Some self-similar solutions of two-temperature hydrodynamic equations with allowance for nonlinear heat conduction and exothermic reactions

1986 ◽  
Vol 21 (2) ◽  
pp. 292-298 ◽  
Author(s):  
A. S. Leibenzon
Keyword(s):  
Heat Conduction ◽  
Hydrodynamic Equations ◽  
Nonlinear Heat Conduction ◽  
Exothermic Reactions ◽  
Self Similar ◽  
Similar Solutions ◽  
Two Temperature

Self-similar solutions of unsteady ablation flows in inertial confinement fusion

2008 ◽  
Vol 603 ◽  
pp. 151-178 ◽  
Author(s):  
C. BOUDESOCQUE-DUBOIS ◽  
S. GAUTHIER ◽  
J.-M. CLARISSE
Keyword(s):  
Heat Conduction ◽  
Inertial Confinement Fusion ◽  
Group Symmetry ◽  
Physical Analysis ◽  
Inertial Confinement ◽  
Nonlinear Heat Conduction ◽  
Starting Point ◽  
Confinement Fusion ◽  
Self Similar ◽  
Similar Solutions

We exhibit and detail the properties of self-similar solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction which are relevant to inertial confinement fusion (ICF). These solutions have been found after several contributions over the last four decades. We first derive the set of ODEs – a nonlinear eigenvalue problem – which governs the self-similar solutions by using the invariance of the Euler equations with nonlinear heat conduction under the two-parameter Lie group symmetry. A sub-family which leaves the density invariant is detailed since these solutions may be used to model the ‘early-time’ period of an ICF implosion where a shock wave travels from the front to the rear surface of a target. A chart allowing us to determine the starting point of a numerical solution, knowing the physical boundary conditions, has been built. A physical analysis of these unsteady ablation flows is then provided, the associated dimensionless numbers (Mach, Froude and Péclet numbers) being calculated. Finally, we show that self-similar ablation fronts generated within the framework of the above hypotheses (electron heat conduction, growing heat flux at the boundary, etc.) and for large heat fluxes and not too large pressures at the boundary do not satisfy the low-Mach-number criteria. Indeed both the compressibility and the stratification of the hot-flow region are too large. This is, in particular, the case for self-similar solutions obtained for energies in the range of the future Laser MegaJoule laser facility. Two particular solutions of this latter sub-family have been recently used for studying stability properties of ablation fronts.

On the correctness of a model for phase transitions in binary alloys

1990 ◽  
Vol 1 (4) ◽  
pp. 327-338 ◽  
Author(s):  
I. G. Götz
Keyword(s):  
Phase Transitions ◽  
Heat Conduction ◽  
Stability Criterion ◽  
Binary Alloys ◽  
The Self ◽  
Self Similar Solution ◽  
Self Similar ◽  
The Stability ◽  
Similar Solutions ◽  
And Diffusion

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

scholarly journals Approximate Solution of the Nonlinear Heat Conduction Equation in a Semi-Infinite Domain

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Jun Yu ◽  
Yi Yang ◽  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Energy Density ◽  
Approximation Method ◽  
Heat Conduction Equation ◽  
Boundary Energy ◽  
Approximate Solutions ◽  
Nonlinear Heat Conduction ◽  
Infinite Domain ◽  
Nonlinear Heat Conduction Equation ◽  
Self Similar

We use an approximation method to study the solution to a nonlinear heat conduction equation in a semi-infinite domain. By expanding an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain, we reduce the partial differential equation to a set of first-order ordinary differential equations in time. We describe a systematic approach to derive approximate solutions using Taylor polynomials of a different degree. For a special case, we derive an analytical solution and compare it with the result of a self-similar analysis. A comparison with the numerically integrated results demonstrates good accuracy of our approximate solutions. We also show that our approximation method can be applied to cases where boundary energy density and the corresponding effective conductivity are more general than those that are suitable for the self-similar method. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions.

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