The Convective Boundary-Layer Flow on a Reacting Surface in a Porous Medium

The convective boundary-layer flow on an impermeable vertical surface in a fluid-saturated porous medium is considered where the flow results from the heat released by an exothermic catalytic reaction on the surface converting a reactive component within the convective fluid to an inert product. The reaction is modelled by first-order kinetics with an Arrhenius temperature dependence. Numerical solutions of the governing equations are obtained for a range of parameter values. These show, for large activation energies, that localized rapid changes in wall temperature and localized high reaction rates occur a little way from the leading edge. Asymptotic expansions, valid at large distances from the leading edge, are derived, the form that these expansions take is qualitatively different depending on whether or not reactant consumption is included in the model.     

The anomalous skin effect and the reflectivity of metals

Summary Series expansions are derived for the integrals determining the surface impedance of a metal when the skin effect is anomalous.     

The stick-slip problem for a round jet

Summary The stick-slip problem for a round jet studied in Part I gives a good approximation for the swell of a low speed jet when the surface tension is large but it fails when the surface tension is small. In this paper a new stick-slip problem (II) is defined and solved using matched eigenfunction expansions. The new problem reduces to that solved in Part I when the surface tension is large and gives good results in the case of zero and small surface tension.With 18 figures     

Effect of surface electric current on the electrohydrodynamic flow inside and outside a spherical drop

The problem of electrohydrodynamic flow of a viscous, low-conducting, polarizable liquid inside and outside a spherical drop in an applied homogeneous constant electric field is analytically solved with account for the effect of both surface conduction current and surface convection current. The influence of the drop deformation on the field and the flow is neglected. The solution is obtained in the form of asymptotic expansions in a small parameter corresponding to weak surface convection electric currents.     

The propagation of elastic surface waves along cylindrical cavities of general cross section

The existence of surface wave modes, propagating along an infinite cylindrical cavity of arbitrary constant cross section in an elastic medium, is investigated theoretically. A general secular equation is derived using the null-field approach and global expansions of the surface displacements. Numerical results for the phase velocities, surface displacements and cut-off frequencies are presented for elliptic cross sections. The largest eccentricity considered is 0.99995 and it is inferred that the flexural mode exists at all frequencies for any eccentricity of the cross section.     

Higher-order effects in natural convection flow over uniform flux inclined flat plates in porous media

Higher-order boundary-layer effects for natural convection flow along inclined flat plates (of both positive and negative inclinations) with a uniform heat flux surface condition in a saturated porous medium are studied. Using the method of matched asymptotic expansions the three terms in inner and outer expansions have been obtained. It is shown that the first eigenfunction for this considered problem coincides with 0(? ²)-term in the inner expansion.     

Asymptotic Theory of the Viscous Interaction Between a Vortex and a Plane

The problem of the viscous interaction between a flow induced by a vortex filament and an orthogonal rigid surface is solved for high Reynolds numbers using the method of matched asymptotic expansions. In view of the impossibility of matching the principal terms of the asymptotic expansions directly for the near-axial boundary layer and the main flow zone, the solution is obtained by introducing two intermediate zones. In this case a logarithmic singularity of the axial velocity arises inevitably on the vortex filament. In the near-axial and intermediate zones the solution is obtained numerically and analytically, respectively, while in the main zone the problem reduces to the problem of the flow induced by a line of weakly swirled vortex-sinks.     

A spectral method for free surface flows of inviscid fluids

In this paper, an efficient numerical method for unsteady free surface motions, with simple geometries, has been devised. Under the potential flow assumption, the governing equation of free surface flows becomes a Laplace equation, which is treated here by means of a series expansions of the velocity potential. The free surface is represented with a height function. The present method is applied to surface gravity waves to test the stability and accuracy of the method. To show the versatility of the method, a model for a dip formation is considered. © 1998 John Wiley & Sons, Ltd.     

The evaluation of the surface impedance in the theory of the anomalous skin effect in metals

Summary A simple method is described for obtaining series expansions for the integrals which represent the surface impedance in the theory of the anomalous skin effect in metals. The results agree with those recently obtained by Dingle¹) by means of an intuitive procedure.     

Scattering of elastic waves by non-planar cracks

A modification of the null field approach is used to study the scattering of elastic waves by non-planar cracks. A fictitious surface is added to the crack so that a convenient closed surface is obtained and the surface fields on this closed surface are expanded in vector spherical harmonics. The edge conditions are introduced into these expansions and this is shown to be essential for the numerical convergence. Total cross sections and backscattering amplitudes as functions of frequency are computed numerically for rotationally symmetric cracks which are part of spherical or spheroidal surfaces. By integration in frequency backscattered pulses are also computed. Some cases with two cracks are also considered.     

Torsion of a cylinder with a shallow external crack

The torsional problem of a finite elastic cylinder with a circumferential edge crack is studied in this paper. An efficient solution to the problem is achieved by using a new form of regularization applied to dual Dini series equations. Unlike the Srivastav approach, this regularization transforms dual equations into a Fredholm integral equation of the second kind given on the crack surface. Hence, exact asymptotic expansions of the Fredholm equation solution, the stress intensity factor and the torque are derived for the case of a shallow crack. The asymptotic expansions are certain power-logarithmic series of the normalized crack depth. Coefficients of these series are found from recurrent relations. Calculations for a shallow crack manifest that the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter.     

Simulation of stress-strain state in SiGe island heterostructures

The problem of simulation of the stress-strain state in SiGe island heterostructures is considered. The analytic-numerical method of multipole expansions is used to obtain an approximate solution. The problem of the stressed state influence on the diffusion mobility of atoms adsorbed on the heterostructure free surface is also discussed.     

Asymptotic modelling of an elastic roller crossing a scratch

The paper presents asymptotic modelling of the generalized Greenwood problem for an elastic cylinder (considered as a bounded two-dimensional elastic body) pressed between two rigid punches one of them with the dented surface having a scratch. The method of matched asymptotic expansions is used to obtain analytic approximations for the mutual displacement of the punches and the relative displacement of the roller's center.     

Investigation of the flows in axisymmetric and plane divergent channels with preseparation boundary layers on their walls

The method of matched asymptotic expansions is used to develop a procedure for constructing separationless plane and axisymmetric diffusers for incompressible high-Reynolds-number flows with a given pressure distribution over the diffuser surface and minimum friction losses. The problem for the region of interaction between the boundary layer and the inviscid outer flow is numerically solved and the effect of variation of the on-design flow parameters in the entry section on the diffuser flow parameters is studied.     

Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity

For high Reynolds numbers asymptotic expansions are constructed of the solution of the axially symmetric wave problem on the surface of a viscous incompressible fluid of infinite depth under the assumption that the tangential stresses on the free surface are of the order 0(1/Re). The principal terms of the asymptotic expansion are solutions of linear partial differential equations. The obtained result is then adapted to the case in which the fluid fills a bounded region whose boundary is a free surface. Some examples are given.     

Asymptotic theory of a wing moving near a solid wall

In this study we use the method of matched asymptotic expansions to obtain an approximate solution of the problem of the nonstationary motion of a lifting surface near a solid wall. The region of flow is provisionally subdivided into characteristic zones, in which, using the appropriate coordinates, we construct asymptotic expansions for the velocity potential, which thereafter coalesce in the regions of common validity. In the first approximation (extremely small heights of flight) the problem reduces to the solution of a Poisson equation in a plane region bounded by the contour of the wing in the horizontal plane with boundary conditions established from the coalescence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 115–124, November–December, 1977.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

The Onset of Fluid Rotation in a Thermogravitational Boundary Layer with Local Cooling of the Free Surface

Axisymmetric regimes of flows of an inhomogeneous fluid in the boundary layer near a free surface are calculated for a nonuniform temperature distribution on this surface. For the fluid motion equations written in the Oberbeck-Boussinesq approximation, the leading terms of asymptotic expansions of solutions of a steady-state problem are constructed. It is shown that in the presence of local cooling of the free surface and a rising outer fluid stream, as a result of a bifurcation, a pair of rotational regimes may develop in a thin boundary layer near the free surface, with no rotation observed outside this layer. No bifurcation of rotation was detected in the case of local heating of the free surface.     

A consistent boundary element method for free surface hydrodynamic calculations

Free surface phenomena are described by equations that exhibit two types of non-linearities. The first is inherent to the equations themselves and the second is caused by the application of boundary conditions at a free surface at an unknown location. Numerical calculations usually do not specifically recognize the second non-linearity, nor treat it in a fashion consistent with the more obvious non-linearities in the boundary conditions. A consistent formulation is introduced in the present paper. The field equation is integrated and the free surface boundary conditions are applied on the unknown geometry by means of appropriate series expansions. The consistent formulation introduces improvements in accuracy and computing speed. The method is demonstrated on several hydrodynamic free surface problems and an error analysis is included.     

CAPILLARY EFFECT ON VERTICALLY EXCITED SURFACE WAVE IN CIRCULAR CYLINDRICAL VESSEL

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.