Instability of the Motion of a Drop in Filtration Flows

The behavior of an inclusion (drop) of foreign fluid in a porous medium saturated with another fluid is considered. A steady-state regime of gravity-induced drop settlement is found and the instability of this regime is demonstrated. The horizontal motion of a liquid inclusion under the action of a stationary reservoir pressure gradient is also studied.     

Instability and waves during generalized Newtonian fluid film flow down a vertical wall

The problem of linear stability of a non-Newtonian fluid film flowing down a vertical plane under the action of gravity is considered. The linear stability of steady-state flow with a plane free boundary and the nonlinear waves that arise if this flow is unstable are investigated. The results obtained for two rheological models, the power-law and Eyring fluids, are compared.     

Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading

Poroelasticity is a theory that quantifies the time-dependent mechanical behavior of a fluid-saturated porous medium induced by the interaction between matrix deformation and interstitial fluid flow. Based on this theory, we present an analytical solution of interstitial fluid pressure in poroelastic materials under uniaxial cyclic loading. The solution contains transient and steady-state responses. Both responses depend on two dimensionless parameters: the dimensionless frequency Ω that stands for the ratio of the characteristic time of the fluid pressure relaxation to that of applied forces, and the dimensionless stress coefficient H governing the solid-fluid coupling behavior in poroelastic materials. When the phase shift between the applied cyclic loading and the corresponding fluid pressure evolution in steady-state is pronounced, the transient response is comparable in magnitude to the steady-state one and an increase in the rate of change of fluid pressure is observed immediately after loading. The transient response of fluid pressure may have a significant effect on the mechanical behavior of poroelastic materials in various fields.     

Flow of a Weakly Conducting Fluid in a Channel Filled with a Porous Medium

We investigate the fully developed flow in a fluid-saturated porous medium channel with an electrically conducting fluid under the action of a parallel Lorentz force. The Lorentz force varies exponentially in the vertical direction due to low fluid electrical conductivity and the special arrangement of the magnetic and electric fields at the lower plate. Exact analytical solutions are derived for fluid velocity and the results are presented in figures. All these flows are new and are presented for the first time in the literature.     

The Onset and Nonlinear Regimes of Convection in a Two-Layer System of Fluid and Porous Medium Saturated by the Fluid

The onset of convection and its nonlinear regimes in a heated from below two-layer system consisting of a horizontal pure fluid layer and porous medium saturated by the same fluid is studied under the conditions of static gravitational field. The problem is solved numerically by the finite-difference method. The competition between the long-wave and short-wave convective modes at various ratios of the porous layer to the fluid layer thicknesses is analyzed. The data on the nature of convective motion excitation and flow structure transformation are obtained for the range of the Rayleigh numbers up to quintuple supercriticality. It has been found that in the case of a thick porous layer the steady-state convective regime occurring after the establishment of the mechanical equilibrium becomes unstable and gives way to the oscillatory regime at some value of the Rayleigh number. As the Rayleigh number grows further the oscillatory regime of convection is again replaced by the steady-state convective regime.     

Generation of an average flow by a pulsating stream near a curved free surface

The generation of an average flow near a curved free surface under the action of small-amplitude harmonic translational vibrations is studied. It is found that, in contrast to the case of a flat non-deformable free surface, an average flow is generated in the viscous boundary layer. This flow, propagating in the bulk fluid, can be described by the steady-state Navier-Stokes equations with effective boundary conditions for the shear stresses. It is shown that, in contrast to the flow generation near a solid surface, the flow generated near the free surface depends on the fluid viscosity and the curvature of the surface.     

Change of the types of instability of a steady two-layer flow in an inclined channel

A plane steady-state two-layer fluid flow under the coupled action of the buoyancy and Marangoni forces is considered. The system is oriented at an arbitrary angle with respect to the gravity force. Exact solutions generalizing the Ostroumov-Birikh solution are obtained and their stability is studied in the framework of a linear theory. On the basis of numerical calculations, the influence of the inclination angle, the thickness of the layers, and the wall heating conditions on the instability mechanisms is investigated.     

Motion of an External Load over a Semi-Infinite Ice Sheet in the Subcritical Regime

The three-dimensional problem of steady-state forced vibrations of fluid and semiinfinite ice sheet under the action of a local external load traveling along the rectilinear sheet edge at a constant velocity is considered. Two cases are analyzed. In the first case the fluid surface outside the ice sheet is free and in the second the fluid is confined by a rigid vertical wall and the ice sheet edge adjacent to the wall can be both clamped and free. The ice sheet is simulated by a thin elastic isotropic plate floating on the surface of fluid of finite depth. The load traveling velocity is assumed to be not higher than the minimum phase velocity of the flexural-gravity waves (subcritical regime). The solution to the linear problem is obtained by means of the integral Fourier transform and matching the expansions of the velocity potential in the vertical eigenfunctions. Examples of the numerical investigation of the ice sheet and fluid displacements are given.     

Simulation of the Onset of Nonstationarity and Chaos in a Hydrodynamic System Governed by a Small Number of Degrees of Freedom

The hypothesis of the onset of nonstationarity and chaos in a hydrodynamic system as a result of the nonlinear interaction of a small number of degrees of freedom is verified experimentally with reference to fluid convection in a toroidal channel. Regimes of motion of a fluid medium which correspond qualitatively to the Lorenz model are obtained experimentally. These include steady-state regimes, their bifurcations, nonuniqueness and instability, unsteady periodic and stochastic regimes. The spectral and statistical characteristics of the and unsteady processes are investigated, the nature of the onset of chaos is analyzed, and the results are compared with calculations. The mathematical model of the problem is refined.     

Temperature field of heat sources during fluid injection in an anisotropic inhomogeneous reservoir

The problem of the temperature field produced by sources whose position does not depend on the vertical coordinate and which are concentrated in a horizontal permeable layer surrounded by a heat-conducting medium with radial steady-state fluid flow. The problem is solved using an averagely accurate asymptotic method. Analytical expressions for the zero-order approximation and the first coefficient of the expansion. A condition is determined under which the averaged problem for the remainder term has a trivial solution.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

Modeling wave-induced pore pressure and effective stress in a granular seabed

The response of a sandy seabed under wave loading is investigated on the basis of numerical modeling using a multi-scale approach. To that aim, the discrete element method is coupled to a finite volume method specially enhanced to describe compressible fluid flow. Both solid and fluid phase mechanics are upscaled from considerations established at the pore level. Model’s predictions are validated against poroelasticity theory and discussed in comparison with experiments where a sediment analog is subjected to wave action in a flume. Special emphasis is put on the mechanisms leading the seabed to liquefy under wave-induced pressure variation on its surface. Liquefaction is observed in both dilative and compactive regimes. It is shown that the instability can be triggered for a well-identified range of hydraulic conditions. Particularly, the results confirm that the gas content, together with the permeability of the medium are key parameters affecting the transmission of pressure inside the soil.     

Sediment-discharge vector in a turbulent flow above an eroded bottom

The problem of determination of sediment discharge by a turbulent flow of a fluid above an eroded surface of an arbitrary relief with a finite slope of the bottom is considered. The surface of the bottom separates a stationary granular medium (sand) from a moving two-phase mixture of a fluid and solid particles. The medium is set into motion under the action of shear stress of the fluid. The medium obeys Coulomb's friction law for a granular medium and Prandtl's law of turbulent friction of the fluid. As a result of solving the boundary-value problem for the motion of a two-phase mixture of a fluid and solid particles, a generic formula for sediment discharges is derived. The sediment-discharge vector is expressed through the vector of shear stress on the bottom, the vector of the slope of the bottom, and the distribution function of the solid particles in the bottom layer for an arbitrary relief of the bottom with a finite slope. It is shown that the sediment discharge depends weakly on the detailed distribution of particles in the bottom layer. Conditions of failure of the bottom surface are obtained. The sediment-discharge formula allows one to derive a closed system of equations that determines the process of bottom erosion in the river or channel bed. Institute Problems of Mechanics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 102–112, March–April, 2000.     

Hydroelastic behavior of a complex structure floating on waves

A numerical method is developed to solve the plane problem of the hydroelastic behavior of a complex structure floating on the surface of an ideal incompressible fluid of finite depth. The motion of the structure described by a deflection function is considered steady-state under the action of incident waves. The hydrodynamic part of the problem is solved using the proposed approach based on the normal-mode method for homogeneous plates. The problem is reduced to a system of linear algebraic equations by means of a transition matrix between representations of the required deflection in the form of expansion in the vibration eigenfunctions of the structure and the plate. It is shown that the results of the calculation performed are in good agreement with available calculation results for a two-part hinged structure at wavelengths comparable to the length of the structure.     

Features of deformation of poroelastic media with low structural strength

Many natural and technological processes are associated with deformation and fracture of saturated or being saturated poroelastic media. Among such processes one can mention fluid soaking through a dam, fluid inflow to the cracks of hydraulic fracture, polishing using porous materials and special fluids, flow in catalytic pellets. All these processes are accompanied by deformation and fracture of a matrix with fluid flow. The effects at the interface porous body–fluid are essential for the processes.The specific features of deformation of poroelastic media with low structural strength are considered in this paper. The compressibility of the matrix skeleton is larger as compared to the compressibility of the saturating fluid in such media.It is shown that the oozing of the fluid at the surface of the poroelastic medium occurs in the consolidated flow regime under the action of `fluid piston' like loads if the structural strength of the medium is low. This result is obtained for both plane (deformation of a layer or halfinfinite medium) and centrally symmetric (deformation of a sphere) problems.     

High Velocity Flow in and Around Long Perforation Tunnels

In oil industries, wells are mostly cased and perforated rather than completing the producing formation as open-hole. This study concentrates on the steady-state flow behaviour of a single-phase fluid in and around perforated completion tunnels (up to 50 inches long) including inertial effects (Non-Darcy flow). It is shown that the pressure drop inside long perforated tunnels under high flow velocity conditions is negligible compared to that around the perforated region within the porous medium. The results also indicate that the impact of perforation parameters varies with increasing fluid velocity but approaches an asymptotic value at very high flow velocity. The perforation length is the most important parameter whereas perforation radius, rock and fluid properties have little impact on the perforation performance.     

Vortex-induced vibration of pipes conveying fluid using the method of multiple scales

The nonlinear dynamics of supported pipes conveying fluid subjected to vortex-induced vibration is evaluated using the method of multiple scales. Frequency response portraits for different internal fluid velocities under lock-in conditions are obtained and the stability of steady-state responses is discussed. Results show that the internal fluid velocity has a prominent effect on the oscillation amplitude and that the steady-state responses incorporating unstable solutions in the lock-in region are also obtained. In addition, the effects of two kinds of fluctuating lift coefficients on the steady-state responses are compared with each other.     

Convection of a Freezing Fluid in a Square Cell After Loss of Hydrostatic Stability

The behavior of a freezing (defrosting) fluid in a square cell is studied for three different initial conditions. For a fixed Grashof number Gr, the existence of four steady-state solutions is demonstrated. On a certain range of Gr, an increased solid-phase fraction in the cell, as compared with the pure heat conduction case, is obtained. The critical values of Gr corresponding to passage from one type of solution to another are found     

Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Equilibrium of a crack in a porous medium with injection of the filtering liquid

In connection with the exploitation of petroleum deposits, the article discusses the equilibrium of a porous medium with a crack under conditions of plane deformation, with the steady-state filtration of a liquid injected into the porous medium through a crack. It is assumed that the crack, which has initial zero dimensions, can become wider and longer with a rise in the pressure. The displacement of the sides of the crack is determined on the basis of the theory of elasticity, taking account of the deformation properties of a saturated porous medium. The stress and the displacement are expressed in terms of two analytical Muskhelishvili functions and the complex filtration potential. A change in the volume of the porous medium leads to a discontinuity of the displacements at the feed contour, and to distortion in the filtration region. For a circular stratum, the dimensions of the crack and the mass flow rate of the liquid are determined in the first approximation. The region of values of the pressure in which there exists a stable equilibrium state of the open crack and a steady-state flow of the liquid is found.