The motion of snow avalanches in the conditions of limiting friction

We investigate the equations of motion of large snow avalanches, and in contrast with [1–3] we take into account the fact that the dry friction can reach a critical value above which the snow in the avalanche or the underlaying material cannot sustain the friction. We find asymptotic solutions for long times after the beginning of motion. These solutions describe the avalanche motion in which a part of the snow moves in the conditions of limiting friction over a tilted plane with a uniform layer of snow. The equations which are used to find these asymptotic solutions have the property that for certain depths the flow velocity of small perturbations decreases with increasing depth. This is related to a number of unusual features (from the hydraulic point of view) of the solutions. In particular, on relatively gentle slopes two zones are formed in the avalanche: the forward part, with a large velocity and thickness of the moving layer, and the rear part, which is significantly slower and thinner. The two parts are separated by a narrow region characterized by a sharp decline in velocity and thickness of the moving layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–37, September–October, 1977.     

Reflection and refraction of a planar acoustic wave in an anisotropic inhomogeneous layer

The problem of reflection and refraction of a planar acoustic wave by an inhomogeneous elastic layer whose material possesses general-type anisotropy is considered. The equations of motion of the elastic layer are reduced to a system of ordinary differential equations. The boundary-value problem for this system is solved by two methods: by reduction to problems with initial conditions and by the method of power series. Analytical expressions that describe acoustic fields outside the layer are obtained. Calculation results of the transmission factor for transversely isotropic layers inhomogeneous in thickness are presented. Tula State University, Tula 300600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 179–184, September–October, 1999.     

Solution of one problem of fracture mechanics

This paper considers a model for the opening-mode fracture separation process based on the introduction of an interaction layer. This layer is defined as the region of localization of the fracture process. The stress-strain state of the layer material is uniform in the cross section of the layer. A study is made of the deformation of a double-cantilever beam weakened by a notch whose width is equal to the thickness of the interaction layer. The problem is solved in a linearly geometrical approximation. The thickness of the interaction layer is estimated, and a method for solving the formulated problem is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 121–127, July–August, 2007.     

Well-posedness and Stability of a Multi-dimensional Tumor Growth Model

We study a moving boundary problem modeling the growth of in vitro tumors. This problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure, respectively, and a first-order partial differential equation describing the evolution of the moving boundary. An important feature is that the effect of surface tension on the moving boundary is taken into account. We show that this problem is locally well-posed for a large class of initial data by using analytic semi-group theory. We also prove that if the surface tension coefficient γ is larger than a threshold value γ _* then the unique flat equilibrium is asymptotically stable, whereas in the case γ  < γ _* this flat equilibrium is unstable.     

Control of the Strain Field Near a Crack in Elastic Layer

In the present paper, we consider the strain analysis near the crack parallel to boundary surfaces of a linear isotropic elastic layer of constant thickness. We pose the problem of minimizing the opening of crack faces when a normal force is applied to the boundary of the layer. The control secondary loading is a normal stress uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied force. We show that the most dangerous are tangential deformations, and the normal ones are always by many times less dangerous. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 265–277, April–June, 2005.     

State space approach to thermoelectric fluid with fractional order heat transfer

A new mathematical model for electromagnetic thermofluid equation heat transfer with thermoelectric properties using the methodology of fractional calculus is constructed. The governing coupled equations in the frame 11 of the boundary layer model are applied to variety problems. Laplace transforms and state space techniques (Ezzat Can J Phys Rev 86:1241–1250 in 2008) are used to get the solution of a thermal shock problem, a layer problem and a problem for the semi-infinite space in the presence of heat sources. According to the numerical results and its graphs, a parametric study of time-fractional order 0 < α ≤ 1, on temperature and the thermoelectric figure of merit are conducted.     

Tolerance averaging of heat conduction in transversally graded laminates

It is considered a heat conduction in a layer made of two conductors distributed in the form of laminas with varied thicknesses. Macroscopic (averaged) properties of the layer are continuously “transversally” graded across its thickness (TGL layer), cf. Fig. 1. The aim of the paper is to present and apply an averaged model of the heat conduction, obtained within the tolerance averaging technique, discussed in the book edited by Woźniak et al. (Thermomechanics of microheterogeneous solids and structures. Tolerance averaging approach, Łódź, Wydawnictwo Politechniki Łódzkiej, 2008). It is shown that the proposed model describes the microstructural effect on the heat conduction of the TGL layer. Moreover, results obtained within this model are compared to results by the higher order theory (cf. Aboudi et al., Composites B, 30:777–832, 1999).     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

Investigation of double-diffusive convection during the solidification of a binary mixture (NH4Cl–H2O) in a trapezoidal cavity

The solidification of binary mixture (NH₄Cl–H₂O) inside a trapezoidal cavity is investigated experimentally in this study. The effect of the initial concentration of ammonium chloride (0–19.8%) and boundary temperatures (−30 to 0°C) on the solidification process was investigated. Particle image velocimetry (PIV) technique was used for the visualization of the dynamic field in the melt. Thirty-two thermocouples were used to monitor the temperature distribution inside the cavity and on the cooling walls. The convective flow field, the temperature distribution, the frozen layer thickness and the moving solid/liquid interface were obtained for different initial concentrations of ammonium chloride and various boundary temperatures. The results obtained in the course of this study reveal that: (1) the process of solidification is slower with an increase in initial concentration levels of the binary solution: as the concentration increases, the time needed to get the same thickness of frozen layer increases; (2) an increase in the initial concentration of ammonium chloride solution reduce significantly the temperature in the melt; and (3) the initial concentration play a significant role in the evolution of convection flow patterns.     

Dynamic contact angle for etting of a surface by a van der waals fluid

We consider the creeping motion of a thin layer of a nonvolatile viscous fluid spreading due to capillary forces over a rigid surface covered by a thin homogeneous film (microfilm). The influence of van der Waals forces on the asymptotic slope of the free boundary of the layer is studied in the region of large thickness, where capillary forces dominate. A solution of the problem of the slope angle is obtained for the entire possible range of the microfilm thickness. In the limit of small thickness of the microfilm, this solution is in agreement with the well-known solution of the problem of the dynamics of wetting of a dry surface in the presence of a precursory film and van der Waals forces. The role of the condition at the end of the precursory film is studied. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences. Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika. Vol. 41, No. 4, pp. 101–105, July–August. 2000.     

Modeling of heat transfer in a turbulent flow around a heat-conducting plate with local regions of heating

A solution of the coupled nonstationary boundary-value problem of turbulent flow around a flat heat-conducting plate of finite thickness having local regions with volume heat sources is given. For modeling the heat transfer in the boundary layer, thek-ε turbulence model is used. It is shown that the thermal conductivity of the plate material significantly affects the surface distributions of both temperature and local friction. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 79–86, November–December, 1998. The work received financial support from the International Scientific and Engineering Center (project No.199).     

Conjugate heat transfer study of incompressible turbulent offset jet flows

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.     

Effect of a thin coating on the pressure distribution in contact problems involving frictional heat generation

A thermoelastic problem for a layer of finite thickness one of whose surfaces is subjected to the action of normal pressure and heat flux is studied. A relationship among vertical displacements of the surface of the layer, the surface temperature, and the disturbing factors is obtained. Corresponding relations are obtained for a layer of small thickness. An axisymmetric contact problem for a rigid heat-conducting base whose surface is coated with a thin elastic layer is studied as an example. Franko L'vov State University, L'vov 290602. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 110–118, January–February, 1998.     

Broadwell Model and Conservative Supersonic Boundary

In this paper, we consider an initial-boundary value problem for the Broadwell model with a supersonic physical boundary. Here, we pose a conservative boundary condition which preserves the total number of particles. We show that the solution converges pointwise to a boundary layer exponentially, when the perturbations of the initial data to the equilibrium state are sufficiently small, by using Green’s function as established in Lan et al. (Netw Heterog Media 1:167–183, 2006), weighted energy estimates and by constructing a pair of anti-derivatives to convert the conservative boundary condition into a dissipative boundary condition with conservation laws together with an a priori chosen boundary layer.     

Boundary layer laminarization on a thermally insulated surface with energy supply to the flow

A subsonic stream of gas flowing over a thermally insulated plate and having an elevated temperature in a thin layer adjacent to the surface is considered. This temperature distribution in the flow can be obtained by providing a volume energy supply near the leading edge of the plate. The results of calculating the position of the line of laminar-turbulent transition on the basis of linear stability theory and the e^N method are presented. It is shown that the presence of a heated layer of gas near the surface of the plate leads to an increase in the stability of the laminar flow and an extension of the laminar interval of the boundary layer. A nonmonotonic dependence of the length of the laminar interval on the thickness of the heated layer of gas is detected. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 58–61, September–October, 1988.     

Experimental study of variations in the thickness of a liquid film moving over the inner surface of a rotating cylinder

The behavior of a liquid layer moving in a mass-force field on the inner surface of a rotating vertical cylinder is studied experimentally. Free-surface profiles of the liquid moving under these conditions are constructed. An empirical dependence for the mean thickness of the film is obtained in criterial forms. The presence of a hydraulic jump in the lower part of the cylinder behind the entrance of the liquid onto the vertical surface is revealed. Tomsk, State Architectural-Building University, Tomsk 634003. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 65–71, November–December, 1998.     

A film-based wall shear stress sensor for wall-bounded turbulent flows

In wall-bounded turbulent flows, determination of wall shear stress is an important task. The main objective of the present work is to develop a sensor which is capable of measuring surface shear stress over an extended region applicable to wall-bounded turbulent flows. This sensor, as a direct method for measuring wall shear stress, consists of mounting a thin flexible film on the solid surface. The sensor is made of a homogeneous, isotropic, and incompressible material. The geometry and mechanical properties of the film are measured, and particles with the nominal size of 11 μm in diameter are embedded on the film’s surface to act as markers. An optical technique is used to measure the film deformation caused by the flow. The film has typically deflection of less than 2% of the material thickness under maximum loading. The sensor sensitivity can be adjusted by changing the thickness of the layer or the shear modulus of the film’s material. The paper reports the sensor fabrication, static and dynamic calibration procedure, and its application to a fully developed turbulent channel flow at Reynolds numbers in the range of 90,000–130,000 based on the bulk velocity and channel full height. The results are compared to alternative wall shear stress measurement methods.     

A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

Asymptotic solution of the problem of the action of a stamp on an elastic layer lying on the surface of a compressible fluid of infinite depth

This paper considers a two-dimensional linear unsteady problem of rigid-stamp indentation on an elastic layer of finite thickness lying on the surface of a compressible fluid of infinite depth. The Lamé equations holds for the elastic layer, and the wave equation for the fluid velocity potential. Using the Laplace and Fourier transforms, the problem is reduced to determining the contact stresses under the stamp from the solution of an integral equation of the first kind, whose kernel has a logarithmic singularity. An asymptotic solution of the problem is constructed for large times of interaction. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 131–142, March–April, 2008.