Nonlinear Diffusive Waves

Sachdev, P. L.

doi:10.1017/cbo9780511569449

Cited by 107 publications

(96 citation statements)

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“…Such solutions have been grouped into two categories: self-similar solutions of the first and second kind [Barenblatt, 1979]. Similarity solutions of the first kind may be completely specified using dimensional analysis [Sachdev, 1987]. Similarity solutions of the second kind, also known as intermediate asymptotics, are solutions which are stable over a wide range of propagation distances [Barenblatt, 1979].…”

Section: Similarity Solutionmentioning

confidence: 99%

“…The generalized Burgers' equation, which arose in nonlinear acoustics [Lighthill, 1956], appears in many other contexts and is a canonical equation, describing dispersive and diffusive nonlinear wave propagation in heterogeneous media [Sachdev, 1987;Jeffrey, 1989;Anile et al, 1993]. The long-time asymptotic solutions of the generalized Burgers' equation were classified by Scott [1981].…”

Section: Similarity Solutionmentioning

confidence: 99%

“…to convert (24) to the heat equation [Sachdev, 1987]. This equation may be solved exactly and the explicit expression for the saturation is…”

Section: Similarity Solutionmentioning

confidence: 99%

“…When capillary effects are included in the analysis, the resulting equation governing the evolution of the saturation amplitude along a trajectory is a generalization of Burger's equation [Burgers, 1948]. Burger's equation governs nonlinear diffusive wave propagation and is used in numerous applications [Burgers, 1974;Sachdev, 1987]. Thus the asymptotic formulation reveals a link between two-phase flow and nonlinear wave propagation [Whitham, 1974].…”

Section: Introductionmentioning

confidence: 99%

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An asymptotic solution for two‐phase flow in the presence of capillary forces

Vasco

2004

Water Resources Research

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[1] Under the assumption of smoothly varying background properties I derive an asymptotic solution for two-phase flow. This formulation partitions the modeling of two-phase flow into two subproblems: an arrival time calculation and a saturation amplitude computation. The asymptotic solution itself is defined along a trajectory though the model. If gravitational forces are not important and the flow field is independent of changes in the background saturation, the trajectory may be identified with a streamline, and the asymptotic approach provides a mathematical basis for streamline simulation. The evolution of the saturation amplitude is governed by a generalization of Burgers' equation, defined along the trajectory. If the capillary properties are uniform and gravitational forces are negligible, one can derive self-similar solutions for the three limiting behaviors of a two-phase front. Asymptotic solutions were found to deviate by less than 5% from saturations calculated using a numerical simulator. The numerical results indicate that changes in capillary properties can introduce significant variations in the arrival time of a specific aqueous fraction. A more robust definition of arrival time appears to be the time associated with the peak rate of change in aqueous phase fraction, the greatest slope of the breakthrough curve.

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Section: Similarity Solutionmentioning

confidence: 99%

Section: Similarity Solutionmentioning

confidence: 99%

“…to convert (24) to the heat equation [Sachdev, 1987]. This equation may be solved exactly and the explicit expression for the saturation is…”

Section: Similarity Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An asymptotic solution for two‐phase flow in the presence of capillary forces

Vasco

2004

Water Resources Research

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“…Nonlinear acoustical waves in dissipative media have been extensively studied and model equations describing weakly nonlinear, weakly dissipative waves have been solved [1,2,3]. In most cases the nonlinearity is quadratic and the model equations are Burgers' or generalized Burgers' equations [4].…”

Section: Introductionmentioning

confidence: 99%

Standing and propagating waves in cubically nonlinear media

Enflo¹,

Hedberg²,

Руденко³

2006

AIP Conference Proceedings

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Abstract. The paper has three parts. In the first part a cubically nonlinear equation is derived for a transverse finite-amplitude wave in an isotropic solid. The cubic nonlinearity is expressed in terms of elastic constants. In the second part a simplified approach for a resonator filled by a cubically nonlinear medium results in functional equations. The frequency response shows the dependence of the amplitude of the resonance on the difference between one of the resonator's eigenfrequencies and the driving frequency. The frequency response curves are plotted for different values of the dissipation and differ very much for quadratic and cubic nonlinearities. In the third part a propagating N-wave is studied, which fulfils a modified Burgers' equation with a cubic nonlinearity. Approximate solutions to this equation are found for new parts of the wave profile.

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An improved coarse-mesh nodal integral method for partial differential equations

Rizwan-uddin

1997

Numer. Methods Partial Differential Eq.

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An improved variation of the nodal integral method to solve partial differential equations has been developed and implemented. Rather than treating all of the nonlinear terms as the so-called pseudo-source terms (to be approximated), in this modified version of the nodal integral method, by approximating part of the nonlinear terms in terms of the discrete variable(s) that ultimately result at the end of the formulation process, some or all of the nonlinear terms are kept on the left-hand side in the transverse-integrated equations, which are to be solved analytically. Application of the method to solve the Burgers equation leads to exponential variation within the nodes and shows that the resulting scheme has inherent upwinding. Reconstruction of node interior solution -as a function of one independent variable, and averaged in all others -makes it possible to obtain rather accurate solutions even on a fine scale. Results of the numerical analysis and comparison with results of other methods reported in the literature show that the new method is comparable and sometimes better in accuracy than the currently used schemes. Extension to multidimensional problems is straightforward.

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Nonlinear Diffusive Waves

Cited by 107 publications

References 0 publications

An asymptotic solution for two‐phase flow in the presence of capillary forces

An asymptotic solution for two‐phase flow in the presence of capillary forces

Standing and propagating waves in cubically nonlinear media

An improved coarse-mesh nodal integral method for partial differential equations

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