The present study reveals the turbulence dynamics and morphological adjustments of mobile dune-shaped bedforms in an alluvial stream. Results demonstrate acceleration of flow over the dune crest enhancing streamwise velocity, while the initial and the lee side sections of the dune experience flow circulation. The near-bed regions of the initial and lee sections experience peak Reynolds shear stress, marking zones of higher momentum exchange and active sediment entrainment. Turbulence is dominated by streamwise fluctuations, with spanwise and vertical components reinforcing lateral mixing and particle suspension. Octant analysis indicates that sweep events dominate in the near-bed regions of the initial, crest and lee sections of the dune, driving bedform migration and intensifying scour development on the lee side. Probability distribution functions highlight strong non-Gaussian behaviour and intermittency at crest and lee sections, linked to vortex shedding and flow separation. Higher-order structure functions further confirm the presence of intense turbulent bursts in the near-bed region, underscoring the role of coherent structures in driving sediment motion. Morphological analysis shows progressive scour development at the lee side and downstream crest erosion, resulting in continuous dune migration. These findings advance understanding of turbulence–morphology interactions and their control on sediment transport in alluvial channels.
To determine an optimal contour of holes for a perforated stringer plate weakened by a periodic system of cracks, an inverse problem of fracture mechanics is considered. It is assumed that the material of the plate is elastic or elastic-plastic. The stiffeners (stringers) are symmetrically riveted to the plate. The perforated plate is uniformly stretched at infinity along the stringers. It is assumed that rectilinear cracks are located near the contours of the holes and are perpendicular to the riveted stiffeners. The solution of the formulated inverse problem is based on the principle of equal strength. The optimal shape of the holes satisfies two conditions: the condition for the absence of stress concentration on the hole surface and the condition for the zero stress intensity factors in the vicinity of the crack tips. The unknown contour of holes is looked for in the class of contours close to circular. The action of the stiffeners is replaced by unknown equivalent concentrated forces at the points of their connection with the plate. The sought-for functions (the stresses, displacements, concentrated forces and stress intensity factors) are looked for in the form of expansion in small parameter. The solution to the problem is sought using the apparatus of the theory of analytic functions and the theory of singular integral equations, then the conditional extremum problem is solved. As a result, a closed system of algebraic equations is obtained, which allows to minimize the stress state on the contours of holes and stress intensity factors in the vicinity of the crack tips. The obtained system of algebraic equations allows to determine the form of equal strength contour of holes, the stress-strain state of the perforated stringer plate and also the optimal value of the tangential stress.
This article investigates a fourth-order differential equation defined in a Hilbert space, with an unbounded intermediate coefficient and potential. The key distinction from previous research lies in the fact that the intermediate term of the equation does not obey to the differential operator formed by its extreme terms. The study establishes that the generalized solution to the equation is maximally regular, if the intermediate coefficient satisfies an additional condition of slow oscillation. A corresponding coercive estimate is obtained, with the constant explicitly expressed in terms of the coefficients’ conditions. Fourth-order differential equations appear in various models describing transverse vibrations of homogeneous beams or plates, viscous flows, bending waves, and etc. Boundary value problems for such equations have been addressed in numerous works, and the results obtained have been extended to cases with smooth variable coefficients. The smoothness conditions imposed on the coefficients in this study are necessary for the existence of the adjoint operator. One notable feature of the results is that the constraints only apply to the coefficients themselves; no conditions are placed on their derivatives. Secondly, the coefficient of the lowest order in the equation may be zero, moreover, it may not be unbounded from below.
Flagellated bacteria propel themselves by rotating flexible flagella driven by independent motors. Depending on the rotation direction of the motors and the handedness of the helical filaments, the flagella either pull or push the cell body. Motivated by experimental observations of Magnetococcus marinus, we develop an elastohydrodynamic model to study the locomotion of a bi-flagellated bacterium with one puller flagellum and one pusher flagellum. In this model, the boundary integral technique and Kirchhoff rod model are employed respectively to calculate the hydrodynamic forces on the swimmer and model the elastic deformations of the flagella. Our numerical results demonstrate that the model bacterium travels along a double helical trajectory, which is consistent with the experimental observations. Varying the stiffness, orientations or positions of the flagella significantly changes the swimming characteristics. Notably, when either the applied torque is higher than a critical value or the flagellum stiffness is lower than a critical stiffness, the pusher flagellum exhibits overwhirling motion, resulting in a more complicated swimming style and a lower swimming speed. For a moderate flagellum stiffness, the swimming speed is insensitive to the rest configuration orientation over a wide range of orientation angles as the flagella deform to maintain alignment with the swimming direction.
Beauchard, Karine, Le Borgne, Jérémy, Marbach, Frédéric
Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann’s infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann’s infinite product expansion.Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.
In this paper, we consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.
This paper formulates and solves a generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator. The fractional derivative is understood as the Gerasimov-Caputo derivative. The boundary conditions are given in the form of linear functionals, which makes it possible to cover a wide class of linear local and non-local conditions. A representation of the solution is found in terms of special functions. A necessary and sufficient condition for the solvability of the problem under study is obtained, as well as conditions under which the solvability condition is certainly satisfied. The theorem of existence and uniqueness of the solution is proved.
In this paper, we study the interpolation properties of the net spaces N_p,q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from R^n. The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in R^n, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.
Derivatives in time of higher order (more than two) arise in various fields such as acoustics, medical ultrasound, viscoelasticity and thermoelasticity. The inverse problems for higher order derivatives in time equations connected with recovery of the coefficient are scarce and need additional consideration. In this article the inverse problem of determination is considered which depends on time, lowest term coefficient in differential equation in partial derivatives of fourth order in time with initial and boundary conditions from an additional integral observation is considered. Under some conditions regularity, consistency and orthogonality of data by using of the contraction principle the unique solvability of the solution of the coefficient identification problem on a sufficiently small time interval has been proved.
The theory of embedding of spaces of differentiable functions studies important relations of differential (smoothness) properties of functions in various metrics and has wide application in the theory of boundary value problems of mathematical physics, approximation theory and other fields of mathematics. In this article, we prove the theorems about traces and extensions for functions from Nikolsky-Besov spaces with generalized mixed smoothness and mixed metrics. The proofs of the obtained results is based on the inequality of different dimensions for trigonometric polynomials in Lebesgue spaces with mixed metrics and the embedding theorem of classical Nikolsky-Besov spaces in the space of continuous functions.
This paper considers a difference problem for a mixed-type equation, to which a problem of integral geometry for a family of curves satisfying certain regularity conditions is reduced. These problems are related to numerous applications, including interpretation problem of seismic data, problem of interpretation of Xray images, problems of computed tomography and technical diagnostics. The study of difference analogues of integral geometry problems has specific difficulties associated with the fact that for finite-difference analogues of partial derivatives, basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioned to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Another important feature of the integral geometry problem is the absence of a theorem for existence of a solution in general case. Consequently, the paper uses the concept of correctness according to A.N.Tikhonov, particularly, it is assumed that there is a solution to the problem of integral geometry and its differential-difference analogue. The stability estimate of the difference analogue of the boundary value problem for a mixed-type equation obtained in this work is vital for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc. It also has a great practical significance in solving multidimensional inverse problems of acoustics, seismic exploration.
Konrad A. Goc, Oriol Lehmkuhl, George Ilhwan Park
et al.
While there have been numerous applications of large eddy simulations (LES) to complex flows, their application to practical engineering configurations, such as full aircraft models, have been limited to date. Recently, however, advances in rapid, high quality mesh generation, low-dissipation numerical schemes and physics-based subgrid-scale and wall models have led to, for the first time, accurate simulations of a realistic aircraft in landing configuration (the Japanese Aerospace Exploration Agency Standard Model) in less than a day of turnaround time with modest resource requirements. In this paper, a systematic study of the predictive capability of LES across a range of angles of attack (including maximum lift and post-stall regimes), the robustness of the predictions to grid resolution and the incorporation of wind tunnel effects is carried out. Integrated engineering quantities of interest, such as lift, drag and pitching moment will be compared with experimental data, while sectional pressure forces will be used to corroborate the accuracy of the integrated quantities. Good agreement with experimental $C_L$ data is obtained across the lift curve with the coefficient of lift at maximum lift, $C_{L,max}$, consistently being predicted to within five lift counts of the experimental value. The grid point requirements to achieve this level of accuracy are reduced compared with recent estimates (even for wall modelled LES), with the solutions showing systematic improvement upon grid refinement, with the exception of the solution at the lowest angles of attack, which will be discussed later in the text. Simulations that include the wind tunnel walls and aircraft body mounting system are able to replicate important features of the flow field noted in the experiment that are absent from free air calculations of the same geometry, namely, the onset of inboard flow separation in the post-stall regime. Turnaround times of the order of a day are made possible in part by algorithmic advances made to leverage graphical processing units. The results presented herein suggest that this combined approach (meshing, numerical algorithms, modelling, efficient computer implementation) is on the threshold of readiness for industrial use in aeronautical design.
The article deals with the existence of a generalized solution for the second order nonlinear differential equation in an unbounded domain. Intermediate and lower coefficients of the equation depends on the required function and considered smooth. The novelty of the work is that we prove the solvability of a nonlinear singular equation with the leading coefficient not separated from zero. In contrast to the works considered earlier, the leading coefficient of the equation can tend to zero, while the intermediate coefficient tends to infinity and does not depend on the growth of the lower coefficient. The result obtained formulated in terms of the coefficients of the equation themselves; there are no conditions on any derivatives of these coefficients.
Abstract We consider the variable selection problem of generalized linear models (GLMs). Stability selection (SS) is a promising method proposed for solving this problem. Although SS provides practical variable selection criteria, it is computationally demanding because it needs to fit GLMs to many re-sampled datasets. We propose a novel approximate inference algorithm that can conduct SS without the repeated fitting. The algorithm is based on the replica method of statistical mechanics and vector approximate message passing of information theory. For datasets characterized by rotation-invariant matrix ensembles, we derive state evolution equations that macroscopically describe the dynamics of the proposed algorithm. We also show that their fixed points are consistent with the replica symmetric solution obtained by the replica method. Numerical experiments indicate that the algorithm exhibits fast convergence and high approximation accuracy for both synthetic and real-world data.
S.F. Pylypaka, M.B. Klendii, V.I. Trokhaniak
et al.
Differential equations of material particle movement on an inclined rough plane, which performs oscillatory motion in such a way that its every point describes a circle in the same plane, have been deduced. Peculiarities of relative particle movement on a plane depending on the angle of its inclination to the horizon have been investigated. The equations have been solved using numerical methods. Relative velocities have been found and particle motion trajectories have been constructed. Kinematic characteristics of relative particle movement depending on the angle of plane inclination, angular velocity, the coefficient of particle friction on a plane and the radius of circular motion of plane points have been determined.
Zh.A. Sartabanov, A.Kh. Zhumagaziyev, G.A. Abdikalikova
There is researched existential problem of a unique multiperiodic in all independent variables solution of a linear hyperbolic in the narrow sense system of differential equations with constant coefficients and its integral representation in vector - matrix form. To solve this problem, based on Cauchy’s method of characteristics, a constructing methodology for solutions of initial problem system under consideration with various differentiation operators in vector fields directions of independent variables space has been developed based on projectors. Using this method, Cauchy problems for linear system with integral representation are solved. The introduced projectors by definition characteristic had significant value. By solving the main problem necessary and sufficient conditions for existence of multiperiodic solutions linear homogeneous systems other than trivial are established. The conditions are obtained for absence of nonzero multiperiodic solutions of these systems...
Abstract The mechanism and fracture aperture of thick hard roof failure were theoretically analyzed from the perspective of elastoplastic mechanics. A uniform load cantilever beam was established by considering the voussoir beam structure and stress condition. According to the stress inverse principle in solving the plane strain problem, the internal stress component analytic solution was deduced. Considering the lateral thrust and shear force and by applying the Mohr‐Coulomb failure criterion, the plastic zone boundary equation was obtained. Further analysis of the equivalent stress, equivalent strain, elastic energy density, distortion energy density, and strain energy density was studied with blocks of four different lengths, combined with the rotation angles after failure, to analyze the fissure aperture between straight and curved fracture traces. The theoretical analysis reveals that the deviatoric effect plays the lead role in rupture initiation and propagation; the cracks will deflect during failure, and the fracture trace presents a curved shape. The variation of fracture aperture exhibits a tendency to increase first, then decrease, and then increase along the fracture trace. The analytical approach presented in this paper provides a theoretical basis for determining the main seepage channel flow in underground mining.
A well - known analogy of the flow of viscous incompressible fluid and incompressible elastic medium. According to this analogy, the solution of the equations of the elasticity theory with the Poisson’s ratio v = 0, 5 and for any fixed shear modulus µ can be interpreted as a motion of a viscous incompressible fluid with viscosity µ. Thus, we can consider the usual static linear elasticity task with Hooke’s law at λ → ∞, as a mathematical model of approaching to incompressible medium. In this paper, we obtained the asymptotic λ → ∞. Estimation of the proximity of the solution of an elastic static problem with Hooke’s law to the solution of incompressible medium (Stokes problem). The final estimate allows to use well - known difference schemes and algorithms for an elastic compressible medium to solve incompressible medium. In this paper, an estimate of the proximity of the solutions of these problems is proved at λ → ∞, i.e. u→uH λ→∞ λ div u→-p λ→∞ σ→σH λ→∞. To substantiate this fact in [1-3], various methods for the first boundary value problem were investigated. For the static problem of the theory of elasticity, there is currently a whole series of papers devoted to numerical implementation using difference schemes. In paper [4], the estimate O(λ-α) where k = 0,5 was obtained, in the proposed paper the estimate O(λ-1), and in further work we will show that this estimate is best possible in order.
Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. First, the main strength of our approach is to reduce this intrinsically fourth-order eigenvalue problem to a sequence of second-order problems. Second, an interesting by-product is that the unstable part of the spectrum is computed by making use of its stable part. Third, as an example, we consider a pressure-confined, static, self-gravitating slab of a fluid with an arbitrary polytropic exponent, with either free or rigid boundary conditions. The method can naturally be generalised to analyse the stability of richer, more complex systems. Finally, our analytical results are in excellent agreement with numerical solutions. Their second-order expansions provide a valuable insight into how the rate and wavenumber of maximal instability behave as functions of the polytropic exponent and the external pressure (or, equivalently, the column density of the slab).