Similarity solutions for power-law and exponentially stretching plate motion with cross flow

Laminar boundary layers generated by power-law plate stretching with cross flows are studied. Only the stretching solutions of Banks [10] are considered, those being bounded by exponentially stretched plates. In one case the cross flow is generated by a uniform transverse stream far above the stretching plate or a wall moving with uniform transverse velocity. Two other cases deal with cross flows generated by transverse shearing motions of the surface. Possible two parameter solutions appear, but here we present two one-parameter families of cross flow solutions generated by transverse plate shearing motion. Streamwise and transverse shear stresses and velocity profiles are displayed in graphical form.     

Oblique water entry of a cone by a fully three-dimensional nonlinear method

The hydrodynamic problem of a cone entering the water surface obliquely has been analyzed by the three-dimensional (3-D) incompressible velocity potential theory with the fully nonlinear boundary conditions on the moving free surface and body surface boundary. The time stepping method is used in the stretched coordinate system defined as the ratio of the physical system to the distance that the cone has travelled into water. The boundary element method is used to solve the potential at each time step. Both triangular element mesh and quadrilateral element mesh have been used. Discretisation of the body surface and the free surface is applied regularly during the simulation to account for their change and deformation, and data from the old mesh is transferred into the new one through interpolation. Both the dynamic and kinematic free surface boundary conditions are satisfied through the Eulerian form. In particular the free surface elevation and potential variation are traced at a given azimuth of the cylindrical coordinate system, in the direction parallel to the body surface or perpendicular to the free surface to avoid multi-valued function. Detailed convergence study with respect to time step and element size has been undertaken and high accuracy has been achieved. Results for the cone in vertical entry are compared with those obtained from the 2-D axisymmetric method and good agreement is found. Simulations are made for cones of various deadrise angles and different oblique entries and detailed results are provided.     

The asymptotic behavior of solutions to the Kirchhoff equation with a viscous damping term

We study the asymptotic behavior of solutions to an equation describing non-linear vibration of a string with viscosity. In the case when the string is unstretched (the degenerate case), we determine the decay order of solutions by investigating the dynamics near an infinite-dimensional center manifold. Moreover, we classify the asymptotic behavior of all solutions from a dynamical systems point of view. We also deal with the case where the string is stretched (the nondegenerate case).     

The analytical solutions for the wave propagation in a stretched string with a moving mass

This paper derives the analytical solutions for a stretched string subjected to a concentrated mass moving at a constant velocity. From the derived analytical solutions of the contact force between the string and the mass, the displacement responses of the string can be easily obtained. The solutions cover an infinite, semi-infinite or finite string subjected to a moving mass at subsonic, sonic or supersonic velocities. For the semi-infinite or finite strings, the solutions for different types of boundary conditions are presented in both a unified form and in the form of a series of exponential and polynomial functions. The formula derived is shown to be correct by comparison with the semi-analytical method.     

Three-dimensional laminar boundary layer on a blunt body

We consider the Prandtl laminar boundary layer which occurs with stationary flow about a blunted cone at an angle of attack. The solution of the Prandtl equations is sought using a finite difference method. It is found that a smooth solution of the problem exists only in the vicinity of the rounded nose of the body, while far from the nose the solutions acquire a singularity; in the problem symmetry plane (on the downwind side) there is a discontinuity of the first derivatives of the velocity components and the density. In the study of the Prandtl boundary layer in the problem of stationary flow about a pointed cone at an angle of attack, it has been shown [1] that the self-similar solution (dependent on two independent variables) of the Prandtl equations has a discontinuity of the first derivatives in the problem symmetry plane (on the downwind side of the cone). The suggestion has been made that in the three-dimensional problem of flow about a blunt cone at an angle of attack the solutions of the Prandtl equations may also be discontinuous. The present study was carried out to clarify the nature of the behavior of the solutions of the three-dimensional Prandtl equations. To this end we considered stationary supersonic flow of an ideal gas past a blunted cone. The results of this study (as well as those of [1]) were obtained using a numerical, finite-difference method. However, an analysis of the numerical results (investigation of the scheme stability, reduction of step size, etc.) shows that the properties of the solutions of the finite-difference equations are not in this case a result of numerical effects, but reflect the behavior of the solutions of the differential equations. The mathematical problem on the boundary layer which is considered in this study will be formulated in §2; this formulation is due to K. N. Babenko.     

The role of solution structure in apparent thickening behavior of dilute peo/water systems

An experimental investigation was undertaken to determine the role of solution structure on the apparent thickening exhibited by “dilute” polyethylene oxide/water solutions in extensional flow. Measurements of apparent relative viscosity were obtained as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm internal diameter tube. As the wall shear rate was increased, slightly shear thinning behavior was observed up until a critical wall shear rate was exceeded at which point a large increase in relative viscosity was seen. Other researchers have observed these apparent thickening effects and have interpreted them in terms of individual polymer molecules undergoing coil-stretch transitions. However, in the systems used in this study, the critical wall shear rate and the degree to which relative viscosity increased wer both seen to be strong functions of solution aging time and concentration. These results are inconsistent with the simple picture of individual polymer coils undergoing a coil-stretch transition and instead are consistent with the picture of aggregated systems or micronetworks being stretched from their equilibrium configurations.     

On the stretching and swelling of an elastic liquid extruded from a capillary die

A theoretical model for calculating the profile of a polymer jet extruded from a circular die and stretched by an axial force is elaborated. The model exploits the idea of matching the solutions for the polymer flow inside and outside the capillary by equating the elastic energy flux at the die outlet. The rheological equations used to obtain the two solutions have been previously tested in many situations. Comparison of the results of computations made according to the theoretical model with those of some special experiments revealed satisfactory agreement.     

Natural Convection on a Horizontal Cone in a Porous Medium with Non-Uniform Wall Temperature/Concentration or Heat/Mass Flux and Suction/Injection

The steady natural convection flow on a horizontal cone embedded in a saturated porous medium with non-uniform wall temperature/concentration or heat/mass flux and suction/injection has been investigated. Non-similar solutions have been obtained. The nonlinear coupled differential equations under boundary layer approximations governing the flow have been numerically solved. The Nusselt and Sherwood numbers are found to depend on the buoyancy forces, suction/injection rates, variation of wall temperature/concentration or heat/mass flux, Lewis number and the non-Darcy parameter.     

Asymmetric bifurcations in a pressurised circular thin plate under initial tension

It has been known for some time that under certain circumstances the axisymmetric solution describing the deformation experienced by a stretched circular thin plate or membrane under sufficiently strong normal pressure does not represent an energy-minimum configuration. By using the method of adjacent equilibrium a set of coordinate-free bifurcation equations is derived here by adopting the Föppl–von Kármán plate theory. A particular class of asymmetric bifurcation solutions is then investigated by reduction to a system of ordinary differential equations with variable coefficients. The localised character of the eigenmodes is confirmed numerically and we also look briefly at the role played by the background tension on this phenomenon.     

Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén–Lindel?f result as well as a principle of positive singularities in certain Lipschitz domains.     

The Characteristic Response of Hollow Cones

The response of a solid cone to a pure bending moment applied at its tip was found recently in terms of spherical polar coordinates by a using a suitable change of parameter leading to series solutions. By a very simple further parameter change, the additional solutions have been found which are needed to extend the analysis to give the response of a hollow cone to any resultant load applied to its tip. This revised version was published online in August 2006 with corrections to the Cover Date.     

磁电弹性圆锥顶端作用集中荷载的解析解

当磁电弹性材料特征根互异时，用5个势函数表示的通解出发，对圆锥顶端作用集中扭矩Mx的扭转、集中力Px和点电荷Q的压缩、集中力Px和集中力矩My的弯曲变形问题，用一些调和函数的线性组合分别构造了势函数，并根据边界条件求出了势函数中的待定系数从而确定势函数，再将势函数代入通解得到磁电弹性圆锥顶端作用集中载荷时的位移、电势、磁势、应力、电位移和磁感应强度的三维解析解。此解形式简单便于应用。当圆锥角2α=π时，可退化得到半空间问题的解。     

Combined heat and mass transfer by hydromagnetic natural convection over a cone embedded in a non-Darcian porous medium with heat generation/absorption effects

 The problem of combined heat and mass transfer by natural convection over a permeable cone embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The cone surface is maintained at either constant temperature and constant concentration or uniform heat and mass fluxes. In addition, the cone surface is assumed permeable in order to allow for possible fluid wall suction or blowing. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved numerically by an implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement between the results is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the temperature and concentration profiles as well as the local Nusselt number and the Sherwood number is illustrated graphically to show special trends of the solutions. Received on 5 June 2000 / Published online: 29 November 2001     

On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine

In the present note we study the motion of a particle under uniform gravity on a tilted smooth cone. We point out some integrable cases of that problem. In particular, we establish that motion on the cone with semi-vertical angle π/6 and one generator vertical is integrable. Moreover, we show that the swinging Atwood machine is equivalent to a special version of the motion on the tilted cone. This analogy makes it easier to analyze motion of SAM from a geometrical point of view.     

Improving the accuracy of discontinuous Galerkin schemes at boundary layers

The resolution of boundary layers typically requires fine grids in the wall normal direction, which leads to anisotropic elements being refined towards the wall. Best practice guidelines for the mesh generation of stretched boundary layer grids exist for the finite volume or finite difference discretizations. A similar resolution of boundary layers with DG schemes can be achieved with a coarser grid because of the subgrid resolution of the DG scheme. High order schemes incorporate the possibility of high order element mappings, resulting in different resolution properties inside the element. In this paper, we show that the use of an internal element mapping in combination with a stretched grid can be used to reduce the error of the boundary layer approximation by an order of magnitude in comparison with the classical linear internal element mapping. The boundary layer is modeled by a one‐dimensional singular perturbation problem. In addition, we discuss the construction of the element mappings by interpolation and investigate the limits of the stretching function such that the resulting element Jacobian remains positive. A parameter study shows the influence of the element mapping for different polynomial degrees on the solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Nontrivial equilibrium solutions for a semilinear reaction-diffusion system

By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.     

Diffraction of a spherical acoustic wave on a cone

An exact analytic solution of the problem of diffraction of a plane acoustic wave on a cone of arbitrary aperture angle was obtained and studied in [1]. For the case of spherical wave diffraction on a cone a formula is known [2] which relates the solutions of the spherical and plane wave diffraction problems. This study will employ the results of [1, 2] to derive and investigate an exact analytical solution of the problem of diffraction of a spherical acoustic wave on a cone of arbitrary aperture angle. Results of numerical calculations will be presented and compared with analogous results for a plane wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 200–204, March–April, 1976.The author is indebted to S. V. Kochura for her valuable advice.     

Breakup of a liquid bridge as a method of rheological testing of biological fluids

A method for the on-line testing of small volumes of biological fluids is presented. The method is based on the ability of some liquids to form, when stretched, long thinning filaments, liquid bridges. Measuring the filament thinning rate makes it possible to determine the rheological parameters of the fluid. The method was tested on synthetic polymer solutions and biological fluids, such as saliva, bronchial sputum, follicular fluid, and chicken egg albumen and yolk. The results evidence that this technique can be used in various biomechanical applications, in particular, for disease diagnostics and treatment monitoring.     

Experimental estimation of elongational stresses of dilute polymer solutions and a related examination of some constitutive equations

Elongational stresses of dilute polymer solutions have been estimated by utilizing the flow through small orifices under the condition of no vortex upstream of the orifice plane. The flow was approximated with a linearly converging flow towards an apex of a cone, its validity being partially confirmed by the measured center velocities, and the elongational stresses are determined from the measured thrusts of dilute polymer solutions. On the other hand, elongational stresses were theoretically obtained with the modified Maxwell model and the second order fluid. A comparison was made between the experimental and the theoretical results and the following points were clarified; below an elongational rate of 2 × 10⁴ s⁻¹ the modified Maxwell model gives elongational stresses close to the experimentally determined ones, but above that elongational rate it deviates from the experimental results. The second order fluid is not sufficient to describe the stresses in this kind of elongational flow and an acceleration term such as δ²e^ij/δt² may be necessary in this case.     

New Transforms for the Resolving Equations in Elastic Theory and New Integral Transforms, with Applications to Boundary-Value Problems of Mechanics

This paper reviews results from an analysis of exact boundary-value solutions of static and dynamic elasticity obtained by the method of integral transforms. Consideration is given to the solutions of problems for a half-space and a blunted hollow cone, the heat conduction problem for a cone, and the diffraction problem for a cone with a hole along the generatrix. Solutions of mechanics problems are analyzed and the method of integral transforms is generalized