on multiple solutions for berman s problem we study berman s problemsubjects to the conditions f 0 f 0 f 1 f 1 1 0 which arises from the study of laminar flows in channels with porous walls positive negative re denotes the case of suction injection flows preliminary numerical studies indicated the existence of three different types of solutions and a small portion of mathematical evidence was provided in this paper we are able to show that there exist connected sets in the re k plane on which different types of solutions occur in particular our result verifies that the problem possesses all three types of suction solutions for sufficiently large re moreover the limiting injection solution is also obtained

on steady rotational high speed flows the compressible taylor culick profile we consider the compressible flow analogue of the well known taylor culick profile we first present the compressible euler equations for steady axisymmetric isentropic flow assuming uniform injection of a calorically perfect gas in a porous chamber we then apply the rayleigh janzen expansion in powers of m 2 w where mw is the wall mach number we solve the ensuing equations to the order of m 4 w and apply the results up to the sonic point in a nozzleless chamber area averaging is also performed to reconcile with one dimensional theory our solution agrees with the existing theory to the extent that it faithfully captures the steepening of the taylor culick profile with downstream movement based on the closed form expressions that we obtain the main flow attributes are quantified parametrically and compared to the existing incompressible and quasi one dimensional theories verification by computational fluid dynamics is also undertaken comparison with two turbulent flow models shows excellent agreement particularly in retracing the streamwise evolution of the velocity regardless of the mach number we observe nearly identical trends in chambers that are rescaled by the critical sonic length ls using a suitable transformation we prove the attendant similarity and provide universal criteria that can be used to assess the relative importance of gas compression in solid rocket motors owing to sharper velocity gradients at the wall we find that an incompressible model underestimates the skin friction along the wall and underpredicts the centreline speed by as much as 13 at the sonic point in practice such deviations become appreciable at high injection rates or chamber aspect ratios

analysis of steady flow in a channel with one porous wall or with accelerating walls a similarity solution of the navier stokes equations that describes the two dimensional flow of a viscous incompressible fluid through a channel with one porous wall is analysed the analysis which determines the number and character of solutions in various cases according to the higher order boundary conditions at the impermeable wall of the channel agrees with asymptotic results in the limits of small and large reynolds numbers flow in a channel with one or two accelerating walls is similarly analysed

on the asymptotic solution of a high order nonlinear ordinary differential equation we examine berman s similarity solution for steady flow of a viscous incompressible fluid in a channel driven by suction or injection of the fluid through the channel walls we convert the two point boundary value problem for the similarity function into an initial value problem in which the position of the upper channel wall and the reynolds number are determined as part of the solution and consider two cases in the first case equal suction or injection takes place at each wall and solutions are sought that are symmetrical about the mid line of the channel in the second case fluid can pass through only the upper channel wall in each case analysis of a single initial value problem yields two distinct solutions one corresponding to injection the other to suction and our analysis is valid in the limit of large suction or injection the injection solutions are well known for the first symmetrical case the suction solution has previously been given only incompletely while for the second case the asymptotic structure of the solution for large suction has been an open problem we complement our asymptotics with numerical integrations of the ordinary differential equation governing the similarity function and find excellent agreement between the two

on the existence of solutions of an equation arising in the theory of laminar flow in a uniformly porous channel with injection we establish the existence of concave solutions of berman s equation which describes the laminar flow in channels with injection through porous walls it is found that the unique concave solutions exist for all injection reynolds number r 0

two dimensional viscous flow between slowly expanding or contracting walls with weak permeability abstract since the transport of biological fluids through contracting or expanding vessels is characterized by low seepage reynolds numbers the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls the scope is limited to two dimensional symmetrical solutions inside a simulated channel with moving porous walls in seeking an exact solution similarity transformations are used in both space and time the problem is first reduced to a nonlinear differential equation that is later solved both numerically and analytically the analytical procedure is based on double perturbations in the permeation reynolds number r and the wall expansion ratio results are correlated and compared via variations in r and under the auspices of small r and the analytical result constitutes a practical equivalent to the numerical solution we find that when suction is coupled with wall contraction rapid flow turning is precipitated near the wall where the boundary layer is formed conversely when injection is paired with wall expansion the flow adjacent to the wall is delayed in this case the viscous boundary layer thickens as injection or expansion rates are reduced furthermore the pressure drop along the plane of symmetry increases when the rate of contraction is increased and when either the rate of expansion or permeation is reduced as nonlinearity is retained our solutions are valid from a large cross section down to the state of a completely collapsed system

approximate high order dynamic theory of a fluid layer in between two thick solids a thin layer of heavy fluid with bound faces is considered first for the sake of simplicity the effects of vortex and viscosity are neglected two cases of compressible and incompressible fluids are treated assuming that a thickness over wavelength ratio is a small parameter for the internal state of medium the relations between the quantities on the layer surfaces are deduced in contrast to the standard approach which expands the propagator matrix into power series of a small wave number the asymptotic integration of 3d equations and boundary conditions of fluid dynamics is performed respective relations are represented in a recurrent form and permit one to obtain the high order components of the displacements and pressure rather simply when considering two thick solids with a fluid in between this result is used to derive the so called impedance boundary conditions ibc with a relative asymptotic error up to tenth order tests show their validity until the first quasiresonance frequency of a layer so the analysis is not just long wave but essentially low frequency thus these ibc are applicable to reduce the dimension in the analysis of challenged multicomponent system of fluid coupled solids in a reasonable frequency range

lie group method for unsteady flows in a semi infinite expanding or contracting pipe with injection or suction through a porous wall the unsteady incompressible laminar flow in a semi infinite porous circular pipe with injection or suction through the pipe wall whose radius varies with time is considered the present analysis simulates the flow field by the burning of inner surface of cylindrical grain in a solid rocket motor in which the burning surface regresses with time we apply lie group method for determining symmetry reductions of partial differential equations lie group method starts out with a general infinitesimal group of transformations under which given partial differential equations are invariant then the determining equations are derived ibragimov elementary lie group analysis and ordinary differential equations wiley new york 1999 hydon symmetry methods for differential equations cambridge university press cambridge 2000 olver applications of lie groups to differential equations springer new york 1986 seshadri na group invariance in engineering boundary value problems springer new york 1985 yi fengxiang lie symmetries of mechanical systems with unilateral holonomic constraints chinese sci bull 45 2000 1354 1358 moritz schwalm uherka finding lie groups that reduce the order of discrete dynamical systems j phys a math 31 1998 7379 7402 nucci clarkson the nonclassical method is more general than the direct method for symmetry reductions an example of the fitzhugh nagumo equation phys lett a 164 1992 49 56 basarab lahno group classification of nonlinear partial differential equations a new approach to resolving the problem proceedings of institute of mathematics of nas of ukraine vol 43 2002 pp 86 92 burde expanded lie group transformations and similarity reductions of differential equations proceedings of institute of mathematics of nas of ukraine vol 43 2002 pp 93 101 gandarias bruzon classical and nonclassical symmetries of a generalized boussinesq equation j nonlinear math phys 5 1998 8 12 hill solution of differential equations by means of one parameter groups pitman publishing co 1982 the determining equations are a set of linear differential equations the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables after the group has been determined a solution to the given partial differential equation may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables in the system effect of the crossflow reynolds number re and the dimensionless wall expansion ratio on velocity flow streamlines axial and radial pressure drop and wall shear stress has been studied both analytically and numerically and the results are plotted

existence of solutions for berman s equation from laminar flows in a porous channel with suction abstract we study the berman problem re 2 k re 0 d d subject to the conditions 0 0 1 1 1 0 which arises from laminar flows in a porous channel with suction the existence of nonnegative concave solutions for each re is verified by applying a topological method with an a priori estimate the uniqueness for small re is also shown

approximate dynamic boundary conditions for a thin piezoelectric layer abstract approximate dynamic boundary conditions of different orders are derived for the case of a thin piezoelectric coating layer bonded to an elastic material the approximate boundary conditions are derived using series expansions of the elastic displacements and the electric potential in the thickness coordinate of the layer all the expansion functions are then eliminated with the aid of the equations of motion and boundary interface conditions of the layer this results in boundary conditions on the elastic material that may be truncated to different orders in the thickness of the layer to obtain approximate boundary conditions the approximate boundary conditions may be used as a replacement for the piezoelectric layer and thus simplify the analysis significantly numerical examples show that the approximate boundary conditions give good results for low frequencies and or thin piezoelectric layers

on the lagrangian optimization of wall injected flows from the hart mcclure potential to the taylor culick rotational motion the taylor culick solution for a porous cylinder is often used to describe the bulk gas motion in idealized representations of solid rocket motors however other approximate solutions may be found that satisfy the same fundamental constraints in this vein steeper or smoother profiles may be observed in either experimental or numerical tests particularly in the presence of intense levels of acoustic energy in this study we use the lagrangian optimization principle to arrive at multiple solutions that can help to explain the practically observed motions we then search for the extreme states that display either the least or the most kinetic energy these are derived and found to be dependent on the chamber s aspect ratio by assuming a slender case simple expressions are retrieved from which both the rotational taylor culick and the irrotational hart mcclure solutions are recovered as special cases at the outset a new family of flow approximations is derived extending from purely potential to highly rotational fields these are constructed verified and catalogued based on their kinetic energies to help understand the tendency of a motion to shift energy states a second law analysis is performed and used to rank these solutions based on their entropy content interestingly the taylor culick profile is found to represent an equilibrium state entailing the most energy and entropy among those starting from the rest a formal extension of kelvin s minimum energy theorem to an open region is then pursued followed by a direct implementation to the problem at hand three other flow motions with open boundaries are examined to further exemplify the application of kelvin s extended criteria our efforts illustrate the overarching harmony that exists between lagrange s minimization principle and kelvin s minimum energy theorem both converge on the potential solution as the least kinetic energy bearer even when the velocity at the open barrier is finite

on timoshenko reissner type theories of plates and shells abstract linear tr theories timoshenko reissner theories of isotropic plates and shells are discussed these theories take into account the transversal shear deformation and rotation inertia the main subject under consideration is the construction of these theories by the asymptotic method and the related error estimates for static and dynamic problems in the dynamic case a method is suggested for the extension of the range of applicability of the tr theory

analysis of the acoustical edge flexural mode in a plate using refined asymptotics the dispersion properties of bending wave localized near the stress free edge of a thin isotropic plate are investigated for relatively high frequency in order to clarify the nature of the wave attention is focused to the sensibility to the improved boundary conditions and to the iterations of the main differential operator when describing by two dimensional 2 d approximate plate theories as shown the boundary conditions of next asymptotic order do not change too much the leading part of the desired wave but the corrections of the main operator lead to dramatical changes finally beginning with its simplest kirchooff s plate model only the leading part of wave may be subjected to asymptotic correction for applications the wave speed calculation is reduced to two simple analytical formulas the results agree well with experiments and finite element testing the alternative approach related to the timoshenko reissner mindlin theory is discussed

high order approximate low frequency theory of elastic anisotropic lining and coating a problem of the dynamic behavior of an elastic layer coupled to one or two thick elastic solids is considered all the materials may possess a general anisotropy and the layer is assumed to be thin enough with respect to the characteristic wavelength introducing the asymptotic power series with respect to the thickness over wavelength ratio for the main quantities and using the asymptotic integration method the displacements and stresses on the layer surfaces are related thus the so called impedance boundary conditions ibc are deduced for three cases for a coated substrate with given displacements or with given stresses on the surface and for two substrates with a layer in between in contrast to previous papers these ibc are obtained for the most general situation with the asymptotic accuracy up to the sixth order uniform with respect to the representation of the displacements and stresses presented theory can be used for studying the surface and interface phenomena as well as for calculating fie

faber krahn inequalities for the robin laplacian a free discontinuity approach isoperimetric inequalities for the principal eigenvalues of the robin laplacian are interpreted as free discontinuity problems of unusual type we prove a full range of faber krahn inequalities in a nonlinear setting and for non smooth domains including the open case of the torsional rigidity the key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation as a byproduct we obtain the best constants for a class of poincare inequalities with trace terms in mathbb r n

numerical solution of steady viscous flow of a micropolar fluid driven by injection between two porous disks abstract steady laminar incompressible and two dimensional flow of a micropolar fluid between two porous coaxial disks is considered to describe the working fluid we use the micropolar model given by eringen the governing equations of motion are reduced to a set of non linear coupled ordinary differential equations using berman s similarity transformation the sor iterative method is used to solve the differential equations numerically for higher order accuracy the results obtained are further refined and enhanced by richardson s extrapolation method the results of micropolar fluids are compared on different grid sizes

symmetric solutions for the oscillatory channel flow with arbitrary suction this paper considers a porous channel in which a suction driven flow is modulated by arbitrary levels of fluid extraction acting uniformly along its porous boundaries when small longitudinal oscillations are enabled a rotational wave motion is established that this study attempts to analyse for an elongated channel two asymptotic methods are used the first technique is based on a two variable multiple scale expansion that takes into account the thin boundary layer near the wall while retaining generality of expression the multiple scale procedure is carried out until a closed form solution for the velocity field is obtained for an arbitrary mean flow function an alternative approach based on wkb exponentials is also employed the wkb expansion is then pursued to arbitrary order these asymptotic formulations are shown to agree with one another and with numeric simulations of the problem for three specific mean flow functions

extensional channel flow revisited a dynamical systems perspective extensional self similar flows in a channel are explored numerically for arbitrary stretching shrinking rates of the confining parallel walls the present analysis embraces time integrations and continuations of steady and periodic solutions unfolded in the parameter space previous studies focused on the analysis of branches of steady solutions for particular stretching shrinking rates although recent studies focused also on the dynamical aspects of the problems we have adopted a dynamical systems perspective analysing the instabilities and bifurcations the base state undergoes when increasing the reynolds number it has been found that the base state becomes unstable for small reynolds numbers and a transitional region including complex dynamics takes place at intermediate reynolds numbers depending on the wall acceleration values the base flow instabilities are constitutive parts of different codimension two bifurcations that control the dynamics in parameter space for large reynolds numbers the restriction to self similarity results in simple flows with no realistic behaviour but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic navier stokes solutions

the oscillatory channel flow with large wall injection in the presence of smallamplitude pressure oscillations the linearized navierstokes equations are solved to obtain an accurate description of the timedependent field in a channel having a rectangu

similarity solutions of steady flows in a channel with accelerating walls abstract we study the boundary layer equation f r ff f 2 0 subject to the conditions f 0 f 0 f 1 f 1 1 0 and f 1 f 1 f 1 f 1 1 0 respectively which arises from the study of two dimensional steady flows in a channel with two equally accelerating walls or one accelerating wall the preliminary classification of possible solutions was introduced by cox 1 in this paper we are able to verify the existence of families of solutions as indicated from the numerical computations moreover multiple solutions for some positive r and local uniqueness are also obtained

symmetry of minimizers for some nonlocal variational problems abstract we present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems this approach which generalizes the reflection method is based on the existence of some integral identities we study the identities that lead to symmetry results the functionals that can be considered and the function spaces that can be used then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of laplacian for the generalized choquard functional and for the standing waves of the davey stewartson equation

a technique for the numerical verification of asymptotic expansions a technique is presented that uses numerical solutions to verify the order of the accuracy of an asymptotic expansion for several types of problems this technique may be introduced in any beginning course on asymptotic or perturbation methods but is equally suited to verifying advanced asymptotic results

higher mean flow approximation for solid rocket motors with radially regressing walls the bulk gas motion in a circular port rocket motor is described using a rotational incompressible and viscous flow model that incorporates the effect of wall regression the mathematical idealization developed is also applicable to semi open porous tubes with expanding walls based on mass conservation a linear variation in the mean axial velocity is ascertained this relationship suggests investigating a spatial transformation of the proudman johnson form with the use of similar arguments a temporal transformation is also introduced when these transformations are applied in both space and time the navier stokes equations are reduced to a single nonlinear fourth order differential equation following this exact navier stokes reduction the resulting problem is solved using variation of parameters and small parameter perturbations the asymptotic solutions for the velocity pressure vorticity and shear are obtained as function of the injection reynolds number re and the dimensionless regression ratio a by way of verification it is shown that as re 0 yuan and finkelstein s solutions can be restored from ours similarly as re 0 culick s inviscid profile is recovered it is demonstrated that for a range of small re inviscid solutions are practical however for fast burning propellants under development the inviscid assumption deteriorates because it is applicable over a broader range of operating parameters the current analysis leads to a closed form mean flow solution that can be used instead of the inviscid profile to 1 prescribe an adjusted aeroacoustic field 2 describe the so called acoustic boundary layer 3 evaluate the viscous and rotational contributions to the acoustic stability growth rate factor 4 track the evolution of hydrodynamic instability and 5 accurately simulate the internal gasdynamics in rapidly regressing motors and cold flow experiments with medium to high levels of injection

steady flows in the slender noncircular combustion chambers of solid propellant rockets an analysis of the steady flows with axial vortices in slender cylindrical nonaxisymmetric cavities generated by injection from their porous walls is presented this problem is encountered in the description of the flow in slender combustion chambers of solid propellant rocket motors associated with the gasification of the solid propellant surrounding the combustion chamber nonreacting flow can be described in terms of self similar solutions of the navier stokes equations and the solution calculated numerically for noncircular grain configurations shows strong axial vortices with a viscous vortex core that has been analyzed asymptotically for large reynolds numbers it was found that the important property of the flow namely c 1 2 s where c is a constant determining the axial pressure gradient s and n are the cross section area and its perimeter respectively becomes unexpectedly close to 2 at large reynolds numbers independently on the geometry of the cross section of the cavity one can suppose that c 1 2 s 2 for any cavities in the invicsid limit a conjecture that also obtained a support from numerical calculations of flows in rectangular cavities generated by injection from their porous walls

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

traveling waves for nonlinear schrodinger equations with nonzero conditions at infinity for a large class of nonlinear schrodinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension n 3 our results are valid as well for the gross pitaevskii equation and for nls with cubic quintic nonlinearity

turbularization of an acoustic boundary layer on a transpiring surface turbularization of an acoustic boundary layer stokes layer on impermeable and permeable surfaces is analytically considered the theoretical approach utilizes a second order closure model of turbulence both an approximate closed form solution and a more comprehensive finite difference solution of the time dependent parabolic one dimensional governing equations are obtained for simple acoustic boundary layers on impermeable surfaces both the approximate solution and the numerical results for the critical acoustic mach number required for turbulent transition are qualitatively confirmed by experiment the calculations for acoustic boundary layers with transpiration injection indicate a substantial reduction of the acoustic mach number required for transition up to a limiting injection velocity that is frequency dependent the results may provide a mechanism for flow related combustion instability in practical systems particularly solid propellant rockets since turbularization of the near surface combustion zone could result at relatively low acoustic mach numbers

the oscillatory pipe flow with arbitrary wall injection the linearized navierstokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube asymptotic solutions of these equations have been found and here

multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall abstract in this paper the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method however this solution neglects exponentially small terms in the matching process to take into account these exponentially small terms a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically the series involving the exponentially small terms and expansion ratio predicts dual solutions furthermore the result indicates that the expansion ratio has much important influence on the solutions when the expansion ratio is zero it is a special case that terrill has discussed

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

vortical and acoustical mode coupling inside a porous tube with uniform wall suction this paper considers the oscillatory motion of gases inside a long porous tube of the closed open type in particular the focus is placed on describing an analytical solution for the internal acoustico vortical coupling that arises in the presence of appreciable wall suction this unsteady field is driven by longitudinal oscillatory waves that are triggered by small unavoidable fluctuations in the wall suction speed under the assumption of small amplitude oscillations the time dependent governing equations are linearized through a regular perturbation of the dependent variables further application of the helmholtz vector decomposition theorem enables us to discriminate between acoustical and vortical equations after solving the wave equation for the acoustical contribution the boundary driven vortical field is considered the method of matched asymptotic expansions is then used to obtain a closed form solution for the unsteady momentum equation developing from flow decomposition an exact series expansion is also derived and shown to coincide with the numerical solution for the problem the numerically verified end results suggest that the asymptotic scheme is capable of providing a sufficiently accurate solution this is due to the error associated with the matched asymptotic expansion being smaller than the error introduced in the navier stokes linearization a basis for comparison is established by examining the evolution of the oscillatory field in both space and time the corresponding boundary layer behavior is also characterized over a range of oscillation frequencies and wall suction velocities in general the current solution is found to exhibit features that are consistent with the laminar theory of periodic flows by comparison to the sexl profile in nonporous tubes the critically damped solution obtained here exhibits a slightly smaller overshoot and depth of penetration these features may be attributed to the suction effect that tends to attract the shear layers closer the wall

rarefaction pulses for the nonlinear schrodinger equation in the transonic limit we investigate the properties of finite energy travelling waves to the nonlinear schrodinger equation with nonzero conditions at infinity for a wide class of nonlinearities in space dimension two and three we prove that travelling waves converge in the transonic limit up to rescaling to ground states of the kadomtsev petviashvili equation our results generalize an earlier result of f bethuel p gravejat and j c saut for the two dimensional gross pitaevskii equation and provide a rigorous proof to a conjecture by c jones and p h roberts about the existence of an upper branch of travelling waves in dimension three