The effect of the slip condition on flows of an Oldroyd 6-constant fluid

The steady flows of a non-Newtonian fluid are considered when the slippage between the plate and the fluid is valid. The constitutive equations of the fluid are modeled by those for an Oldroyd 6-constant fluid. They give rise to non-linear boundary value problems. The analytical solutions are obtained using powerful, easy-to-use analytic technique for non-linear problems, the homotopy analysis method. It is shown that solutions exist for all values of non-Newtonian parameters. The solutions valid for no-slip condition for all values of non-Newtonian parameters can be derived as the special cases of the present analysis. Finally, graphs are plotted and critical assessment is made for the cases of slip and no-slip conditions.     

Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders

This paper deals with the unsteady flows of a viscoelastic fluid between two infinitely long concentric circular cylinders. The fractional calculus approach in the constitutive relationship model of a Burgers’ fluid is introduced. With the help of integral transforms (the Laplace transform and the Weber transform), exact solutions are constructed for the following two problems: (i) when the outer cylinder makes a simple harmonic oscillation; and (ii) when the outer cylinder suddenly begins rotating while the inner cylinder remains stationary. Some previous and classical results can be recovered from the presented results, such as starting solutions for second grade, Maxwell, Oldroyd-B, and Burgers’ fluids.     

Thermal analysis of moving induction heating of a hollow cylinder with subsequent spray cooling: Effect of velocity,initial position of coil,and geometry

In this paper, temperature analysis of the complete process of moving induction heat treatment is performed using numerical methods. A non-linear and transient magneto-thermal coupled problem with a moving coil which is considered as moving heat source, is investigated by an efficient finite-element procedure. A vertical hollow circular cylinder is heated by the moving coil at a given velocity along it, and the heated parts then quenched by a moving water–air spray. The effects of natural convection with air on the both inner and outer surfaces of cylinder, and also radiation of outer surface of cylinder with ambient are taken into account. For quenching of work-piece, a specific kind of atomized spray cooling which utilizes a mixture of water and air with different mass fractions is used. This procedure includes moving boundary conditions, temperature-dependent properties, and change in magnetic permeability of specified alloy at the Curie temperature. Obtained numerical results have been verified by comparison with analytical solutions using Green’s function methods. Also, the effect of velocity, initial position of inductor and inner to outer radius ratio on temperature distribution are investigated.     

Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading

Thermoelasticity problem in a thick-walled cylinder is solved analytically using the finite Hankel transform. Time-dependent thermal boundary conditions are assumed to act on the inner surface of the cylinder. For the mechanical boundary conditions two different cases are assumed: Traction–displacement problem (traction is prescribed on the inner surface and the fixed displacement boundary condition on the outer one) and Traction–Traction problem (tractions are prescribed on both the inner and outer surfaces of the hollow cylinder). The quasi-static solution of the thermoelasticity problem is derived analytically, i.e., the transient thermal response of the cylinder is derived and then, quasi-static structural problem is solved and closed form relations are extracted for the thermal stresses in the two problems. The results show to be in accordance with that cited in the literature in the special cases.     

Slow motion of a sphere away from a wall: Effect of surface roughness on the viscous force

An asymptotic analysis is given for the effect of roughness exhibited through the slip parameter β on the motion of the sphere, moving away from a plane surface with velocityV. The method replaces the no-slip condition at the rough surface by slip condition and employs the method of inner and outer regions on the sphere surface. For β > 0, we have the classical slip boundary condition and the results of the paper are then of interest in the microprocessor industry.     

Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Laplace方程有孔椭圆区域边值问题的解析解

本文利用相似流动替换方法 ,解决了中心有圆孔的椭园形区域上 Laplace方程第一类边值问题 ;采用分区域解法 ,给出了中心有椭园孔的椭园形区域上 Laplace方程第一类边值问题的解析通解 .这一结果在许多工程领域有重要应用 ,本文给出了油藏工程实例     

Axisymmetric flow over a backward-facing step in an annular pipe

The incompressible flow of a Newtonian fluid over a backward-facing step is investigated numerically. The geometry is an annular pipe in which the radius of the inner cylinder decreases suddenly. Keeping the radial expansion ratio fixed axisymmetric flows are computed for outlet radius ratios from 0.1 to 1 (ratio of the inner to the outer outlet radius). The Reynolds number at which the flow separates from the outer cylinder decreases as the outlet radius ratio decreases for constant inlet geometry. The growth with Reynolds number of the recirculation zone on the inner outlet cylinder just behind the step is strongly reduced when the recirculation zone on the outer cylinder is established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder with an imperfect interface

This paper studies the influence of the imperfectness of the contact condition on the torsional wave propagation in the initially stressed (stretched) bi-material compounded circular cylinder. The investigation is carried out within the scope of the piecewise homogeneous body model with the use of the Three-dimensional Linearized Theory of Elastic Waves in Initially Stresses Bodies. The mathematical formulation of the corresponding eigen-value problem is formulated and the solution method for that is developed. The two cases considered are the bi-material compounded cylinder consists of the solid inner and surrounding hollow outer cylinders (Case 1); the bi-material compounded cylinder consists of the hollow inner and surrounding hollow outer cylinders (Case 2). The mechanical relations of the cylinders’ materials are written through the Murnaghan potential. It is proven that the imperfectness of the contact condition does not influence the asymptotic-limit values of the wave propagation velocity. Moreover, the numerical results on the effects of the imperfectness of the boundary condition on the influence of the initial stresses on the wave propagation velocity are presented and discussed.     

A general approach to get series solution of non-similarity boundary-layer flows

An analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM), is applied to give convergent series solution of non-similarity boundary-layer flows. As an example, the non-similarity boundary-layer flows over a stretching flat sheet are used to show the validity of this general analytic approach. Without any assumptions of small/large quantities, the corresponding non-linear partial differential equation with variable coefficients is transferred into an infinite number of linear ordinary differential equations with constant coefficients. More importantly, an auxiliary artificial parameter is used to ensure the convergence of the series solution. Different from previous analytic results, our series solutions are convergent and valid for all physical variables in the whole domain of flows. This work illustrates that, by means of the homotopy analysis method, the non-similarity boundary-layer flows can be solved in a similar way like similarity boundary-layer flows. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of non-linear partial differential equations with variable coefficients in science and engineering.     

Motion of a non-Newtonian liquid in the annular channel of an insulation-coating tool

The axisymmetric flow of a non-Newtonian liquid with an exponential rheological law between two coaxial cylinders is considered for the case when the outer cylinder is fixed and the inner cylinder moves freely together with the liquid. At the outer cylinder the liquid moves either with or without slip. The individual theoretical conclusions are confirmed by the experimental results.Mekhanika Polimerov, Vol. 3, No. 1, pp. 151–155, 1967     

A singular perturbation approach to integral equations occurring in poroelasticity

Singular perturbation theory is used to solve the integral equationswhich occur when treating finite-length crack problems in porouselastic materials. The method provides the stress intensityfactors which characterize the near crack tip stress and displacementfields for small times. The method also gives the stress andpore pressure fields on the fracture plane for small times relativeto the diffusive time scale. In this paper, the authors treatcrack problems which are unmixed in the pore pressure boundarycondition on the fracture plane. The Abelian result that smalltimes correspond, in Laplace transform space, to large valuesof the transform variable is used to formulate the problemsin terms of a small parameter. Rescaling on this small parameterleads to inner problems which are eigensolutions of the semi-infiniteproblems treated earlier by the authors. The outer solutionsare given by elastic eigensolutions together with appropriatefluid dipole responses. These outer solutions give the completestress and pore pressure fields except in the neighbourhoodof the crack tips; in this region the outer solutions are asymptoticallymatched with inner solutions. The full outer solutions are givenhere as an asymptotic expansion for small times and enable thedevelopment of the outer fields to be followed in real time.A reciprocal theorem in Laplace transform space is used to checkthe small-time solutions. The inner problem is rescaled to asemi-infinite crack problem, so eigensolutions of this semi-infiniteproblem are used together with the known asymptotic behaviourof the real solution to identify the stress intensity factor.The stress intensity factor is then related to an integral involvingthe inner limit of the outer solution together with the eigensolutionof the semi-infinite problem. Using this integral, we recoverthe result for the stress intensity factor found using singularperturbation theory. A ‘nearly’ invariant integralanalogous to the invariant M integral used in elastostaticsis derived. Unfortunately, the poroelastic analogue is not invariant,although it is used to verify the small-time results.     

Wave front sets of the Riemann function of elastic interface problems

We obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley & Sons, Ltd.     

Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative

This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Numerical treatment of singularly perturbed two point boundary value problems

We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort.     

Analytic solutions for a rotating radial fin of rectangular and various convex parabolic profiles

The homotopy analysis method (HAM) is used to develop an analytical solution for the thermal performance of a radial fin of rectangular and various convex parabolic profiles mounted on a rotating shaft and losing heat by convection to its surroundings. The convection heat transfer coefficient is assumed to be a function of both the radial coordinate and the angular speed of the shaft. Results are presented for the temperature distribution, heat transfer rate, and the fin efficiency illustrating the effect of thickness profile, the ratio of outer to inner radius, and the angular speed of the shaft. Comparison of HAM results with the direct numerical solutions shows that the analytic results produced by HAM are highly accurate over a wide range of parameters that are likely to be encountered in practice.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.