the three dimensional flow past a stretching sheet and the homotopy perturbation method an approximate analytical solution is obtained of the steady laminar three dimensional flow for an incompressible viscous fluid past a stretching sheet using the homotopy perturbation method hpm proposed by he the flow is governed by a boundary value problem bvp consisting of a pair of non linear differential equations the solution is simple yet highly accurate and compares favorably with the exact solutions obtained early in the literature the methodology presented in the paper is useful for solving the bvps consisting of more than one differential equation

anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field inertial range asymptotic behavior of a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian not correlated in time with the pair correlation function of the form t t k d 1 where k k and k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda commun math phys 131 381 1990 the stochastic advection diffusion equation for the transverse divergence free vector field includes as special cases the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized navier stokes equation in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the dependence on the integral turbulence scale l has a logarithmic behavior instead of powerlike corrections to ordinary scaling determined by naive canonical dimensions the anomalies manifest themselves as polynomials of logarithms of l the key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized the detailed proof of this fact is given for the correlation functions of arbitrary order

the effect of scaling physiological cross sectional area on musculoskeletal model predictions abstract personalisation of model parameters is likely to improve biomechanical model predictions and could allow models to be used for subject or patient specific applications this study evaluates the effect of personalising physiological cross sectional areas pcsa in a large scale musculoskeletal model of the upper extremity muscle volumes obtained from mri were used to scale pcsas of five subjects for whom the maximum forces they could exert in six different directions on a handle held by the hand were also recorded the effect of pcsa scaling was evaluated by calculating the lowest maximum muscle stress max a constant for human skeletal muscle required by the model to reproduce these forces when the original cadaver based pcsa values were used strongly different between subject max values were found max 106 1 39 9 n cm 2 a relatively simple uniform scaling routine reduced this variation substantially max 69 4 9 4 n cm 2 and led to similar results to when a more detailed muscle specific scaling routine was used max 71 2 10 8 n cm 2 using subject specific pcsa values to simulate an shoulder abduction task changed muscle force predictions for the subscapularis and the pectoralis major on average by 33 and 21 respectively but was

self similar decay in the kraichnan model of a passive scalar we study the two point correlation function of a freely decaying scalar in kraichnan s model of advection by a gaussian random velocity field that is stationary and white noise in time but fractional brownian in space with roughness exponent 0 2 appropriate to the inertial convective range of the scalar we find all self similar solutions by transforming the scaling equation to kummer s equation it is shown that only those scaling solutions with scalar energy decay exponent a d 1 are statistically realizable where d is space dimension and 2 an infinite sequence of invariants jp p 0 1 2 is pointed out where j0 is corrsin s integral invariant but the higher invariants appear to be new we show that at least one of the invariants j0 or j1 must be nonzero possibly infinite for realizable initial data initial datum with a finite nonzero invariant the first being jp converges at long times to a scaling solution p with a d p p 0 1 the latter belongs to an exceptional series of self similar solutions with stretched exponential decay in space however the domain of attraction includes many initial data with power law decay when the initial datum has all invariants zero or infinite and also it exhibits power law decay then the solution converges at long times to a nonexceptional scaling solution with the same power law decay these results support a picture of a two scale decay with breakdown of self similarity for a range of exponents d a d 2 analogous to what has recently been found in the decay of burgers turbulence

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

harmonic wave generation in non linear thermoelasticity by variational iteration method and adomian s method this paper applies the variational iteration method and adomian s decomposition method to solve numerically the harmonic wave generation in a nonlinear one dimensional elastic half space model subjected initially to a prescribed harmonic displacement the results show that the variational iteration method is much easier more convenient and more stable and efficient than adomian decomposition method

scalings of scalar structure functions in a velocity field with coherent vortical structures in planar turbulence modeled as an isotropic and homogeneous collection of two dimensional noninteracting compact vortices the structure functions s p r of a statistically stationary passive scalar field have the following scaling behavior in the limit where the peclet number pe s n r const in r l pe 1 3 for l pe 1 3 r l s p r r l pe 1 3 6 1 d for l pe 1 2 r l pe 1 3 where l is a large scale and d is the fractal codimension of the spiral scalar structures generated by the vortices d 2 3 note that l pe 1 2 is the scalar taylor microscale that stems naturally from our analytical treatment of the advection diffusion equation the essential ingredients of our theory are the locality of interscale transfer and lundgren s time average assumption a phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields k ln k 1 generalized power spectra for the advected scalar fields

approximate analytical solution for the zakharov kuznetsov equations with fully nonlinear dispersion in this paper variational iteration method vim is used to obtain numerical and analytical solutions for the zakharov kuznetsov equations with fully nonlinear dispersion comparisons with exact solution show that the vim is a powerful method for the solution of nonlinear equations

heat equilibrium distribution in a turbulent flow we consider a shear flow of a scale invariant gaussian random velocity field that does not depend on the coordinates in the direction of the flow we investigate a heat advection coming from a gaussian random homogeneous source we discuss a relaxation at large time of a temperature distribution determined by the forced advection diffusion equation we represent the temperature correlation functions by means of the feynman kac formula jensen inequalities are applied for lower and upper bounds on the correlation functions we show that at finite time there is no velocity dependence of long range temperature correlations low momentum asymptotics in the direction of the flow but the equilibrium heat distribution has large distance correlations low momentum behavior with an index depending on the scaling index of the random flow and of the index of the random forcing if the velocity has correlations growing with the distance a turbulent flow then the large distance correlations depend in a crucial wa

intuitive approach to the approximate analytical solution for the blasius problem for the blasius problem we propose an approximate analytical solution in the form of a logarithm of the hyperbolic cosine function which satisfies the given boundary conditions and some known properties of the exact solution furthermore adding some hyperbolic tangent functions to this solution we obtain much more accurate approximate solution with the relative error less than 0 16 over the whole region the superiority of the proposed solutions is shown by comparison with the existing approximate analytical solution

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

axisymmetric flow due to a stretching sheet with partial slip the steady laminar axisymmetric flow of a newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet has been investigated the flow is governed by a third order non linear boundary value problem whose exact numerical solution has been obtained non iteratively in terms of the non dimensional slip parameter l a perturbation solution valid for small l and an asymptotic solution valid for large l have been derived finally a solution based upon he s homotopy perturbation method has been developed the latter being analytical is elegant but is sufficiently accurate for all values of l

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

a numerical method for solving nth order boundary value problems in this paper we consider the solution of an nth order boundary value problem we solve this problem by changing the problem to a system of two integral differential equations and using the variational iteration method by giving three examples and comparing with the other methods the efficiency of the method will be shown

particle dispersion in a multidimensional random flow with arbitrary temporal correlations abstract we study the statistics of relative distances r t between fluid particles in a spatially smooth random flow with arbitrary temporal correlations using the space dimensionality d as a large parameter we develop an effective description of lagrangian dispersion we describe the exponential growth of relative distances r 2 t exp 2 t at different values of the ratio between the correlation and turnover times we find the stretching correlation time which determines the dependence of r 1 r 2 on the difference t1 t2 the calculation of the next cumulant of r2 shows that statistics of r2 is nearly gaussian at small times as long as d 1 and becomes log normal at large times when large d approach fails for high order moments the crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t we establish the dispersion of the ln r2 in the log normal statistics

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

effects of hip center location on the moment generating capacity of the muscles we have developed a three dimensional biomechanical model of the human lower extremity to study how the location of the hip center affects the moment generating capacity of four muscle groups the hip abductors adductors flexors and extensors the model computes the maximum isometric force and the resulting joint moments that each of 25 muscle tendon complexes develops at any body position abduction adduction flexion and extension moments calculated with the model correspond closely with isometric joint moments measured during maximum voluntary contractions we used the model to determine 1 the hip center locations that maximize and minimize the moment generating capacity of each muscle group and 2 the effects of superior inferior anterior posterior and medial lateral displacement of the hip center on the moment arms maximum isometric muscle forces and maximum isometric moments generated by each muscle group we found that superior inferior displacement of the hip center has the greatest effect on the force and moment generating capacity of the muscles a 2 cm superior displacement decreases abduction force 44 moment arm 12 and moment 49 while a 2 cm inferior displacement increases abduction force 20 moment arm 7 and moment 26 similarly a 2 cm superior displacement decreases flexion force 27 moment arm 6 and moment 22 while inferior displacement increases all three variables anterior posterior displacement alters the moment generating capacity of the flexors and extensors considerably primarily due to moment arm changes medial lateral displacement has a large effect on the moment generating capacity of the adductors only a 2 cm medial displacement decreases adduction moment arm 20 force 26 and moment 40 these results demonstrate that the force and moment generating capacities of the muscles are sensitive to the location of the hip center

exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method the purpose of this study is to introduce a modification of the homotopy perturbation method using laplace transform and pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations two test examples are given the coupled nonlinear system of burger equations and the coupled nonlinear system in one dimensional thermoelasticity the results obtained ensure that this modification is capable of solving a large number of nonlinear differential equations that have wide application in physics and engineering

radiation effects for the blasius and sakiadis flows with a convective surface boundary condition abstract this study aims to analyze the effects of thermal radiation on the laminar boundary layer about a flat plate in a uniform stream of fluid blasius flow and about a moving plate in a quiescent ambient fluid sakiadis flow both under a convective surface boundary condition the resulting similarity energy equation is solved numerically and the variations of dimensionless surface temperature and fluid solid interface characteristics for different values of prandtl number pr radiation parameter nr and parameter a which characterizes our convection processes are graphed and tabulated quite different and interesting behaviours were encountered for a blasius flow compared with a sakiadis flow a comparison with previously published results on special case of the problem shows excellent agreement

exact results for conditional means of a passive scalar in certain statistically homogeneous flows for a passive scalar t r t randomly advected by a statistically homogeneous flow the probability density function pdf of its fluctuation can in general be expressed in terms of two conditional means 2 t t and t 2 t we find that in some special cases either one of the two conditional means can be obtained explicitly from the equation of motion in the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping 2 t t t 2 t 2 t the linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear on the other hand when the scalar is supported by a homogeneous white in time external source t 2 t t 2

two different rapid decorrelation in time limits for turbulent diffusion a turbulent diffusion model in which the velocity field is gaussian and rapidly decorrelating in time grdt has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations the utility of the grdt model is the fact that correlation functions of the passive scalar field solve closed partial differential equations the usual moment closure obstacle is averted we study here the sense in which the grdt model describes turbulent diffusion by a general non gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero when the velocity field is rescaled in a particular manner in this rapid decorrelation limit then a limit theorem of khas minskii indeed shows that the passive scalar statistics are described asymptotically by the grdt model for a broad class of velocity field models we provide however an explicit example of a poisson blob model velocity field which has two different well defined rapid decorrelation in time limits in one the passive scalar correlation functions converge to those of the grdt model and in the other they converge to a distinct nontrivial limit in which the correlation functions do not solve closed pde s we provide both mathematical and heuristic explanations for the differences between these two limits the conclusion is that the grdt model provides a universal description of the rapid decorrelation in time limit of general non gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

subject specific knee joint geometry improves predictions of medial tibiofemoral contact forces estimating tibiofemoral joint contact forces is important for understanding the initiation and progression of knee osteoarthritis however tibiofemoral contact force predictions are influenced by many factors including muscle forces and anatomical representations of the knee joint this study aimed to investigate the influence of subject specific geometry and knee joint kinematics on the prediction of tibiofemoral contact forces using a calibrated emg driven neuromusculoskeletal model of the knee one participant fitted with an instrumented total knee replacement walked at a self selected speed while medial and lateral tibiofemoral contact forces ground reaction forces whole body kinematics and lower limb muscle activity were simultaneously measured the combination of generic and subject specific knee joint geometry and kinematics resulted in four different opensim models used to estimate muscle tendon lengths and moment arms the subject specific geometric model was created from ct scans and the subject specific knee joint kinematics representing the translation of the tibia relative to the femur was obtained from fluoroscopy the emg driven model was calibrated using one walking trial but with three different cost functions that tracked the knee flexion extension moments with and without constraint over the estimated joint contact forces the calibrated models then predicted the medial and lateral tibiofemoral contact forces for five other different walking trials the use of subject specific models with minimization of the peak tibiofemoral contact forces improved the accuracy of medial contact forces by 47 and lateral contact forces by 7 respectively compared with the use of generic musculoskeletal model

anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow a model of the passive vector field advected by the uncorrelated in time gaussian velocity with power like covariance is studied by means of the renormalization group and the operator product expansion the structure functions of the admixture demonstrate essential power like dependence on the external scale in the inertial range the case of an anomalous scaling the method of finding independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions the constructed operator bases which include the powers of the dissipation operator and the enstrophy operator provide the possibility of calculating the exponents of the anomalous scaling

statistics of fourier modes in a turbulent flow fourier series are often used to discuss the properties of a homogeneous turbulent field we investigate the statistics of fourier modes of the turbulent velocity field and of a passive scalar the statistics of individual fourier modes is known to be gaussian when the size of the system l is much larger that the integral correlation size l 0 the case where the integral size is of the order of the system size l approximately l 0 is studied by direct numerical simulations in the range 20 or approximately rlambda or approximately 80 at a given rlambda we find that the probabilities of large fluctuations become larger when the wave number increases in qualitative agreement with the notion of intermittency as the reynolds number increases however the probability density functions become closer to gaussian in sharp contrast with the behavior of velocity increments we also show that in a simple model of cascade the fourier series decomposition is not appropriate to capture intermittency effects last we discuss other issues related to our results

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

finite time lyapunov exponents for products of random transformations i show how continuous products of random transformations constrained by a generic group structure can be studied by using iwasawa s decomposition into angular diagonal and shear degrees of freedom in the case of a gaussian process a set of variables adapted to the iwasawa decomposition and still having a gaussian distribution is introduced and used to compute the statistics of the finite time lyapunov spectrum of the process the variables also allow to show the exponential freezing of the shear degrees of freedom which contain information about the lyapunov eigenvectors

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

statistics of a passive scalar advected by a large scale two dimensional velocity field analytic solution steady statistics of a passive scalar advected by a random two dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale this corresponds to the so called batchelor regime where the velocity is replaced by its large scale gradient the probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate the description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant we have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics the proof is valid for any finite correlation time of the velocity field whatever the statistics of the velocity field the statistics of the passive scalar averaged over time locally in space is shown to approach gaussian statistics with increase in the p eclet number pe the pumping to diffusion scale ratio the first nsimultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field the mean stretching rate which can be calculated analytically in limiting cases non gaussian tails of the probability distributions at finite pe are found to be exponential

a simple perturbation approach to blasius equation in this paper we couple the iteration method with the perturbation method to solve the well known blasius equation the obtained approximate analytic solutions are valid for the whole solution domain comparison with howarth s numerical solution reveals that the proposed method is of high accuracy the first iteration step leads to 6 8 accuracy and the second iteration step yields the 0 73 accuracy of initial slop

rational scaled generalized laguerre function collocation method for solving the blasius equation in this paper we propose a collocation method for solving the blasius equation the blasius equation is a third order nonlinear ordinary differential equation this approach is based on a rational scaled generalized laguerre function collocation method we also present the comparison of this work with some well known results and show that the present solution is accurate

the blasius function computations before computers the value of tricks undergraduate projects and open research problems the blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick semi infinite flat plate the blasius function is the solution to 2 f xxx f f xx 0 on x in 0 infty subject to f 0 f x 0 0 f x infty 1 we use this famous problem to illustrate several themes first although the flow solves a nonlinear partial differential equation pde toepfer successfully computed highly accurate numerical solutions in 1912 his secret was to combine a runge kutta method for integrating an ordinary differential equation ode initial value problem with some symmetry principles and similarity reductions which collapse the pde system to the ode shown above this shows that pde numerical studies were possible even in the precomputer age the truth both a hundred years ago and now is that mathematical theorems and insights are an arithmurgist s best friend and they can vastly reduce the computational burden second we show that special tricks applicable only to a given problem can be as useful as the broad general methods that are the fabric of most applied mathematics courses the importance of particularity in spite of these triumphs many properties of the blasius function f x are unknown we give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician

new perturbation iteration solutions for bratu type equations perturbation iteration theory is systematically generated for both linear and nonlinear second order differential equations and applied to bratu type equations different perturbation iteration algorithms depending upon the number of taylor expansion terms are proposed using the iteration formulas derived using different perturbation iteration algorithms new solutions of bratu type equations are obtained solutions constructed using different perturbation iteration algorithms are contrasted with each other as well as with numerical solutions it is found that algorithms with more taylor series expansion terms yield more accurate results

level of subject specific detail in musculoskeletal models affects hip moment arm length calculation during gait in pediatric subjects with increased femoral anteversion biomechanical parameters of gait such as muscle s moment arm length mal and muscle tendon length are known to be sensitive to anatomical variability nevertheless most studies rely on rescaled generic models rgmo constructed from averaged data of cadaveric measurements in a healthy adult population as an alternative deformable generic models dgmo have been proposed these models integrate a higher level of subject specific detail by applying characteristic deformations to the musculoskeletal geometry in contrast musculoskeletal models based on magnetic resonance mr images mrmo reflect the involved subject s characteristics in every level of the model this study investigated the effect of the varying levels of subject specific detail in these three model types on the calculated hip mal during gait in a pediatric population of seven cerebral palsy subjects presenting aberrant femoral geometry our results show large percentage differences in calculated mal between rgmo and mrmo furthermore the use of dgmo did not uniformly reduce inter model differences in calculated mal the magnitude of these percentage differences stresses the need to take these effects into account when selecting the level of subject specific detail one wants to integrate in musculoskeletal furthermore the variability of these differences between subjects and between muscles makes it very difficult to a priori estimate their importance for a biomechanical analysis of a certain muscle in a given subject

singularly perturbed third order boundary value problem for nonlinear systems abstract this paper deals with the existence of solutions to a singularly perturbed boundary value problem for third order nonlinear differential systems the authors construct an appropriate generalized lower solution upper solution pair a concept that is defined in this paper and employ the method of descent nagumo conditions algebraic boundary layer functions and volterra type integral operator to guarantee the existence of solutions of the problem the uniformly valid asymptotic estimate of the solutions is given as well the differential systems has nonlinear dependence on all over order derivatives of the unknown

application of fixed point method to obtain semi analytical solution to blasius flow and its variation the classical blasius flow and its variation sakiadis flow are investigated by the fixed point method fpm and the semi analytical solutions are obtained different from early perturbation series solutions the present semi analytical solutions are valid in the whole region without using matching procedure moreover these solutions can correctly provide the asymptotic property of the governing equations the comparison of the present results with numerical results is carried out which reveals that fpm is a convenient and effective tool for nonlinear problems

numerical transformation methods blasius problem and its variants blasius problem is the simplest nonlinear boundary layer problem we hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems with this motivation we review the so called topfer transformation which allows us to find a non iterative numerical solution of the blasius problem by solving a related initial value problem and applying a scaling transformation the applicability of a non iterative transformation method to the blasius problem is a consequence of its partial invariance with respect to a scaling group several problems in boundary layer theory lack this kind of invariance and cannot be solved by non iterative transformation methods to overcome this drawback we can modify the problem under study by introducing a numerical parameter and require the invariance of the modified problem with respect to an extended scaling group involving this parameter then we apply initial value methods to the most recent developments involving variants and extensions of the blasius problem

the optimal homotopy asymptotic method for solving blasius equation starting from the reality that many known methods fail in the attempt to obtain analytic solutions of blasius type equations in this work a new procedure namely optimal homotopy asymptotic method oham is proposed to obtain an explicit analytical solution of the blasius problem comparison with howarth s numerical solution as well as the obtained residual reveals that the proposed method is highly accurate this proves the validity and great potential of the proposed procedure oham as a new kind of powerful analytical tool for nonlinear problems

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

stirring up trouble multi scale mixing measures for steady scalar sources abstract the mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring relative to the variance in the absence of stirring these variances can be weighted at various spatial scales leading to a family of multi scale mixing measures and efficiencies we derive a priori estimates on these efficiencies from the advection diffusion partial differential equation focusing on a broad class of statistically homogeneous and isotropic incompressible flows the analysis produces bounds on the mixing efficiencies in terms of the peclet number a measure of the strength of the stirring relative to molecular diffusion we show by example that the estimates are sharp for particular source sink and flow combinations in general the high peclet number behavior of the bounds scaling exponents as well as prefactors depends on the structure and smoothness properties of and length scales in the scalar source and sink distribution the fundamental model of the stirring of a monochromatic source sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis the large scale mixing efficiency follows the upper bound scaling within a logarithm at high peclet number but the intermediate and small scale efficiencies are qualitatively less than optimal the peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi static model

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

towards a geometrical classification of statistical conservation laws in turbulent advection abstract the paper revisits the compressible kraichnan model of turbulent advection in order to derive explicit quantitative relations between scaling exponents and lagrangian particle configuration geometry

he s parameter expanding methods for strongly nonlinear oscillators ji huan he s work on asymptotic techniques is briefly reviewed and his parameter expanding methods including modified lindstedt poincare method and bookkeeping parameter method are discussed in detail some remarkable virtues of the methods are exploited and their applications are illustrated

scaling properties of passive scalars in one dimension this paper is a set of notes about work in progress since the work is directed toward scaling it seems quite appropriate to report this in a context devoted to ben widom s scientific contributions model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived all these models follow the approach set up by kraichnan in which the driving velocities have gaussian correlations with scaling properties in space but zero range correlations in time the simplest and most natural versions of the model fail to satisfy an incompressibility condition in another version the model does have an incompressible flow to permit the incompressibility there are two pipes each of which is described by temperature and velocity fields which are functions of a single coordinate x and time flow from one pipe to the other transfers both mass and heat this model has two conserved quantities in the limit of zero dissipation in still another version fluid elements are interchanged in position so that there are an infinite number of conservation laws in the no dissipation limit scaling properties of the resulting two point functions are derived

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field