Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field

A similarity analysis was performed to investigate the laminar free-convection boundary-layer flow in the presence of a transverse magnetic field over a vertical down-pointing cone with mixed thermal boundary conditions. Boundary layer velocity and temperature profiles were determined numerically for various values of the magnetic parameter and the Prandtl number. The results show that the magnetic field suppresses the velocity profiles and increases the skin friction. The temperature profiles were expanded with increasing values of the magnetic parameter resulting in higher surface temperatures. A transformation relating the similarity solutions of the boundary-layer velocity and temperature profiles associated with different values of the mixed thermal boundary condition parameter was obtained.     

Analytical and numerical simulation investigation in effects of radiation and porosity on a non-orthogonal stagnation-point flow towards a stretching sheet

The steady two-dimensional non-orthogonal flow near the stagnation point on a stretching sheet embedded in a porous medium in the presence of radiation effects is studied. Using similarity variables, the nonlinear boundary-layer equations are solved analytically by homotopy perturbation method (HPM) employing Padé technique. Comparison between the results of HPM-Padé solution and numerical simulation as well as some other results which are available in the literature, demonstrates a very good agreement between them and the HPM-Padé solution provides a convenient way to control and adjust the convergence region of a system of nonlinear boundary-layer problems with high accurate. The effect of involved parameters such as striking angle, radiation parameter, porosity parameter and the Prandtl number on flow and heat transfer characteristics have been discussed with more details.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Navier-Stokes equations on an exterior circular domain: construction of the solution and the zero viscosity limit

In this Note, we consider the limit of Navier-Stokes equations on a circular domain. By an explicit construction of the solution, it is proved that, when viscosity goes to zero, solution converges to the Euler solution outside the boundary layer and to the Prandtl solution inside the boundary layer.     

A scaling and energy equality preserving approximation for the 3D Navier-Stokes equations in the finite energy case

We discuss a new model (inspired by the work of Vishik and Fursikov) approximating the 3D Navier-Stokes equations, which preserves the scaling as in the Navier-Stokes equations and thus allows the study of self-similar solutions. Using some energy estimates and Leray’s limiting process, we show the existence of a solution of this model in the finite energy case, and the energy equality and inequality fulfilled by it. This approximation can be shown to converge to the Navier-Stokes equations using a mild approach based on the approximated pressure, and the solution satisfies Scheffer’s local energy inequality, an essential tool for proving Caffarelli, Kohn and Nirenberg’s regularity criterion. We also give a partial result of self-similarity satisfied by the approximated solution in the infinite energy case.     

Newton methods for a class of nonlinear hypersingular integral equations

Different iterative schemes based on collocation methods have been well studied and widely applied to the numerical solution of nonlinear hypersingular integral equations (Capobianco et al. 2005). In this paper we apply Newton’s method and its modified version to solve the equations obtained by applying a collocation method to a nonlinear hypersingular integral equation of Prandtl’s type. The corresponding convergence results are derived in suitable Sobolev spaces. Some numerical tests are also presented to validate the theoretical results.     

Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to p-system which consists of two different families of shocks interacting at some positive time,we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.     

A note on the similarity equations arising in free convection boundary layers with blowing and suction

The similarity equations for the free convection boundary-layer flow on a vertical plate with prescribed wall temperature and transpiration velocity are considered. The range of existence of solution is discussed first. For blowing this is seen to be independent of the transpiration parameter , depending only on the Prandtl number. For suction this range of existence of solutions is seen to depend on as well. Asymptotic solutions for strong suction and strong blowing are derived and compared with numerical solutions of the similarity equations.     

Series Solutions of Unsteady Boundary-Layer Flows over a Stretching Flat Plate

An analytic technique, namely, the homotopy analysis method, is applied to give series solution of the unsteady boundary-layer flows over an impermeable stretching plate. Different from all previous perturbation solutions, our series solutions are convergent in the whole time region 0 ≤τ < +∞. To the best of our knowledge, such kind of series solution has never been reported for this problem. Besides, two kinds of new similarity transformations about dimensionless time are proposed. Using these two different similarity transformations, we obtain the same convergent solution valid in the whole time region 0 ≤τ < +∞. Furthermore, it is shown that a nonlinear initial/boundary-value problem can be replaced by an infinite number of linear boundary-value subproblems.     

Examples of boundary layers associated with the incompressible Navier-Stokes equations

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. A meta theorem is then presented which covers all the cases considered here.     

二维带跳Navier-Stokes方程解的大偏差原理

在带泊松跳二维随机Navier-Stokes方程解的解的存在唯一性的基础上,利用弱收敛的方法证明了带泊松跳二维随机Navier-Stokes方程解的Freidlin-Wentzell型的大偏差原理.     

A note on Laplace’s equation inside a cylinder

Two difficulties connected with the solution of Laplace’s equation around an object inside an infinite circular cylinder are resolved. One difficulty is the non-convergence of Fourier transforms used, in earlier publications, to obtain the general solution, and the second difficulty concerns the existence of apparently different expressions for the solution. By using a Green’s function problem as an easily analyzed model problem, we show that, in general, Fourier transforms along the cylinder axis exist only in the sense of generalized functions, but when interpreted as such, they lead to correct solutions. We demonstrate the equivalence of the corrected solution to a different general solution, also previously published, but we point out that the two solutions have different numerical properties.     

Natural convection on a vertical stretching cylinder

The flow and natural (or mixed) convection due to a vertical stretching cylinder is studied. Using similarity transforms, the Navier-Stokes and energy equations reduce to a set of nonlinear ordinary differential equations. Asymptotic analysis for large Reynolds numbers shows the relation between axisymmetric flow and two-dimensional flow. Due to the algebraic decay of the similarity functions, numerical integration is performed using a compressed coordinate. The axial velocity is composed of forced convection due to stretching and natural convection from the heated cylinder. The heat transfer increases with both the Reynolds number and the Prandtl number. The result is also a rare similarity solution of the free convection and Navier-Stokes equations.     

Regularization of a two-dimensional singularly perturbed parabolic problem

A regularized asymptotic expansion of the solution to a singularly perturbed two-dimensional parabolic problem in domains with boundaries containing corner points is constructed. The asymptotics of solutions to such problems contain ordinary boundary-layer functions, parabolic boundary-layer functions, and their products, which describe a corner boundary layer.     

Hybrid,numerical solutions,for three-dimensional,compressible Navier-Stokes layer

The author proposes new hybrid solutions for the three-dimensional, compressible Navier-Stokes layer (NSL) over a flying configuration (FC), which use the analytical potential solutions, of the same FC, two times, namely: to reinforce the numerical solutions by multiplying them with these analytical solutions and as outer flow (instead of the parallel flow, used by Prandtl in his boundary layer theory). These hybrid solutions fulfill the last behavior, have correct jumps along the singular lines (like subsonic leading edges, junction lines wing-fuselage, etc.), are split, accurate and rapid convergent. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Compressible boundary layer and the Boltzmann equation

We consider the flow of ideal gas in half space described by the system of compressible Navier-Stokes equations. We apply the Prandtl scaling and we obtain the system of compressible Prandtl equations. In this article, a modification of the classical Chapman-Enskog method is proposed, which allows us to derive the system of compressible Prandtl equations directly from the Boltzmann equation without the use of the Knudsen-layer correction. Different types of boundary conditions are discussed.     

Homotopy analysis method for the asymmetric laminar flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of the disks contracting, rotation, heat transfer and different permeability on the viscous fluids and temperature distribution between two heated contracting rotating disks. Two cases are considered. For the first case, we neglect the viscous dissipation effects in the energy equation and reduce the Navier-Stokes equations and energy equation into nonlinear coupled ODEs by introducing the Von Kármán type similarity transformations. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number and rotation ratio on the velocity and temperature are discussed in detail. The second and more general case is that we consider the viscous dissipation in the energy equation. Under this assumption, the energy equation is reduced to a ordinary differential equation including the Eckert number, whose solution also is solved by HAM.     

Maximization of the lift/drag ratio of airfoils with a turbulent boundary layer: Sharp estimates,approximation, and numerical solutions

The lift/drag ratio of an airfoil placed in an incompressible attached flow is maximized taking into account the viscosity in the boundary-layer approximation. An exact solution is constructed. The situation when the resulting solutions are not in the admissible class of univalent flows is discussed. A procedure is proposed for determining physically feasible airfoils (with a univalent flow region) with a high lift/drag ratio. For this purpose, a class of airfoils is constructed that are determined by a twoparameter function approximating the found exact solution to the variational problem. For this class, the ranges of free parameters leading to physically feasible flows are found. The results are verified by computing a turbulent boundary layer using Eppler’s method, and airfoils with a high lift/drag ratio in an attached flow are detected.     

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-?verák’s criterion. We also show that if a weak solution u satisfies $\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $ for some 3 < p < ∞, then the number of singular points is finite.     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.