A method for calculating the behavior and formation of smoke particle agglomerates in crossed ultrasonic fields is proposed. The proposed method takes into account: the convergence of agglomerates with allowance for the moments of flow forces from the gas flow, the rotation of agglomerates due to the moments of forces from the flow around the gas flow, and a phase shift between ultrasonic cross fields, leading to the rotation of the resulting vibrational velocity vector and, consequently, to an increase in the effective collision cross-sectional area due to the rotation of the agglomerates. A description of the morphology and position of agglomerates is constructed when representing the agglomerate as a solid body, typical for solid-phase aerosols in the form of smoke. An equation for the dynamics of translational and rotational motion of an agglomerate is obtained taking into account the interaction of particles. The statement about the presence of rotational motion with a limited angular velocity proportional to the sound pressure level is proven, with a phase shift of cross fields equal to 90 deg. By means of numerical experiments a critical value of the phase shift angle is established at which the pseudo-rotational motion (change in the rotation angle in a limited range the width of which is less than 180 deg) transforms into rotational motion (the agglomerate makes a full revolution through 360 deg). The critical phase shift depends (weakly) on the sound pressure level and ranges from 82 to 85 deg. The transition to rotational motion increases the efficiency of forming larger agglomerates due to an increase in the average cross section during particle collisions, caused by the difference in the sizes of agglomerates along different axes. It was established that for the practical implementation of increasing the efficiency of smoke deposition, the most appropriate and feasible is ultrasonic action in cross fields with a frequency difference much lower than the fundamental frequency (from approximaterly 100 to 200 Hz). Such action is much easier to implement compared to maintaining a constant phase shift between the cross fields and increases the collision cross-sectional area up to several times.
This study focuses on investigating an optimal control problem that represents the unilateral frictional contact between a thermo-electro-elastic body and a foundation with electrical and thermal conductivity. The problem is considered in a static setting, and the material behavior is described using linear thermo-piezoelectric constitutive laws. Tresca’s friction law is used to model the contact, along with regularized conditions for electrical and thermal conductivity. The mathematical properties of the problem, including its variational formulation, existence, and uniqueness of weak solutions, are discussed. Additionally, an optimal control problem is formulated, and the existence of at least one solution is established. A regularized version of the unilateral contact and Tresca’s law is then incorporated into the model. The existence of a weak solution to the regularized control problem is proven, and its convergence to the solution of the original optimal control problem is verified. Finally, an optimality condition is presented for this type of problem.
The paper considers the following problem of scheduling theory. Jobs are serviced by a single machine. Each job is characterized by a positive weight and a release date. The service durations are the same. Interruptions are allowed in service. Time is assumed to be discrete. It is necessary to find a schedule for servicing jobs that minimizes the weighted sum of moments of completion of processing jobs. The complexity status of the problem is currently unknown. The paper presents a new Boolean linear programming model for this problem that includes some necessary conditions for the optimality of schedules. These conditions are presented as linear inequalities. As a result of the computational experiment, it was noted that the model produces an integer solution even without imposing integer conditions on the variables.
A nonpendant vertex of a graph is called a support vertex if it is adjacent to any leaf. This paper examines how two classes of extremal trees among all trees with a prescribed degree sequence can relate to each other: the class of trees minimizing the number of support vertices and the class of trees minimizing the domination number. Two specially defined values play important role here: the Slater number, proposed by Slater as a lower bound on domination number, and the lower bound on the number of the support vertices of a tree proposed by Kurnosov. In the paper, the problem of comparing the classes is completely solved in the case where the maximum of two values is the latter.
In this note, we single out some promising classes of differential-algebraic equations (DAEs) with nonlinearity of hysteresis type modeled by a sweeping process. DAEs is a well recognized and extensively studied area of the modern applied mathematics, arisen as a natural generalization of the concept of ordinary differential equations (ODEs). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. The analysis is carried out under the assumption of the existence of a structural form with separated “differential” and “algebraic” subsystems. This structural form is equivalent to the initial system in the sense of solution, and the operator that transformes the DAE into the structural form possesses the left inverse operator. Finding the structural form is constructive and does not use a change of variables. In addition, the problem of consistency of initial data is solved automatically. Systems of DAEs are attracting more and more attention due to mathematical modeling problems in many applied domains: automated control theory, optimal control with mixed constraints, mechanics, chemical kinetics, hydrodynamics, thermal engineering, etc. The systems under investigation arise in modeling various physical processes, in particular, in electrical circuits with hysteresis phenomena. For such a DAE, we design an equivalent structural form (in the sense of solutions). Necessary and sufficient conditions for the existence and uniqueness of a solution to an initial value problem and controllability are proved. Illustrative examples are given in the conclusions.
