Contact problem for a plane containing a slit of variable width : PMM vol. 38, n 6, 1974, pp. 1084–1089

The problem of compression of an elastic plane with a slit of variable width commensurate to the elastic strains is considered. The case of the origination of several contact sections of the slit edges is investigated. Adhesion of the edges hence occurs at some part of the contact area, while slip is possible at the rest of this area. A solution of the problem is obtained in quadratures by the Muskhelishvili method using the apparatus of linear conjugates of analytic functions. The stress and displacement potentials are found, the magnitudes of the contact sections and the adhesion zones are determined. A specific example is analyzed and numerical computations are carried out.The contact problem for a plane weakened by a constant-width rectilinear slit has been considered in [1 – 3].     

Convergence of finite-difference techniques for a harmonic mixed boundary value problem

The paper deals with the convergence of finite-difference approximations for a harmonic mixed boundary value problem. In particular the case when the boundary contains a slit is considered.Work on this paper was supported in part by NSF Grant GP-8442 and U.S. Army Grant DA-ARO(D)-31-124-G1050 at The University of Texas at Austin.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Computation of expansion coefficients in a singular harmonic problem

The eigenfunction expansion of a harmonic function in a slit domain around a singular point (end of the slit) is investigated and a boundary integral equation is used for the computation of the coefficients, in the case of a model problem.     

The nonsimilar laminar wall jet along a moving wall,in a free stream and in a free stream/moving wall

The flow of a laminar wall jet along either a moving plate, or in a free stream, or in combined moving plate and free stream is considered. The plate is isothermal and its temperature is different from that of the ambient fluid. The governing boundary-layer equations are converted into non-dimensional form and are solved numerically. Velocity and temperature profiles as well as the variation of wall shear stress and wall heat transfer are presented for all cases considered. For the case of a moving plate new results have been found although this problem has been investigated in the past. For the case of the free stream and the combined moving plate/free stream all the results are new and are presented for the first time in the literature. The three cases have been tackled with a unified way.     

Free-surface shapes of a jet injecting through a porous wall

For a jet incident on a porous wall at which the normal fluidspeed is specified, it is found that the problem of determiningthe free surface of the jet is governed by a system of nonlinearintegral equations relating the flow angles on the boundary,on the free surface, and on the porous wall. With a constantnormal speed at the porous wall, the system reduces to an integralequation for the flow angle, which is solved numerically; anda comparison with previous results is made. Numerical results,corresponding to different nonconstant normal jet speeds alongthe porous wall, are also presented. The extension of this formulationto include the effect of gravity is also given.     

Fast spectral solutions of the double-gyre problem in a turbulent flow regime

Several semi-analytical models are considered for a double-gyre problem in a turbulent flow regime for which a reference fully numerical eddy-resolving solution is obtained. The semi-analytical models correspond to solving the depth-averaged Navier–Stokes equations using the spectral Galerkin approach. The robustness of the linear and Smagorinsky eddy-viscosity models for turbulent diffusion approximation is investigated. To capture essential properties of the double-gyre configuration, such as the integral kinetic energy, the integral angular momentum, and the jet mean-flow distribution, an improved semi-analytical model is suggested that is inspired by the idea of scale decomposition between the jet and the surrounding flow.     

Simulation of Jet Flows in Ideal Fluids by Variable Finite Elements

A free-boundary problem involved in computation of the shapeof a jet of an ideal fluid exiting through an orifice is formulatedas finding a stationary point of a certain functional definedon a variable domain. The stationary point of the functionaldiscretized in terms of linear finite elements is then locatednumerically by using the Newton-Raphson procedure. This numericalapproach is much faster than the previous attempts; it can accuratelyreproduce the exact results known for planar jet flows whenthe orifice is an infinite slit; and it can easily be appliedto the computation of axisymmetric jets.     

Creeping flow analysis of slightly non-Newtonian fluid in a uniformly porous slit

This paper provides the analysis of the steady, creeping flow of a special class of slightly viscoelastic, incompressible fluid through a slit having porous walls with uniform porosity. The governing two dimensional flow equations along with non-homogeneous boundary conditions are non-dimensionalized. Recursive approach is used to solve the resulting equations. Expressions for stream function, velocity components, volumetric flow rate, pressure distribution, shear and normal stresses in general and on the walls of the slit, fractional absorption and leakage flux are derived. Points of maximum velocity components are also identified. A graphical study is carried out to show the effect of porosity and non-Newtonian parameter on above mentioned resulting expressions. It is observed that axial velocity of the fluid decreases with the increase in porosity and non-Newtonian parameter. The outcome of this theoretical study has significant importance both in industry and biosciences.     

A scheme for the electrochemical machining of metals by a cathode tool with a curvilinear part of the boundary

The solution of a non-linear plane problem in the theory of the electrochemical machining of metals, associated with the determination of the shape of a surface (the anode) during its treatment with a cathode tool with a curvilinear part of the boundary, is obtained by methods developed for problems of jet flow past curvilinear obstacles. A condition is obtained which is identical to the well known Brillouin-Villat condition in fluid dynamics for smooth separation, the use of which enables one to determine the position of the transition point from the zone of steady treatment conditions into a region where the dissolution of the metal does not occur. The fixed shapes of the anode boundary are found for two cathode configurations.     

Transient and steady-state analysis of a manufacturing system with setup changes

This paper deals with the optimal scheduling of a one-machine two-product manufacturing system with setup, operating in a continuous time dynamic environment. The machine is reliable. A known constant setup time is incurred when switching over from a part to the other. Each part has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a production flow control problem. The objective is to minimize the sum of the backlog and inventory costs incurred over a finite planning horizon. The global optimal solution, expressed as an optimal feedback control law, provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady-state, the optimal cyclic schedule (Limit Cycle) is determined. This is equivalent to solving a one-machine two-product Lot Scheduling Problem. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region is associated an optimal control policy. A novel algorithm (Direction Sweeping Algorithm) is developed to obtain the optimal state trajectory (optimal policy that minimizes the sum of inventory and backlog costs) for this last case.     

Gas dynamics applications of a characteristic Cauchy problem

Some initial-boundary-value problems for a system of quasilinear partial differential equations of gas dynamics with the initial data prescribed on the characteristic surface (characteristic Cauchy problem) are considered. The following three-dimensional flow problems are investigated: the flow produced by a motion of an impermeable piston; the flow produced by a permeable piston with a given pressure; and the flow produced by the moving free boundary. In the first two problems, the piston motion is prescribed; in the last problem, the free boundary motion cannot be prescribed in advance and must be determined as a part of the problem. It is shown that those problems can be reduced to a characteristic Cauchy problem of a certain standard type that satisfies the analog of Cauchy-Kowalewski's existence theorem in the class of analytical functions (Differential Equations 12 (1977) 1438-1444). Thus, it is proved that, in the case of the analyticity of the input data, the considered problems have unique piecewise analytic solutions which may be expressed by infinite power series (the procedure of constructing the power series solution is described in Differential Equations 12 (1977) 1438-1444 as a part of the proof of the theorem).     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

Free convection from a point heat source in a stratified fluid

A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established.The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/.The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest.The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.     

Temperature distribution in a laminar plane wall jet due to variably heated wall

The heat transfer in a laminar incompressible plane wall jet due to a variably heated wall has been studied. It is assumed that the difference of temperatures between the wall and the issuing jet is inversely proportional to an arbitrary exponent of the distance from the slit. A similar solution of the energy equation is possible. The solutions, for arbitrary values of the Prandtl number and of the exponent are obtained. It is found that in some cases the heat transfer at the wall may become zero or negative.     

Curvilinear airfoil optimization of extremal drag force in hydro-aerodynamics

Direct and inverse boundary value problems are solved and the solution of an optimization problem is obtained analytically in the case of some nonlinear integral functionals. The plane potential flow of an inviscid fluid is considered in the absence of mass forces (Hyp). The flow – unlimited jet – encounters a symmetrical curvilinear obstacle (the Helmholtz scheme). For inverse problems there are derived singular integral equations and the movement is obtained in the auxiliary canonical half plane. Next, an optimization problem is solved analytically. The design of the optimal airfoil is performed. The drag coefficient and other geometrical parameters are computed in the case of a given distribution of the velocity on the profile. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.     

The dynamics of a meandering coastal jet in the lee of a cape

In a barotropic model of an oceanic channel, bounded to the north by a straight coast indented by a Gaussian cape, the evolution of a coastal jet is studied numerically. In the absence of the cape, the barotropic instability of the jet is determined. In the presence of the cape, a regular row of meanders develops downstream of this feature, and becomes stationary for a particular range of parameters. The relevant parameters are the velocity and width of the jet, size of the cape, and beta effect. The formation of meanders occurs first via the instability of the jet, then via the generation of vorticity anomalies at the cape, which are advected both downstream by the flow and offshore by the radiation of Rossby waves. Once the meanders are established, they remain stationary features if the propagation velocity of the meanders (due to the dipolar effect at the coast) opposes the jet velocity and the phase speed of the wave on the vorticity front. Finally, a steady state of a regular row of meanders is also obtained via a matrix method and is similar to that obtained in the time-dependent case.     

Periodic solutions of elasticity and electroelasticity theories for a cylinder in R3

General representations are constructed for periodic solutions of the theories of elasticity and electroelasticity for a cylinder in R³. These solutions are utilized to reduce the first boundary value problem in an unbounded isotropic medium weakened by tunnel slit cavities to a system of singular integral equations. The solvability of the characteristic part of the obtained system is proved.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 13–20, 1990.