Solutions of Higher Class and Their Computations for Polymer Flows Using Giesekus Constitutive Model

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Research Work ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Stick Slip ◽  
Flow Rates ◽  
Conservation Of Mass ◽  
Graded Meshes ◽  
Linear Algebraic Equations

Abstract This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two dimensional viscoelastic fluid flows in u, v, p, τijp, τijs with Giesekus constitutive model using p-version least squares finite element formulation (LSFEF). The main thrust of the research work presented in the paper is to employ ‘right classes of interpolations’ and the ‘best computational strategy’ 1) to obtain numerical solutions of governing differential equations (GDEs) for increasing Deborah numbers 2) to investigate the nature of the computed solutions with the aim of establishing limiting values of the flow parameters beyond which the solutions may be possible to compute, but may not be meaningful. The investigations presented in this paper reveal the following: a) The manner in which the stresses are non-dimensionalized significantly influences the performance of the iterative procedure of solving non-linear algebraic equations. b) Solutions of the class C00 are always the wrong class of solutions of GDEs in variables u, v, p, τijp and τijs and thus spurious. c) C11 class of solutions are the right class of solutions of the GDEs in variables u, v, p, τijp and τijs. d) In the flow domains, containing sharp gradients of the dependent variables, conservation of mass is difficult to achieve at lower p-levels (worse for coarse meshes). e) An augmented form of GDEs are proposed that always ensure conservation of mass at all p-levels regardless of the mesh and the nature of the solution gradients. f) Stick-slip problem is used as a model problem. Dimensions, fluid properties and flow rates used correspond to MIT experiments [20]. We demonstrate that converged solutions are possible to compute for all flow rates reported in ref [20] and that the detailed examination of the solution characteristics reveals them to be in agreement with all the physics of the flow, g) Numerical studies with graded meshes and high p-levels presented in this paper are aimed towards establishing and demonstrating detail behavior of local as well as global nature of the computed solutions, h) Various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior i) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently use definitions j) A significant aspect and strength of our work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used.

Numerical Computations of Higher Class Solutions for 2D Polymer Flows Using PTT Constitutive Model

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Research Work ◽  
Detailed Examination ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Stick Slip ◽  
Conservation Of Mass ◽  
Graded Meshes ◽  
Linear Algebraic Equations

Abstract This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two dimensional viscoelastic fluid flows in u, v, p, τijp, τijs with Phan-Thien-Tanner (PTT) constitutive model using p-version least squares finite element formulation (LSFEF). The main thrust of the research work presented in the paper is to employ ‘right classes of interpolations’ and the ‘best computational strategy’ 1) to obtain numerical solutions of governing differential equations (GDEs) for increasing Deborah numbers 2) investigate the nature of the computed solutions with the aim of establishing limiting values of the flow parameters beyond which the solutions may be possible to compute, but may not be meaningful. The investigations presented in this paper reveal the following: a) The manner in which the stresses are non-dimensionalized significantly influences the performance of the iterative procedure of solving non-linear algebraic equations. b) Solutions of the class C00 are always the wrong class of solutions of GDEs in variables u, v, p, τijp and τijs and thus spurious. c) C11 class of solutions are the right class of solutions of the GDEs in variables u, v, p, τijp and τijs. d) In the flow domains, containing sharp gradients of the dependent variables, conservation of mass is difficult to achieve at lower p-levels (worse for coarse meshes). e) An augmented form of GDEs are proposed that always ensure conservation of mass at all p-levels regardless of the mesh and the nature of the solution gradients. f) Stick-slip problem is used as a model problem. We demonstrate that converged solutions are possible to compute for all flow rates reported and that the detailed examination of the solution characteristics reveals them to be in agreement with all the physics of the flow, g) Numerical studies with graded meshes and high p-levels presented in this paper are aimed towards establishing and demonstrating detail behavior of local as well as global nature of the computed solutions, h) Various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior i) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently used definitions j) A significant aspect and strength of our work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used.

Solutions of Higher Class and Their Computations for Polymer Flows: Oldroyd-B Constitutive Model

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Constitutive Model ◽  
Research Work ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Parallel Plates ◽  
Computational Process ◽  
Stick Slip ◽  
Conservation Of Mass ◽  
Flow Physics ◽  
Graded Meshes

Abstract This paper presents formulations, computations and investigations of the solutions of class C00 and C11 for two dimensional viscoelastic fluid flows in u, v, p, τijp, τijs with Oldroyd-B constitutive model using p-version Least Squares Finite Element Formulation (LSFEF). The main thrust of the research work presented in the paper is to employ ‘right class of interpolations’ and the best computational strategy to establish: i) when does Oldroyd-B model begins to fail in simulating the correct physics of flow ii) when and why does the proposed computational process fail iii) is there a correlation between i) and ii). Fully developed flow between parallel plates and the stick-slip problems are used as model problems. The investigations presented in this paper reveal the following. a) The manner in which the stresses are non-dimensionalized significantly influences the performance of the iterative procedure of solving nonlinear algebraic equations. b) Solutions of the class C00 are always the wrong class of solutions of GDEs in variables u, v, p, τijp and τijs and thus spurious. c) C11 class of solutions are the right class of solutions of the GDEs in variables u, v, p, τijp and τijs. d) In the flow domains, containing sharp gradients of the dependent variables, conservation of mass is difficult to achieve at lower p-levels (worse for coarse meshes). e) An augmented form of GDEs are proposed that always ensure conservation of mass at all p-levels regardless of the mesh and the nature of the solution gradients. f) We demonstrate that Oldroyd-B model describes correct physics of dilute polymer solutions of constant viscosity only for a limited range of Deborah numbers. Beyond this range, the computed solutions are not in agreement with the flow physics (thus spurious) even though the proposed computational process works exceptionally well. g) Numerical studies with graded meshes and high p-levels presented in this paper are aimed towards establishing and demonstrating detail behavior of local as well as global nature of the computed 10 solutions, h) Various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior 1) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently used definitions j) A significant aspect and strength of our work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used.

Computations of Numerical Solutions of Higher Class in 2D Polymer Flows: Upper Convected Maxwell Model

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Constitutive Model ◽  
Polymer Solutions ◽  
Research Work ◽  
Deborah Number ◽  
Stick Slip ◽  
Conservation Of Mass ◽  
Flow Physics ◽  
Numerical Studies ◽  
Dilute Polymer Solutions ◽  
Constant Viscosity

Abstract This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two-dimensional viscoelastic fluid flows in primitive variables u, v, p, τxx, τxy, τyy with Upper Convected Maxwell (UCM) constitutive model using p-version least squares finite element formulation (LSFEF). The main emphasis of the investigations, undertaken in this research work, is to employ the right classes of interpolations and the best computational strategy to address, to illuminate on and perhaps, to answer and resolve whether the “continued obsession of developing newer and newer computational strategies to obtain solutions of Maxwell constitutive equations for ever increasing Deborah number (De) is as meritorious as currently believed. The stick-slip problem is used as a model problem in all investigations. Our investigations reveal the following: a) The manner in which the governing differential equations (GDEs) are non-dimensionalized influences the performance of the iteration procedure for solving nonlinear algebraic equations and thus, computational efficiency achieved. b) C00 class of solutions are always the wrong class of solutions and are spurious. c) In the flow domains, containing sharp gradients of dependant variables, conservation of mass is difficult to achieve specially at lower p-levels. d) C11 solutions are in conformity with the continuity considerations in GDEs. e) An augmented form of the GDEs and associated C00 and C11 formulations are proposed that always ensure conservation of mass regardless of mesh, p-levels and the nature of solution gradients. This approach yields the most desired classes of C11 solutions. f) We demonstrate that UCM constitutive model cannot describe the flow physics at any Deborah number for dilute polymer solutions of constant viscosity due to the fact that I) in UCM model, solvent stresses are a function of Deborah number. Thus, solvent stresses are wrong at any non-zero Deborah number. II) and solvent shear stresses produce elastic normal stresses. Numerical studies presented in the paper support theses findings. UCM model only generates correct flow physics for De = 0 (Newtonian flow). g) Numerical studies, with graded meshes and high p-levels presented in the paper, are aimed towards establishing and demonstrating detailed behavior of local as well as global nature of the computed solutions. Numerical studies are carefully designed and conducted to establish failure of UCM model in describing flow physics as well as failure of computational process. h) various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior. i) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently used definition. j) A very significant feature of our research work is that we utilize straight forward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used. Fully developed flow between parallel plates and stick-slip problems are used as model problems. It is concluded that UCM constitutive model always simulates incorrect behavior for dilute polymer solutions of constant viscosity regardless of Deborah number (except De = 0) and thus, development of newer computational schemes to achieve success at higher Deborah number is of no consequence.

Analysis of Dynamic Systems With Periodically Varying Parameters via Chebyshev Polynomials

1991 ◽  
Author(s):  
S. C. Sinha ◽  
Der-Ho Wu ◽  
Vikas Juneja ◽  
Paul Joseph
Keyword(s):  
Dynamic Systems ◽  
Chebyshev Polynomials ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Varying Parameters ◽  
Approximate Analytical Solutions ◽  
Linear Algebraic Equations ◽  
Mathieu’S Equation ◽  
Runge Kutta

Abstract In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three blade helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton and Gear algorithms. The results obtained in the both examples indicate that the suggested approach extremely accurate and is by far the most efficient one.

scholarly journals A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 904 ◽  
Author(s):  
Afshin Babaei ◽  
Hossein Jafari ◽  
S. Banihashemi
Keyword(s):  
Order Of Convergence ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Test Problems ◽  
Heat Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Stochastic Heat Equations ◽  
Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

scholarly journals Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Neda Khaksari ◽  
Mahmoud Paripour ◽  
Nasrin Karamikabir
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Unknown Function ◽  
Numerical Solutions ◽  
Functional Integral ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

A Novel Finite Difference Method for Flow Calculation on Colocated Grids

2008 ◽  
Author(s):  
Z. Z. Xia ◽  
P. Zhang ◽  
R. Z. Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Algebraic Equations ◽  
Duct Flow ◽  
Staggered Grids ◽  
Linear Algebraic Equations ◽  
Incomplete Lu Factorization

A new finite difference method, which removes the need for staggered grids in fluid dynamic computation, is presented. Pressure checker boarding is prevented through a dual-velocity scheme that incorporates the influence of pressure on velocity gradients. A supplementary velocity resulting from the discrete divergence of pressure gradient, together with the main velocity driven by the discretized pressure first-order gradient, is introduced for the discretization of continuity equation. The method in which linear algebraic equations are solved using incomplete LU factorization, removes the pressure-correction equation, and was applied to rectangle duct flow and natural convection in a cubic cavity. These numerical solutions are in excellent agreement with the analytical solutions and those of the algorithm on staggered grids. The new method is shown to be superior in convergence compared to the original one on staggered grids.

scholarly journals Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 463-472 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
First Time ◽  
Linear Fractional Differential Equations

AbstractIn this work, we propose a new operational method based on a Genocchi wavelet-like basis to obtain the numerical solutions of non-linear fractional order differential equations (NFDEs). To the best of our knowledge this is the first time a Genocchi wavelet-like basis is presented. The Genocchi wavelet-like operational matrix of a fractional derivative is derived through waveletpolynomial transformation. These operational matrices are used together with the collocation method to turn the NFDEs into a system of non-linear algebraic equations. Error estimates are shown and some illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed technique.

Numerical Solution of Non-Linear Algebraic Equations by Modified Genetic Algorithm

2008 ◽  
Vol 3 (1) ◽  
Author(s):  
Mohammad Danish ◽  
Shashi Kumar ◽  
Surendra Kumar
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Research Work ◽  
Difficulty Level ◽  
Algebraic Equations ◽  
Industrial Case Study ◽  
True Solution ◽  
Solution Vector ◽  
Linear Algebraic Equations ◽  
Non Linear

Numerous unit operations in chemical and process engineering can be represented as a system of non-linear algebraic equations, when modeled for steady state operation ,e.g. isothermal and non-isothermal operations of a series of CSTRs, batteries of evaporators, networks of various separation operations (flash drum, mixers), distillation, extraction and absorption columns, and pumps and piping networks etc. These governing equations are sometimes very difficult to solve due to the nonlinear and uneven nature associated with them. The difficulty level increases when the resulting set of equations become flat near their zeros and thus derivatives based schemes mostly diverge or give poor results. Many a times, even a good initial guess in conventional numerical techniques do not guarantee to have a true solution and problem specific methods have to be designed. This whole scenario can also be viewed as an optimization problem having equality constraints only and casting the equations in the form of norm of function vector, which formulates an objective function to be minimized. The true minimum thus found gives us the correct solution vector. Recently, Genetic Algorithms have been quite effectively used to solve many complex engineering optimization problems. In continuation of our earlier research work where an elitist genetic algorithm was developed for the solutions of various difficult MINLP problems (Danish et al. 2006a and b), this research work extends its application for the solution of difficult non-linear algebraic equations. A novel scheme of dynamic mutation parameter as a function of fitness along with dynamic penalty has been proposed. The small value of mutation parameter in initial stages enables the algorithm to search globally, and the solution thus found is refined by keeping its value higher in later generations. This new scheme is found to be very effective in the sense that the algorithm requires very small population size and comparatively lesser number of generations to give reasonably good solutions. To test the efficacy of algorithm we have solved five sets of difficult non-linear algebraic equations (Dennis and Schnabel, 1983). It is worthwhile to mention that one of these equations was having both its Jacobian and Hessian as zero, at its true solution. Applicability of the developed GA was also demonstrated by simulating an industrial case study of triple effect evaporator used for concentrating the caustic soda solution (Zain and Kumar, 1996), which also poses difficulty during numerical simulation by Newton-Raphson method.

An efficient algorithm based on Haar wavelets for numerical simulation of Fokker-Planck equations with constants and variable coefficients

2015 ◽  
Vol 25 (1) ◽  
pp. 41-56 ◽  
Author(s):  
Manoj Kumar ◽  
Sapna Pandit
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Haar Wavelet ◽  
Variable Coefficients ◽  
Haar Wavelets ◽  
Algebraic Equations ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Fokker Planck Equations ◽  
Fokker Planck

Purpose – The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary conditions. Design/methodology/approach – Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method. Findings – The accuracy of the proposed scheme is demonstrated on three test examples. The numerical solutions prove that the proposed method is reliable and yields compatible results with the exact solutions. The scheme provides better results than the schemes [9, 14]. Originality/value – The developed scheme is a new scheme for Fokker-Planck equations. The scheme based on Haar wavelets is expended for nonlinear partial differential equations with variable coefficients.

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