Resonant Generation of Finite-Amplitude Waves by Flow Past Topography on a β-Plane

The forced Korteweg-de Vries (fKdV) equation is the generic equation for resonant flow past an obstacle. However, for flow past topography on a β-plane, the case when the upstream flow is uniform is anomalous in that there is no quadratic nonlinear term in the fKdV equation. Here we show that in this important case an alternative theory is required and obtain a new evolution equation, which has some similarities to the fKdV equation with two significant differences. These are that a small-amplitude topography now produces finite-amplitude waves and the flow response is limited by a wave breakdown characterized by an incipient flow reversal. Various numerical solutions are described.     

Circle theorems for steady Stokes flow

Two circle theorems for two-dimensional steady Stokes flow are presented. The first theorem gives an expression for the stream function for a Stokes flow past a circular cylinder in terms of the stream function for a slow and steady irrotational flow in an unbounded incompressible viscous fluid. The second theorem gives a more general expression for the stream function for another Stokes flow past the circular cylinder in terms of the stream function for a slow and steady rotational flow in the same fluid.     

On the optimal control of viscous incompressible fluid flow

The framework of the Navier-Stokes (N-S) equations is used to study flow past an arbitrary body on whose surface the tangential or normal velocity is under control. The necessary conditions are obtained for the minimum rate of energy dissipation. Exact analytical solutions of the corresponding problems are found for the case of flow past an ellipsoid in the Stokes approximation.     

Application of the small-parameter method to the problem of the spatial flow of a viscous gas past bodies

The solution of the problem of the spatial hypersonic flow of a viscous gas past spherically blunted bodies is considered using the system of equations of a complete viscous shock layer (CVSL). The use of the small-parameter method (SPM) in conjunction with the method of global iterations enables one to reduce the computer time required by a factor of approximately 100 compared with the time needed to calculate similar problems in a strictly spatial formulation by establishment methods [1]. The flow past blunt cones and bicones of long length at low angles of attack is considered as well as the flow past a body, which differs slightly from an axisymmetric one, at zero angle of attack. The applicability of the SPM is confirmed by comparison with experimental and computed data.     

Molecular modellingof flow past an inclined flat plate

Molecular flow past an inclined plate is simulated on a scientific personal computer. Wake flow and turbulent effects are described and discussed for various fluid speeds.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

The effect of gravity on flow past a semi-circular cylinder with a constant pressure wake

The flow past a semi-cylinder with a trailing wake region is considered. In the absence of gravity the only known high Reynolds number solutions have tangential separation from the body and a cusped shape at the back of the wake. This flow can be a simple model for several situations, including the classical approximation of a constant pressure wake and the flow past an object with a region of trapped fluid of different density (or an air cavity) attached on the downstream side. Here we relax the assumption of high flow speeds to examine the effects of gravity. It is shown that there are situations in which a stagnation point can form either on the body or at the tail of the wake and that there is a minimum velocity beneath which a cavity will not form. Non-uniqueness in the parameter space is found in certain cases.     

The topology of the zero-level lines of the components of the acceleration vector in subsonic flows

The steady subsonic flow past bodies of finite dimensions, when the stream is unbounded and uniform at infinity is considered. The structure formed by the stationary points (points where both components of the acceleration vector vanishes), by the zero-level of the components of the acceleration vector emerging from them and the body past which the flow occurs is studied. It is shown that each of the above-mentioned lines must reach the surface of the body past which the flow takes place. This fact, in particular, enables one to estimate the overall number of streamlines with zero curvature emerging from the stationary points in terms of the number of zeros of the curvature of the streamlines on the body around which the flow takes place, including the branch points of a dividing streamline. With a view to refining the above mentioned number of zeros, the known solution for the neighbourhoods of the branch points of a streamline is considered and the singularity of the flow in the neighbourhoods of points of discontinuity of the curvature of the wall around which the flow occurs is investigated. In order to illustrate the above, certain properties of the flow past convex bodies are refined and a fairly broad class of so-called convex-concave bodies with zero angle of tapering of the trailing edge is constructed and considered. It is shown that, for this body, there are not more than four zeros of the curvature of the streamline and, as a consequence, there are no branch points of the isobars and isoclines in the flow field, including at infinity, an infinitely distant point is the sole stationary point and, most important of all, in the case of the flow past the given bodies the values of the circulation and the lifting force cannot vanish. The mathematical apparatus employed is based on the equations of gas dynamics constructed earlier for certain combinations of the components of the acceleration vector.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.     

Uniform Flow Past an Axially Uniform Viscous Vortex

It is believed that the flow past a tornado causes the formation of smaller vortices which produce the “suction spots” observed along the path of destruction. Here we develop a greatly simplified mathematical model to investigate this phenomenon. An axially uniform vortex is developed by visualizing a circular tube with uniform surface suction of fluid possessing circulation at infinity. This vortex is then perturbed by a uniform flow past it. An inner asymptotic expansion of an E^1/3 radial boundary layer is matched to an outer expansion to obtain a solution. The results show that a stagnation point developing into a secondary vortex is formed in a free shear layer at critical flow conditions. However, it is difficult to apply our results quantitatively because of the difficulty in comparing the axially uniform vortex with a real tornado vortex.     

Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

A free boundary problem of elliptic equation arising in supersonic flow past a conical body

We study free boundary value problems of elliptic equation caused by a supersonic flow past a non-symmetric conical body. The flow is described by the potential flow equation. In the self-similar coordinate system the problem can be reduced to a boundary value problem of second order nonlinear elliptic equation with a free boundary. Applying the partial hodograph transformation and the method of nonlinear alternative iteration we proved the existence of solution to this boundary value problem. Consequently, we also proved the conclusion that for the problem of supersonic flow past a conical body, if the conical body is slightly different from a circular cone with its vertex angle less than a given value determined by the parameters of the coming flow, then there exists a weak entropy solution with an attached conical shock.     

Nearly parallel Blasius flow with slip

The steady boundary layer flow past a moving horizontal flat plate with a slip effect at the plate in a free stream with constant speed, slightly different from the plate speed is studied. An analytic perturbation solution of order two is obtained for the velocity. With respect to the parallel flow both the boundary layer and the inverted boundary layer characters of the flow are plotted and discussed. It is observed that under high slip, the flow becomes a nearly parallel flow with an increased speed.     

Hydromagnetic effects on the three-dimensional flow past a porous plate

Hydromagnetic effects on the three-dimensional flow of an electrically conducting viscous incompressible fluid past a porous plate with periodic suction has been analysed. The uniform flow is subjected to a transversely applied magnetic field. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. Approximate solutions for the components of velocity field and the skin-frictions due to them are obtained and discussed with the help of a graph and tables.     

Resonant Flow of a Rotating Fluid Past an Obstacle: The General Case

We consider the flow of a rotating fluid past an antisymmetric obstacle placed on the axis of a cylindrical tube, for the case when the upstream flow is nearly resonant, or critical, so that the speed of a free linear long wave is nearly zero in the frame of reference of the obstacle. The perturbed flow is dominated by the resonant mode, whose amplitude satisfies a forced Korteweg—de Vries equation in this general case when the upstream flow contains radial shear andjor radially dependent angular velocity.     

Compressible Flow Calculations by a Variational Method using Natural Boundary Conditions

The problem of steady inviscid plane subsonic flow past an aerofoilis formulated as a variational principle with the boundary conditionat the surface of the aerofoil treated as a natural boundarycondition. Numerical solutions are obtained for flow past non-liftingellipses by the application of finite differences to two differentforms of the variational principle. The numerical procedureis simpler than when the boundary condition at the surface istreated explicitly, and the solutions are of the same accuracy.     

A Variational Approach Via Bipotentials for a Class of Frictional Contact Problems

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

A note on Hydromagnetic oscillatory flow along an infinite flat plate with variable suction

The two-dimensional, incompressible flow past an infinite plate of a weakly conducting fluid in the presence of a transverse magnetic field is discussed when the suction velocity normal to the plate as well as the external flow velocity vary periodically with time. Expressions for the velocity and the skin-friction in the boundary layer have been obtained in a non-dimensional form.