scholarly journals Approximate Analytical Solutions Using Hyperbolic Functions for the Generalized Blasius Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Beong In Yun
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Hyperbolic Functions ◽  
Blasius Problem ◽  
Approximate Analytical Solutions ◽  
The Given

We propose simple forms of approximate analytical solutions for the generalized Blasius problem based on the given boundary conditions and some known properties of the solution. The efficiency of the proposed solutions is shown for various cases. As a result, one can see that the solutions are uniformly accurate over the whole region.

scholarly journals The scattering of sound waves by a vortex: numerical simulations and analytical solutions

1994 ◽  
Vol 260 ◽  
pp. 271-298 ◽  
Author(s):  
Tim Colonius ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Acoustic Waves ◽  
Stokes Equations ◽  
Spatial Differentiation ◽  
Navier Stokes ◽  
Sound Waves ◽  
Navier Stokes Equations ◽  
Refraction Effect ◽  
Approximate Analytical Solutions

The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier–-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments.

Approximate Analytical Solutions for CO2 Injectivity Into Saline Formations

2013 ◽  
Vol 16 (02) ◽  
pp. 123-133 ◽  
Author(s):  
Ehsan Azizi ◽  
Yildiray Cinar
Keyword(s):  
Boundary Conditions ◽  
Relative Permeability ◽  
Analytical Solutions ◽  
Outer Boundary ◽  
Analytical Models ◽  
Co2 Injection ◽  
Approximate Analytical Solutions ◽  
New Models ◽  
Saline Formation ◽  
Carbon Dioxide Co2

Summary This paper presents new analytical models to estimate the bottomhole pressure (BHP) of a vertical carbon dioxide (CO2) injection well in a radial, homogeneous, horizontal saline formation. The new models include the effects of multiphase flow, CO2 dissolution in formation brine, and near-well drying out on the BHP. CO2 is injected into the formation at a constant rate. The analytical solutions are presented for three types of formation outer boundary conditions: closed boundary, constant-pressure boundary, and infinite-acting formation. The sensitivity of BHP computations to gas relative permeability, retardation factors, and CO2 compressibility is examined. The predictive capability of the analytical models is tested by use of numerical reservoir simulations. The results show a good agreement between the analytical and numerical computations for all three boundary conditions. Variations in gas compressibility, retardation factors, and gas relative permeability in the drying-out zone are found to have moderate effects on BHP computations. It is demonstrated for several hypothetical but realistic cases that the new models can estimate CO2 injectivity reliably.

scholarly journals Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Conditions ◽  
Perturbation Method ◽  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Initial Boundary ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method.

scholarly journals A Homotopy-Analysis Approach for Nonlinear Wave-Like Equations with Variable Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Equations ◽  
Nonlinear Wave ◽  
Analytical Solutions ◽  
Wave Operator ◽  
Series Solution ◽  
Variable Coefficients ◽  
Analysis Approach ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis

We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefficients. We use a wave operator, which provides a convenient way of controlling all initial and boundary conditions. The proposed choice of the auxiliary operator helps to find the approximate series solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method.

scholarly journals Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

Meccanica ◽  
2021 ◽  
Author(s):  
A. Di Matteo ◽  
M. Pavone ◽  
A. Pirrotta
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Line Element ◽  
Scale Effects ◽  
Plate Boundary ◽  
Nonlocal Elasticity Theory ◽  
Algebraic Equations ◽  
Various Boundary Conditions ◽  
Approximate Analytical Solutions ◽  
Pertinent Data

AbstractIn this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries. In addition, several applications are discussed to show the simplicity and applicability of the procedure, and comparison with pertinent data in the literature assesses the accuracy of the proposed approach.

General Method to Calculate the Permeability Weighting Function for Well Testing in Heterogeneous Media

1999 ◽  
Vol 2 (03) ◽  
pp. 281-287 ◽  
Author(s):  
R.K. Romeu ◽  
A.Q. Lara ◽  
Benoit Nœtinger ◽  
Ge´rard Renard
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Heterogeneous Media ◽  
Weighting Function ◽  
Approximate Solutions ◽  
Pressure Solution ◽  
Well Testing ◽  
First Order ◽  
Approximate Analytical Solutions ◽  
General Method

Summary The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The well pressure solution using this method is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As discussed in previous papers, deriving such permeability weighting functions appears to be a complicated calculation, available only for special cases. In this article we present simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc.), the associated permeability weighting function can be derived immediately. Among other examples, we calculate and discuss the well pressure solution for a horizontal well that is producing from a heterogeneous reservoir. Introduction The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media. Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems that involve some simple symmetry: layered reservoirs, single linear discontinuities, radial composite systems etc.1 Rosa and Horne2 computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al.3 proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods. This article is related to these first-order approximate solutions of well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous work in the domain. After that, we present our method to calculate the permeability weighting functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well. Permeability Weighting Function The perturbation method is a well known technique by which to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms. In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The k0 may vary in space, i.e., k0(x→D). What is important is that the background problem has a known exact analytical solution, pD0(x→D,tD). The full problem has the same boundary conditions of the background problem but the permeability k(x→D) differs from k0(x→D) in arbitrary regions of space. Strictly speaking, k(x→D)/k0(∙xD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts up to, say, 10 between these permeabilities, depending on the specific problem. The dimensionless well pressure of the full problem, pwD(tD), is approximated by the sum of two terms: p w D ( t D ) ≅ p w D 0 ( t D ) + p w D 1 ( t D ) , ( 1 ) where pwD0 is the solution of the background problem, which is known, and pwD1 corresponds to the effect of the variation of the permeability. This second term is computed by p w D 1 ( t D ) = ∫ − ∞ + ∞ Δ k D ( x → D ) W ( ∙ x D , t D ) d ∙ x D , ( 2 ) the terms of which will be explained. The dimensionless permeability variation ΔkD may be alternatively defined by Δ k D ( ∙ x D ) = l n ( k ( x → D ) / k 0 ( x → D ) ) , ( 3 a ) Δ k D ( ∙ x D ) = 1 − ( k 0 ( x → D ) / k ( x → D ) ) , ( 3 b ) Δ k D ( ∙ x D ) = ( k ( x → D ) / k 0 ( x → D ) ) − 1 , ( 3 c ) or other equivalent first-order approximations. These three expressions have the same first-order terms of their Taylor series, and produce very close results for k(x→D)/k0(x→D) near 1. However, these definitions are not equally robust for greater permeability contrasts.

scholarly journals Constructing Uniform Approximate Analytical Solutions for the Blasius Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Beong In Yun
Keyword(s):  
Weight Function ◽  
Analytical Solutions ◽  
Series Solution ◽  
Numerical Results ◽  
Infinite Interval ◽  
Constructive Method ◽  
Reference Solution ◽  
Blasius Problem ◽  
Approximate Analytical Solutions ◽  
Uniform Accuracy

We propose a simple constructive method which assures uniform accuracy of the approximate analytical solutions for the Blasius problem on the semi-infinite interval0,∞. The method is based on a weight function having an S-shape to reflect a series solution near the originx=0and a reference solution far from the origin. Numerical results show the efficiency of the proposed method.

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