Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

The study of Heat and Mass Transfer in a Visco elastic fluid due to a continuous stretching surface using Homotopy Analysis Method

In this paper, an approximate analytical solution is derived for the flow velocity and temperature due to the laminar, two-dimensional flow of non-Newtonian incompressible visco elastic fluid due to a continuous stretching surface. The surface is stretched with a velocity proportional to the distance $x$ along the surface. The surface is assumed to have either power-law heat flux or power-law temperature distribution. The presence of source/sink and the effect of uniform suction and injection on the flow are considered for analysis. An approximate analytical solution has been obtained using Homotopy Analysis Method(HAM) for various values of visco elastic parameter, suction and injection rates. Optimal values of the convergence control parameters are computed for the flow variables. It was found that the computational time required for averaged residual error calculation is very very small compared to the computation time of exact squared residual errors. The effect of mass transfer parameter, visco elastic parameter, source/sink parameter and the power law index on flow variables such as velocity, temperature profiles, shear stress, heat and mass transfer rates are discussed.     

Edge Singular Behavior for the Heat Equation on Polyhedral Cylinders in \boldsymbol{\mathbb{R}}^{\bf 3}

We study the Heat equation in the polyhedral cylinder with a non-convex edge. We construct the singularity functions depending on the time and edge axis, and the coefficient of the singularity, called the stress intensity distributions, and show regularity results for the solution and the coefficient. The regularity is achieved in the (not weighted) Sobolev space in the L² and L ^q spaces, respectively. An application to the finite polyhedral cylinder is described.     

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law*

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space, the shock-front coincides with the upwind surface of the body, hence there is an infinite-thin layer of concentrated mass, in which all particles hitting the body move along its upwind surface. By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations, which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces, the authors derive the compressible Euler equations for flows in concentration layers, which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface. As a by-product, they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows, which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.     

Stress concentration/intensity around elliptical/circular cylinder shaped surface flaws in cross-ply plates and validity of St. Venant’s principle in the presence of interacting singularities

Effects of localized elliptical (circular being a special case) cylindrical surface flaws in laminated composite plates are investigated by using C°-type triangular composite plate elements, formulated on the assumptions of transverse inextensibility and layer-wise constant shear-angle theory (LCST). Numerical results for a cross-ply laminate compromised by the presence of an external part-through elliptical/circular cylindrical slot indicate the existence of severe cross-sectional warping in the vicinity of the surface flaw and plate boundaries. Furthermore, three-dimensional nature of the stress concentration factor in the neighborhood of the elliptical or circular cylinder shaped surface flaw boundary is clearly exhibited. Besides, very high stress concentration factors are found in the layer weakened by the surface flaw. Most importantly, the effects of stress singularity in the neighborhood of the circumferential re-entrant corner lines of the elliptical/circular cylindrical surface flaws, weakening laminated composite plates, are numerically assessed, because of their role in crack initiation. Finally, the interaction of this singularity with free edge stress singularity at the plate boundary, and the implication of such interactions (i.e., violation of St. Venant’s principle) in regards to testing of laminated composite specimens are thoroughly investigated.     

Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge–Kutta–Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically.     

The Solution of Two-Dimensional Oseen Flow Problems Using Integral Conditions

A general method of solving Oseen's linearized equations fortwo-dimensional steady flow of a viscous incompressible fluidpast a cylinder in an unbounded field is developed. The analysisis developed in terms of the scalar vorticity and stream functionand it is shown that the vorticity for Oseen flow problems canbe obtained separately from the stream function. The determinationof the vorticity can be effected using conditions of an integralcharacter deduced from the no-slip condition at the cylindersurface together with the conditions at large distances. Theindependent determination of the vorticity seems to be a newstep in Oseen theory. The method enables one to obtain manyproperties of the flow in terms ofthe Reynolds number by usingonly the vorticity without the necessity of finding the streamfunction. The use of integral conditions makes the detailedcalculations straightforward, systematic, and elementary. Themethod is tested by applying it to the case of uniform flowpast an elliptic cylinder at an arbitrary angle of incidenceand also to cases of symmetrical and asymmetrical flows pastcircular cylinders. The leading approximation for small Reynoldsnumber is obtained where possible. In the case of flow pasta rotating cylinder, the only possible solution is the Oseensolution for the nonrotating case with the addition of a potentialvortex.     

Exterior Stokes flows with stick-slip boundary conditions

Steady two-dimensional creeping flows induced by line singularities in the presence of an infinitely long circular cylinder with stick-slip boundary conditions are examined. The singularities considered here include a rotlet, a potential source and a stokeslet located outside a cylinder and lying in a plane containing the cylinder axis. The general exterior boundary value problem is formulated and solved in terms of a stream function by making use of the Fourier expansion method. The solutions for various singularity driven flows in the presence of a cylinder are derived from the general results. The stream function representation of the solutions involves a definite integral whose evaluation depends on a non-dimensional slip parameter l₁\lambda_1. For extremal values, l₁ = 0\lambda_1 = 0 and l₁ = 1\lambda_1 = 1, of the slip parameter our results reduce to solutions of boundary value problems with stick (no-slip) and perfect slip conditions, respectively.¶The slip parameter influences the flow patterns significantly. The plots of streamlines in each case show interesting flow patterns. In particular, in the case of a single rotlet/stokeslet (with axis along y-direction) flows, eddies are observed for various values of l₁\lambda_1. The flow fields for a pair of singularities located on either side of the cylinder are also presented. In these flows, eddies of different sizes and shapes exist for various values of l₁\lambda_1 and the singularity locations. Plots of the fluid velocity on the surface show locations of the stagnation points on the surface of the cylinder and their dependencies on l₁\lambda_1 and singularity locations.     

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant, closed tube, driven by an internal source and sink. Fluid is assumed to be highly viscous with its motion described by Stokes flow. Model geometry is assumed to be axisymmetric, and the governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations. We solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface forcings yield an unsmooth solution, that part of the solution is computed using the immersed interface method. Jump conditions are derived for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. Numerical results are presented that indicate second-order accuracy of the method.     

电磁力控制下圆柱绕流的涡度拟能

Okubo-Weiss函数与流元的容变、畸变以及涡量相关,可以用来评估流场的涡结构．经该文数学证明,对于边界无滑移的,低Reynolds数的二维不可压缩流动,Okubo-Weiss函数的全流场积分为0．还以电磁控制的圆柱绕流为例,通过数值计算,对该结论进行了验证．根据计算结果,依据Okubo-Weiss函数值,对流场进行了划分,讨论了总涡度拟能、总变形率和Okubo-Weiss函数在流场中的分布规律,以及电磁力对分布的影响．     

The interaction between sound and circulatory fluid motion as a problem in matched expansions

A circular cylinder is at rest in a compressible fluid witha given circulation K'. At time t=0 the cylinder is made tomove with low Mach number along a straight line perpendicularto its axis. A modified matching argument is used to describethe sound field induced by the lifting body. The velocity potentialin the sound field can be represented, to leading order, interms of a moving line dipole aligned along the direction ofmotion together with a transverse dipole sheet that extendsfrom the starting location of the centre of the cylinder toits current location. The next-order term is that of a movingline source. The line dipole accounts for the motion of thecylinder. The dipole sheet represents the sound field due tothe circulatory motion. If the circulation is constant, thenso is the strength of the dipole sheet and the time dependencearises from the changing length of the layer. In a more realisticcase, where vorticity is shed to form a wake behind the movingbody, there is a corresponding change in the circulation andin the strength of the evolving dipole sheet.     

Enumeration and limit laws of dissections on a cylinder

We compute the generating function for triangulations on a cylinder, with the restriction that all vertices belong to its boundary and that the intersection of a pair of different faces is either empty, a vertex or an edge. We generalize these results to maps with either constant ({k}-dissections) or unrestricted (unrestricted dissections) face degree. We apply singularity analysis to the resulting generating functions to obtain asymptotic estimates for their coefficients, as well as limit distributions for natural parameters.     

Some Non-standard Sobolev Spaces, Interpolation and Its Application to PDE

We consider some nonstandard Sobolev spaces in one dimension, in which functions have different regularity in different subsets. These spaces are useful in the study of some nonlinear parabolic equations where the nonlinearity is highly degenerate and depends on the smoothness of the solution at a certain subset (that may vary with time). An example of application is a diffusion equation with a smooth free boundary, and a moving source/sink where the solution has singularity. The main new idea here is to characterize the functional space setting that is needed for semigroup theory to apply.     

Modified discrete source method as applied to the simulation of flows over a periodically irregular surface and a body of revolution

The incompressible flow over a stationary periodically irregular surface and a compact body of revolution is computed by applying a modified discrete source method. The method is tested as applied to the flow over a stationary sphere. Numerical results are presented for flows over a sinusoidal surface, a cycloidal surface, a spheroid, a Chebyshev particle, an “analytical” cylinder, and a half ball.     

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.     

Effect of aligned magnetic field on the steady viscous flow past a circular cylinder

The steady viscous incompressible and slightly conducting fluid flow around a circular cylinder with an aligned magnetic field is simulated for the range of Reynolds numbers 100 ? Re ? 500 using the Hartmann number, M. The multigrid method with defect correction technique is used to achieve the second order accurate solution of complete non-linear Navier–Stokes equations. The magnetic Reynolds number is assumed to be small. It is observed that volume of the separation bubble decreases and drag coefficient increases as M is increased. We noticed that the upstream base pressure increases slightly with increase of M whereas downstream base pressure decreases with increase of M. The effect of the magnetic field on the flow is discussed with contours of streamlines, vorticity, plots of surface pressure and surface vorticity.     

Surface waves on steady perfect‐fluid flows with vorticity

This is a theory of two‐dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem: the volume of fluid per period, the circulation per period on the free stream line, and the rearrangement class (equivalently, the distribution function) of the vorticity field. A minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function. This decreasing function can be thought of as an infinite‐dimensional Lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem. (Note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady.) To illustrate the idea with a minimum of technical difficulties, the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved. The theory applies equally to irrotational flows and to flows with locally square‐integrable vorticity. © 2011 Wiley Periodicals, Inc.     

Sound transmission into a thick hollow cylinder with the fixed-end boundary condition

In this paper sound transmission through the air filled finite thick cylinders exposed to the different incident acoustic wave is studied. The effect of end boundary conditions on the noise reduction of finite cylinders is evaluated. The uniform incident wave and the wave radiated from monopole and dipole sources are used in this study. Three positions are considered for the dipole source. Every position for the dipole source causes symmetric or antisymmetric pressure distributions on the external surface of the cylinder in tangential or axial direction. For the purpose of sound transmission analysis the linear three-dimensional theory of elasticity utilizing the technique of variables separation for the infinite circular cylinders is used to analyze the vibration of finite circular cylinder. In these analyzes the stress continuity condition on the inner and outer surfaces of the cylinder is satisfied using orthogonalization technique and velocity continuity condition is exactly satisfied on the interfacial surfaces. The sound transmission evaluation is carried out for cylinders with various half-length to outer-radius ratios. The results show that in the case of the fixed-end cylinder, the effect of boundary conditions on the noise reduction can be neglected for the half-length to outer-radius ratio of more than 10. Comparing between the obtained results from different acoustic sources shows that the obtained noise reductions from the uniform acoustic wave are less than those obtained from the monopole and dipole sources.