, that is, curves in composition space that satisfy the Rankine–Hugoniot conditions but do not contain the reference state. We illustrate how, in water–solvent floods into a medium with mobile water and residual oil (immobile to water), the solvent front and the water Buckley–Leverett front may interact, resulting in a leading water/solvent shock that is stable to viscous fingering. The analytical solutions explain why in these miscible tertiary floods, oil and solvent often break through simultaneously. We discuss the implications of the new solutions in the design of miscible tertiary floods, such as the estimation of the optimum WAG ratio.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

Formation of a gas hydrate due to injection of a cold gas into a porous reservoir partly saturated by water

Specific features of formation of gas hydrates due to injection of a gas into a porous medium initially filled by a gas and water are considered. Self-similar solutions of an axisymmetric problem, which describe the distributions of the basic parameters in the reservoir, are constructed. The existence of solutions is demonstrated, which predict gas hydrate formation both on the frontal surface and in the volume zone. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 137–150, May–June, 2008.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

Initiation of intense pressure fluctuations when a flow is turned in a duct with a blind-ended cavity

The initiation of narrow-band pressure fluctuations in the duct of a pipeline gas-compression station when the flow is turned near a blind-ended cavity was studied in a wind tunnel. The flow pattern under consideration was estimated from visualization of the flow in a water channel using the hydraulic analogy technique. It is believed that the high-intensity narrow-band pressure fluctuations observed in the flow result from the instability and regular rearrangement of large-scale vortex structures in the blind-ended cavity. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–111, March–April, 1998.     

Long-time Solvability of the Navier–Stokes Equations in a Rotating Frame with Spatially Almost Periodic Large Data

We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the ℓ ¹-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping from a global-in-time unique solution to the limit equation.     

A Free Boundary Approach to Two-Dimensional Steady Capillary Gravity Water Waves

We prove that suitable weak solutions of the two-dimensional steady capillary gravity water wave problem (even in the presence of vorticity) are smooth. On a technical level, solutions are closely related to critical points of the Mumford–Shah functional, so that our main task is to exclude cusps pointing into the water phase. We prove that cusps do not exist, and that the free surface is smooth.     

Open-channel waves generated by propagation of a discontinuous wave over a bottom step

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts). __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 31–44, January–February, 2008.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.     

Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T², adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.     

Modified shallow water equations which admit the propagation of discontinuous waves over a dry bed

A method for modeling the propagation of discontinuous waves over a dry bed using the first approximation of shallow water theory is proposed. The method is based on a modified conservation law of total momentum that takes into account the concentrated momentum losses due to the formation of local turbulent vortex structures in the fluid surface layer at a discontinuous-wave front. A quantitative estimate of these losses is obtained by deriving the shallow water equations from the Navier-Stokes equations with allowance for viscosity, which has a rapidly increasing effect in the turbulent flow regions described by discontinuous waves. The stability of the discontinuous waves admitted by the modified system of conservation laws of shallow water theory is examined. As an example, a comparative analysis is performed of the solutions of the dam-break problem obtained for the classical and modified shallow water models. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 22–43, November–December, 2007     

The Analytical Solution of the Boussinesq Equation for Flow Induced by a Step Change of the Water Table Elevation Revisited

The classical problem of flow induced by a sudden change of the piezometic head in a semi-infinite aquifer is re-examined. A new analytical solution is derived, by combining an expression describing the water table elevation upstream, obtained by the Adomian’s decomposition approach, to an existing polynomial expression (Tolikas et al. in Water Resour Res 20:24–28, 1984), adequate for the downstream region; the parameters of both approximations are computed by matching the two solutions at the inflection point of the water table. Although several analytical solutions are available in the literature, we demonstrate that the expression we have developed in this issue is the most accurate for strong or moderate non-linear flows, where the degree of non-linearity is defined as the ratio of the piezometric head elevation at the origin to the initial water table elevation. For this type of flows the perturbation-series solution of Polubarinova-Kochina, characterized by previous studies as the best available analytical solution provides physically unacceptable results, while the analytical solution of Lockington (J Irrig Drain Eng 123:24–27, 1997), used to check the accuracy of numerical schemes, underestimates the penetration distance of the recharging front.     

Mechanism of deep gel plugging emplacement in a nonhomogeneous reservoir

The process of determining the mechanism of deep gel plugging emplacement in the high-permeability zones of a reservoir is analyzed numerically for small characteristic gelling times. It is shown that the efficiency of the process is determined by the sequence and size of the injected reactant plugs, which when they react in the reservoir form the high-viscosity gel. The configuration and site of the plug are determined by the size of the water slug separating the reactant solutions. Tyumen'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 98–103, January–February, 1994.     

Flows Resulting from the Incidence of a Discontinuous Wave on a Bottom Step

The solvability of the problem of the flows resulting from the incidence of a discontinuous wave on a bottom step is studied using a single-layer shallow water model. Solutions in which the total energy of the flow is conserved at the step and those in which it is lost at the step are considered. Regions of double and triple hystereses in the obtained self-similar solutions are found. An analogy is drawn with the problem of single-layer flow over a bottom obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 8–22, March–April, 2006.     

From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System

We establish a Navier–Stokes–Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna–Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point is governed by a weak solution of a Navier–Stokes–Fourier system for all time.     

In situ force-balance tensiometry

 Although a fundamental physical parameter, surface tension is difficult to measure. Common tensiometry inaccuracy comes from failure to control air–liquid–solid contact conditions, or account for liquid meniscus geometry and buoyancy corrections. This paper describes an in situ tensiometry technique, based on withdrawal of a thin-walled tube from the liquid interface, that enforces a known air–liquid–solid contact condition. This technique can be pursued at any level of experimental hygiene. Experimental results for filtered tap water, an alcohol–water solution, and a surfactant–water solution show that results repeatable to three significant digits are obtained with modest effort for a variety of geometrical parameters. Received: 7 October 1997/Accepted: 23 April 1998     

Finite volume element method for analysis of unsteady reaction–diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.