Half-planes without coupling under contact loading

Summary Nonrotating half-planes in contact under oblique loading are investigated in this paper. The solution is based on the influence integrals of the Flamant solution. The problem is determined by two integral equations for the normal and tangential stresses, which are uncoupled for special cases, as bodies of similar material in contact. In order to simplify the singular integrals, the method of superposition of flat punches is used. The result for the symmetric case is almost identical with the axisymmetric solution. For polynomial profiles of the form x ^s , the Muskhelishvili potentials can be written in terms of a complex hypergeometric function. The interior stress field is illustrated for an example. Accepted for publication 13 July 1996     

Boundary Stabilization of Stokes System in Exterior Domains

The problem of stabilizing a solution to the 2D Stokes system defined in the exterior of the bounded domain with smooth boundary is investigated, i.e. for a given initial velocity field and prescribed positive number k > 0 one has to construct a control function defined on the boundary such that the solution stabilizes to zero at the rate of 1/t ^k .     

Radiation from a sphere in compressible plasma

Summary The Hansen's vector wave functions have been modified so it could apply directly for the solution of general exterior boundary value problems in compressible plasma for a spherical geometry. The vector wave functionL has been included to represent an acoustic wave and the three angular orthogonal functions have been defined using complex Fourier series in the azymuthal direction. Using this modified method of vector wave functions, the exterior spherical boundary value problem in compressible isotropic plasma has been solved. The boundary conditions prescribed over the surface of the sphere have been the tangential electric field (or the tangential magnetic field) and the radial component of the velocity vector (or the pressure). From those four basic boundary value problems the coefficients have been derived and several particular cases has been discussed.The research reported in this paper was supported in part by the National Science Foundation, U.S.A.     

Non-traditional crack problem for transversely-isotropic body

The published traditional crack problem solutions usually consider cracks located in the planes, parallel to the plane of isotropy, which is usually denoted as z = 0. We consider here case of a crack located in the plane x = 0 and subjected to arbitrary normal or tangential loading. The case of elliptic crack is considered in detail. Complete solution for the fields of displacements and stresses is presented single contour integrals of elementary integrands. Stress intensity factors are computed explicitly.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Integral estimates for null Lagrangians

Sobolev spaces of differential forms are studied, L ^p-projections onto exact forms are introduced as a tool to obtain integral estimates for null Lagrangians. New results on compensated compactness are given and mean-coercive variational integrals are found. Existence of minima of certain mean-coercive functionals is established.     

Painlevé paradox during oblique impact with friction

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan^?1[?V₁(0)/V₃(0)] where V₁(0)and V₃(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.     

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W ^k, p (Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beir?o da Veiga and Crispo in J Math Fluid Mech, 2009, doi:). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L ^p -spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.     

Magnetohydrodynamic flow in a rectangular duct

The magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M, have been avoided. It is found that, as M increases, boundary layers are formed near the non-conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.     

Drag measurements on long thin cylinders at small angles and high Reynolds numbers

Measurements of the drag caused by turbulent boundary layer mean wall shear stress on cylinders at small angles of attack and high length Reynolds numbers (8×10⁶<Re_L<6×10⁷) are presented. The use of a full-scale, high-speed towing tank enabled the development of turbulent boundary layers on cylinders made of stainless steel, aluminum, titanium, and polyvinyl chloride. The diameter of all cylinders in this experiment was 12.7 mm; two cylinder lengths, 3.05 m and 6.10 m, were used, corresponding to aspect ratio values L/a=480 and 960, respectively. Materials of various densities were towed at critical angles, resulting in linear cylinder geometry for tow speeds ranging from 2.6 m/s to 20.7 m/s and angles between 0° and 12°. Towing angles were measured with digital photography, and streamwise drag was measured with a strut-mounted load cell at the tow point. The measured tangential drag was very sensitive to small increases in angle at all tow speeds. A momentum thickness length scale is proposed to scale the tangential drag coefficient. The effects of the cross-flow resulting from the small angles of tow have a significant effect on the tangential drag coefficient values. A scaling for the orthogonal force on the cylinders was determined and provides a correction to published normal drag coefficient values for pure cross-flow. The presence of the axial turbulent boundary layer has a significant effect on these orthogonal forces.     

Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L ₁ model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic L ^p theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.     

Optimal convergence rates for three-dimensional turbulent flow equations

In this paper,the convergence rates of solutions to the three-dimensional turbulent flow equations are considered.By combining the Lp-Lq estimate for the linearized equations and an elaborate energy method,the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the equilibrium state is small in the H3-framework.More precisely,the optimal convergence rates of the solutions and their first-order derivatives in the L2-norm are obtained when the Lp-norm of the perturbation is bounded for some p ∈[1,6/5).     

Asymptotic Analysis for a Mixed Boundary-Value Contact Problem

The variational solution of the nonlinear Signorini contact problem determines also the active contact zone Γ ^c . If the latter is known, then the elastic field is a solution of a linear mixed boundary value problem in which on Γ ^c the normal displacement and tangential traction are given, while on the non-contact part the total traction is zero. Such mixed boundary conditions in general generate singularities of the solution's stress field at the points P ^{( k )} where the boundary conditions change. For smooth data, however, the variational solution of the Signorini contact problem actually belongs to H ²(Ω)², which implies the disappearance of these singularities, i.e., that the corresponding stress intensity factors vanish. This paper is devoted to the characterization of the active contact zone Γ ^c by the vanishing stress intensity factors including their sensitivity with respect to varying Γ ^c for two-dimensional problems provided that Γ ^c consists of a finite number of intervals. We use the method of asymptotic expansions and derive an explicit formula for the sensitivity, which is rigorously justified by employing weighted Sobolev spaces with detached asymptotics. These results can be used to determine the points P ^{( k )} with a corresponding Newton iteration. Accepted July 6, 2000?Published online January 22, 2001     

Magnetohydrodynamic flow on a half-plane

We investigate the magnetohydrodynamic flow (MHD) on the upper, half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour.