Poiseuille Flow of Reactive Phan–Thien–Tanner Liquids in 1D Channel Flow

2010 ◽  
Vol 132 (11) ◽  
Author(s):  
T. Chinyoka
Keyword(s):  
Finite Difference ◽  
Strong Dependence ◽  
Viscoelastic Fluids ◽  
Thermal Runaway ◽  
Finite Difference Schemes ◽  
Fluid Temperature ◽  
Transient Problems ◽  
Nonisothermal Flows ◽  
Implicit Finite Difference ◽  
Viscoelastic Liquids

We investigate, using direct numerical simulations, the effects of viscoelasticity on pressure driven flows of thermally decomposable liquids in channels. A numerical algorithm based on the finite difference method is implemented in time and space with the Phan–Thien–Tanner as the model for the viscoelastic liquids. The strong dependence of fluid temperature on the Frank–Kamenetskii parameter is shown for various fluid types and the phenomenon of thermal runaway is demonstrated. It is shown that viscoelastic fluids have in general delayed susceptibility to thermal runaway as compared with corresponding inelastic fluids. This paper demonstrates the efficiency of using semi-implicit finite difference schemes in solving transient problems of coupled nonlinear systems. It also provides an understanding of nonisothermal flows of viscoelastic fluids.

3-D implicit finite‐difference migration by multiway splitting

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 554-567 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Finite Difference ◽  
Power Series ◽  
Taylor Expansion ◽  
Optimization Technique ◽  
Optimization Procedure ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Square Root ◽  
Implicit Finite Difference ◽  
Root Operator

We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

2016 ◽  
Vol 27 (09) ◽  
pp. 1650097 ◽  
Author(s):  
A. H. Encinas ◽  
V. Gayoso-Martínez ◽  
A. Martín del Rey ◽  
J. Martín-Vaquero ◽  
A. Queiruga-Dios
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Nonlinear Phenomena ◽  
Klein Gordon ◽  
Implicit Finite Difference ◽  
The Stability ◽  
Stability And Consistency ◽  
New Algorithms

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

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