In this study, we analyze the β -conformable space-time Burgers equation by employing a wave transformation that reduces the original partial differential equation into a nonlinear ordinary differential equation. Two distinct scenarios are examined: the free particle case with vanishing potential and the case involving a quadratic potential. The resulting nonlinear ODE is solved using the tanh-function method, which yields exact traveling wave solutions. The application of the β -conformable derivative framework enables the incorporation of memory effects while preserving the local structure of the equation. The obtained solutions highlight the role of the conformable order β and the potential function in shaping the dynamics of the system, contributing valuable insight into the analytical treatment of nonlinear conformable partial differential equations.
In this paper, using a numerical model, the processes of oscillation of a gas suspension in an acoustic resonator, which is an open tube, are investigated. The mathematical model utilizes a continuum technique for simulating the dynamics of multiphase media in Euler coordinates, accounting for the interaction between the gas and the dispersed phase.When solving the Navier–Stokes system of equations modeling the flow of the gas phase, the interaction of gas and dispersed particles was taken into account—the exchange of heat and momentum. The interphase momentum exchange forces included the aerodynamic drag force, the added mass force, and the dynamic Archimedes force. The dispersed phase dynamics are described by a system of equations including the continuity equation for the average density, the conservation equations for the spatial components of the dispersed phase momentum, and the thermal energy conservation equation, all written taking into account interphase thermal interaction and momentum exchange between the phases. The system of equations for the dynamics of a multi-velocity, multi-temperature, monodisperse system was integrated using an explicit, second-order finite-difference method. A spatial-direction splitting scheme was used to implement the finite-difference method. A nonlinear correction scheme ensured the monotonicity of the solution. Using a numerical model, the oscillation process of a gas suspension in an open acoustic resonator was studied for various piston stroke amplitudes at the frequency of the first linear resonance. The numerical calculation results were compared with the physical experiment. The comparison showed acceptable agreement between the numerical solution and the physical experiment data. It was also found that for the particle dispersion used in the physical experiment, the processes of changing the velocity of the gas phase and the dispersed phase do not differ significantly. As the piston oscillation amplitude increases, the amplitude of the oscillations of the average density of the dispersed phase and the intensity of the dispersed phase drift increase.
Nowadays, mathematical modeling is currently the most effective tool for the development of advanced technology of thermomechanical processing of metals and alloys. The key element of mathematical models is constitutive models (or constitutive relations). Since the physical and mechanical properties of alloys are largely determined by their structure at various structural-scale levels, in recent decades the emphasis was placed on the development of physically-oriented multi-level constitutive models allowing an explicit description of the material structure evolving in the process of material treatment. The disadvantage of the existing models is inadequate attention to the influence of intergrain boundaries and free boundaries of crystallites located on the surfaces on the behavior of polycrystalline materials. The present article offers a direct physically-oriented elastoviscoplastic model for describing the deformation of multiphase polycrystalline alloys, in which special attention is focused on the contribution of intergrain boundaries to the deformation behavior of these materials. A description of the constitutive model and the algorithm for its implementation are considered. The applicability of the model for studying the deformation of a representative volume (analogue of the macro-sample) of duplex steel is confirmed by some examples.
This paper presents a detailed algorithm for constructing a mathematical model of two-phase flow in porous media within fixed streamtubes for heterogeneous oil and gas reservoirs penetrated by wells. The approach decomposes the original three-dimensional problem into a set of two-dimensional problems posed within streamtubes of variable cross-sectional area, whose geometry is assumed to remain time-invariant over the simulation period. This substantially reduces computational cost while preserving high spatial resolution compared with full 3D simulation. Depth-averaging over the reservoir thickness yields a two-dimensional problem for the vertically averaged pressure and the associated streamline field, which is used to construct the streamtubes. In streamfunction-aligned coordinates, the governing equations are derived in an elementary streamtube, and the concept of a finite effective streamtube is introduced, with equivalent parameters selected to reproduce integral flow characteristics. Using a finite-volume discretization, convergence with respect to grid spacing and the number of streamtubes is investigated for a test problem of two-phase flow between two wells in a heterogeneous domain within the Buckley–Leverett approximation, with compressibility, capillarity, and gravity neglected. Estimates of the minimum grid resolution and the number of streamtubes required between a source–sink pair to achieve a prescribed accuracy are provided.
The equilibrium position of a momentless cable with known total length at a given lifting boom is considered. Determination of the length of the section lifted above the horizontal foundation is the subject of an inverse problem. The exact nonlinear equation is presented in various approximations, more convenient for different problem for-mulations. A solution is provided for low, medium, and high lifting boom and the results are compared. The required lifting force is determined.
The propagation of perturbation along the boundary of porous and fractured—porous media is numerically investigated. The problem is considered in a two-dimensional formulation. The study is carried out using a two-phase model of a porous medium and a three-phase model of a fractured porous medium. The condition ‘‘open pores’’ is used at the interface ‘‘porous medium—fractured-porous medium’’. The cases of the location of a point source of perturbation at or near the interface of media are considered. It is established that longitudinal and transverse waves are formed in each medium during the propagation of perturbation. A surface wave may also occur at the boundary. Estimates of the velocity and attenuation of waves propagating along the boundary are obtained. The penetration depth of the perturbation from the boundary is estimated for the surface wave.
The theoretical discovery of oscillatory flow regimes in anomalously thermoviscous fluids has revealed a large number of factors influencing their occurrence. For example, the influence of heat transfer conditions and the relative width of the fluid’s temperature anomaly were established. In this paper, we investigated the influence of the base viscosity value, determined by the temperature at the channel inlet, and established the difference between the oscillation patterns in the cases of full and truncated Gaussian viscosity-temperature dependencies, as well as the pressure drop and relative width of the annular channel itself. It is shown that when the ratio of the channel width to the radius of the inner cylinder is equal to or greater than unity, a fairly clear symmetry violation occurs during the evolution of the viscous barrier due to the differences in curvature of the outer and inner channel walls. Self-oscillation also occurs in these cases. A diagram of the oscillation regime boundaries for the truncated viscosity-temperature dependence was previously constructed, which was dictated by experimental data available in scientific journals. The results of the presented work indicate that while maintaining the tendency of changing these boundaries in the space of problem parameters, in the case of a complete configuration of the Gaussian curve, the region of existence of self-oscillatory modes significantly expands.
The paper presents a multibody dynamic model of an ornithopter with elastic flapping wings of high aspect ratio. The ornithopter fuselage is modeled as a rigid body, and the wings as elongated orthotropic composite plates of rod type. The orthotropy axes of the plate material may not coincide with the axes of the Cartesian coordinate system chosen for the wing, which allows describing coupled bending-torsion vibrations. A refined geometrically nonlinear Timoshenko shear model is employed to describe wing deformation. Using the d’Alembert–Lagrange variational principle, the equations of motion for the ornithopter elements are derived, the kinematic and force conditions for wing-fuselage coupling are formulated, as well as the equations of perturbed motion of the fuselage. To illustrate the model’s capabilities, an exact analytical solution for the linearized problem of static wing bending under a transverse follower load is obtained. The solution demonstrates that oblique reinforcement ensures the coupling of bending and torsional deformations, leading to the emergence of longitudinal displacements and transverse shear forces along the wing chord. It is shown that these effects are fundamental to thrust generation in an unsteady dynamic regime. The obtained closed-form expressions for kinematic and force factors can serve as a basis for parametric analysis and verification of numerical methods in ornithopter design.
The dynamics of gas (air) bubbles in a circular cluster in a liquid (water) under harmonic variation of the liquid pressure is studied numerically in the conditions close to those of the SBSL phenomenon. The cluster consists of equally-sized spherical bubbles, the centers of which are located at the nodes of a square mesh (one of the bubbles is at the cluster center). The excitation frequency is 20 kHz, the static liquid pressure is 1 bar. The following parameter values are taken as basic: 1.075 bar for the excitation amplitude, 5 μ m for the equilibrium radius of all bubbles, 300 μ m for the equilibrium radius of the cluster, and 121 for the number of bubbles. The study is performed varying one of these parameters, while the others are set equal to their basic values. The number of bubbles is limited to 195. The main attention is focused on the maximum pressures that are achieved inside the bubbles until some of them begin to disintegrate or coalesce with some others. A particle model of joint bubble dynamics allowing for the bubble translations and deformations is applied. It has been shown that the maximum pressure achievable inside the cluster’s bubbles under the imposed constraints is about 8 kbar.
Strong compression of water vapor with the transition from molecular to dissociated (atomic) and ionized states is considered, using wide-range equations of state by Nigmatulin and Bolotnova (EOS NB). The cases of quasi-static isentropic compression, isentropic compression at a rate of Rayleigh’s collapse of a vapor cavity with a constant internal pressure in an inviscid incompressible liquid, and shock compression are analyzed. Such compression scenarios may occur, for example, during cavitation bubble collapse. Specifically, in the course of most part of cavitation bubble collapse in water, the vapor in its central region is compressed at a rate of the Rayleigh’s collapse. If the collapse is sufficiently strong, radially convergent shock waves may form inside the collapsing bubble. Their radial convergence is accompanied by rapid growth of their intensity, so that the dissociation and ionization processes may arise. In EOS NB, the processes of dissociation and ionization are described by a linear kinetics model. In the problems considered, the areas with partial dissociation of the water vapor molecules and partial ionization of their atoms are revealed. The influence of dissociation and ionization on the values of the vapor pressure and temperature and the degrees of their overestimation/underestimation resulted from neglecting the dissociation and ionization processes are demonstrated.
This paper considers the problem of the estimation of population variance of the study character in two-occasion successive sampling in presence of non-response. A conventional estimator for estimating population variance is proposed for this situation. The conventional estimator was further modified using auxiliary variable information through calibration approach to reduce the nuisance effect of non-response in sample surveys. The proposed estimators aim to enhance the precision and validity of inferences drawn from successive sampling surveys, where non-sampling errors can significantly affect the quality of the data and the resulting population parameter estimates. The expressions for the mean squared errors (MSEs) and estimated MSEs of the proposed estimators were derived to quantify their statistical properties and performance. The empirical results through simulation studies demonstrated that the proposed estimators outperformed the proposed traditional variance estimator especially when non-response is substantial with exception of few cases.
Numerical and analytical modeling of the passage of local aerodynamic vortex formation through a long cylindrical tube of circular cross-section is carried out. At the first stage, an analytical study of axisymmetric flows in the form of stationary weakly attenuated solitary waves was carried out and three classes of solutions were obtained, each of which is a discrete set according to the Reynolds number in the approximation of purely stationary waves and waves with weak attenuation. The solutions are characterized by a multi-fold more complex structure—vortex multi-rings or multilayer solutions. At the second stage, a grid model was constructed and an algorithm for the direct numerical solution of the Navier–Stokes equation (using the implicit scheme of the alternating directions method) was implemented for the evolution of vorticity in the axisymmetric case for various variants of the initial configurations. The results of the first stage were used as the initial configuration. For one of the classes of the initial configurations, the author previously demonstrated the fundamental possibility of passing a cascade of vortex formations through a tube with a length of 800 calibers. In this article, the initial configurations of the other two classes of solutions are used for comparison.
The dynamical processes caused by the detonation effects of cylindrical explosive in a layer of aqueous foam surrounding the charge are investigated. A detailed analysis of shock wave formation are carried out, taking into account published experimental measurements of pressures by transducers located from center of explosion at specified distances for charge C4 with a mass of M_HE=250 g is coated with layer of aqueous foam with radius r_foam=0.3 m and liquid volume content α_10≈ 6% . Numerical modeling of the problem was carried out using the twoPhaseEulerFoam solver from OpenFOAM software package modified for the presented study, which implements the two-phase gas-liquid mixture model, taking into account momentum and heat transfer between phases. To calculate the thermodynamic properties of gas-drop mixture, with transition to single-phase state corresponding to gas products of detonation, the equation of state GasLHT was developed and embedded to the library thermophysicalModels of the new solver. Additional processing of calculated data and acceleration of computational stages was carried out by the software package in the GNU Octave. Choice of parameters for initial pulse on pressure and temperature was based on the conditions for matching of calculated shock wave with experimental data. When analyzing the solutions obtained for temperature and gas volume content, the reasons for formation and preservation of a thin layer of gas-droplet structure in the considered time ranges were established. A comparative analysis of numerical simulation results with experimental data proved the reliability of used model and obtained results. Estimates of safe distance for humans from the center of explosion, with protective barrier of aqueous foam for the conditions of experimental data under consideration are given.
This paper presents an exact analytical solution of the Navier–Stokes equations for steady filtration flow of a viscous incompressible fluid in a conical channel. Assuming axisymmetric radial motion, we derive a self-similar velocity profile and the corresponding pressure distribution. The analysis indicates that the solution exists only if the channel opening angle is below a critical value. If the opening angle exceeds this value, flow separation occurs even as the Reynolds number approaches zero. We also examine the conditions under which the governing equations can be linearized. This analysis leads to an additional constraint on the Reynolds number that depends on the channel geometry. The solution is then expressed in cylindrical coordinates and compared with numerical simulations. The comparison shows good quantitative agreement within the range where the analytical solution is valid. The results can be used in a variety of microfluidic applications. In particular, the results can be used to develop improved capillary models of porous media that account for variable pore-channel cross-sections.
The inverse coefficient problem of determining the permeability field of a three-dimensional three-phase reservoir opened by production and injection wells is considered. The permeability field of each reservoir layer laterally is defined as a spline surface. Measurements of the liquid flow rate and bottom hole pressure at wells known from the history of reservoir development, as well as a priori information on the distribution of permeability at wells obtained by geophysical methods, are used as initial information. The number and position of the interpolation nodes of spline surfaces, as well as the permeability values at these nodes, are considered unknown. Various methods of searching of location interpolation nodes are proposed, and their comparison is carried out on synthetic tasks.
This paper focuses on a proposed novel modification to the classical RSA cryptosystem by deriving the encryption key dynamically from a shared secret established by the Diffie–Hellman Key Exchange Protocol (DHKEP) adopted by communicating parties. Moreover, the decryption key is also computed secretly by the same communicating parties, and it represents the inverse the encryption key modulo a large positive integer, a product of two primes. This approach enhances key flexibility since the keys generated differently in each session mitigates attacks targeting public keys. The proposed cryptosystem preserves RSA’s security, that is represented by the Integer Factorization Problem (IFP) while incorporating the security strengths of the Discrete Logarithm Problem (DLP).
Khasminskii (Stochastic Stability of Differential Equations, 2nd Ed. (Springer, 2012), Theorem 6.14) established a limit theorem for the log-norm of solutions to linear stochastic differential equations with constant coefficients, from which it follows that weak and strong stochastic stability are equivalent within this class of systems. We improve this result by showing that one of the assumptions in the theorem is redundant.
The application of Darcy’s law for modeling the technology of impregnating a porous fibrous structure with a binding material in the production of composite products is considered. For the numerical solution of filtration equations in a variable domain, the finite element method is applied, and the position of the front boundary is determined using the control volume method in a liquid. The performance of the numerical method was verified by comparing the results with analytical solutions for linear and radial filling the areas of simple geometry with polymer liquid. A comparison of the numerical results of the penetration process of a complex-shaped area with experimental data was performed. The presented results made it possible to establish regularity of influence of the technological mode of molding on the dynamic characteristics of the liquid impregnation process of porous filler. The developed numerical method can serve as a basis for optimizing the technological processes of manufacturing structures from composite materials using a liquid-phase approach.
This paper investigates the propagation of acoustic waves in a moving medium with simultaneous (multifractional) dispersed inclusions of liquid and solid particles. A system of linearized integro-differential equations describing the motion and thermodynamic state of such mixtures has been derived. Non-stationary effects of interphase interaction caused by friction and heat and mass transfer between the bulk phase and dispersed inclusions are taken into account. A dispersion equation describing the propagation of acoustic waves in moving multifraction vapor-gas-droplet mixtures containing polydisperse liquid and solid inclusions has been obtained.
We consider the local Yang–Baxter equation for a noncommutative version of a Darboux matrix for the Derivative nonlinear Schrödinger equation (DNLS) and we construct a map. We prove that this map is a noncommutative solution to the set-theoretical Zamolodchikov tetrahedron equation.
