Selection of kernel functions used in a new constitutive equation for viscoelastic fluids

1986 ◽  
Vol 25 (3) ◽  
pp. 214-224
Author(s):  
B. Tremblay ◽  
J. M. Piau ◽  
P. J. Carreau
Keyword(s):  
Constitutive Equation ◽  
Viscoelastic Fluids ◽  
Kernel Functions ◽  
Selection Of

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

scholarly journals Acoustic Wave Propagation in a Trifurcated Lined Waveguide

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ayub ◽  
M. H. Tiwana ◽  
A. B. Mann ◽  
H. Zaman
Keyword(s):  
Wave Propagation ◽  
Acoustic Wave ◽  
Closed Form ◽  
Integral Transform ◽  
Closed Form Solution ◽  
Form Solution ◽  
Kernel Functions ◽  
Acoustic Wave Propagation ◽  
Proper Selection ◽  
Selection Of

The diffraction of sound from a semi-infinite soft duct is investigated. The soft duct is symmetrically located inside an acoustically lined but infinite duct. A closed-form solution is obtained using integral transform and Jones' method based on Wiener-Hopf technique. The graphical results are presented, which show how effectively the unwanted noise can be reduced by proper selection of different parameters. The kernel functions are factorized with different approaches. The results may be used to design acoustic barriers and noise reduction devices.

Continuum Theories for the Viscoelasticity of Flexible Homogeneous Polymeric Liquids

2007 ◽  
Author(s):  
Chang Dae Han
Keyword(s):  
Constitutive Equation ◽  
Flow Field ◽  
Polymer Blend ◽  
Constitutive Equations ◽  
Viscoelastic Fluids ◽  
Phenomenological Approach ◽  
The State ◽  
Momentum Balance ◽  
Molecular Approach ◽  
Balance Equations

There are two primary reasons for seeking a precise mathematical description of the constitutive equations for viscoelastic fluids, which relate the state of stress to the state of deformation or deformation history. The first reason is that the constitutive equations are needed to predict the rheological behavior of viscoelastic fluids for a given flow field. The second reason is that constitutive equations are needed to solve the equations of motion (momentum balance equations), energy balance equations, and/or mass balance equations in order to describe the velocity, stress, temperature, and/or concentration profiles in a given flow field that is often encountered in polymer processing operations. There are two approaches to developing constitutive equations for viscoelastic fluids: one is a continuum (phenomenological) approach and the other is a molecular approach. Depending upon the chemical structure of a polymer (e.g., flexible homopolymer, rigid rodlike polymer, microphase-separated block copolymer, segmented multicomponent polymers, highly filled polymer, miscible polymer blend, immiscible polymer blend), one may take a different approach to the formulation of the constitutive equation. In this chapter we present some representative constitutive equations for flexible, homogeneous viscoelastic liquids that have been formulated on the basis of the phenomenological approach. In the next chapter we present the molecular approach to the formulation of constitutive equations for flexible, homogeneous viscoelastic fluids. In the formulation of the constitutive equations using a phenomenological approach, emphasis is placed on the relationship between the components of stress and the components of the rate of deformation (or strain) or deformation (or strain) history, such that the responses of a fluid to a specified flow field or stress can adequately be described. The parameters appearing in a constitutive equation are supposed to represent the characteristics of the fluid under consideration. More often than not, the parameters appearing in a phenomenological constitutive equation are determined by curve fitting to experimental results. Thus phenomenological constitutive equations shed little light on the effect of the molecular parameters of the fluid under investigation to the rheological responses of the fluid.

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