[Paper Review] Transport and concentration processes in the multidimensional zero-pressure gas dynamics model with the energy conservation law

Sergio Albeverio, Olga Rozanova+1 more|arXiv (Cornell University)|Jan 30, 2011

Computational Fluid Dynamics and AerodynamicsEngineering17 references16 citations

TL;DR

This paper introduces integral identities to define δ-shock wave solutions in multidimensional zero-pressure gas dynamics with energy conservation, deriving Rankine-Hugoniot conditions and balance laws for mass, momentum, and energy transport onto the δ-shock front. The key contribution is a rigorous framework showing that total mass, momentum, and energy are conserved while the δ-shock front accumulates mass and energy, with kinetic energy fully converting into internal energy during propagation.

ABSTRACT

We introduce integral identities to define delta-shock wave type solutions for the multidimensional zero-pressure gas dynamics Using these integral identities, the Rankine-Hugoniot conditions for delta-shocks are obtained. We derive the balance laws describing mass, momentum, and energy transport from the area outside the delta-shock wave front onto this front. These processes are going on in such a way that the total mass, momentum, and energy are conserved and at the same time mass and energy of the moving delta-shock wave front are increasing quantities. In addition, the total kinetic energy transfers into the total internal energy. The process of propagation of delta-shock waves is also described. These results can be used in modeling of mediums which can be treated as a {pressureless continuum} (dusty gases, two-phase flows with solid particles or droplets, granular gases).

Motivation & Objective

• To define δ-shock wave solutions in multidimensional zero-pressure gas dynamics with energy conservation using integral identities.
• To derive Rankine-Hugoniot conditions for δ-shocks in the presence of energy conservation.
• To establish balance laws describing transport of mass, momentum, and energy from the surrounding medium onto the δ-shock front.
• To analyze the dynamics of δ-shock propagation and energy transformation processes.
• To provide a mathematical foundation for modeling pressureless continua such as dusty gases and granular flows.

Proposed method

• Derives integral identities in the $L^∞$-generalized solution framework to define weak solutions for the multidimensional zero-pressure gas dynamics system.
• Introduces a surface-based distributional approach using the Heaviside and Dirac delta functions on moving hypersurfaces to model discontinuities.
• Applies integration-by-parts formulas on moving surfaces $\Gamma_t$ with normal velocity $U_\delta = \nu G$, incorporating mean curvature and normal derivatives.
• Uses the surface transport theorem to derive evolution laws for quantities defined on the δ-shock front.
• Derives the adjoint operator $\delta^*/\delta t$ to handle time derivatives on moving surfaces.
• Establishes balance laws for mass, momentum, and energy transport by integrating the governing PDEs over regions bounded by the δ-shock front.

Experimental results

Research questions

• RQ1 How can δ-shock wave solutions be rigorously defined in multidimensional zero-pressure gas dynamics with energy conservation?
• RQ2 What are the Rankine-Hugoniot conditions for δ-shocks in this system, and how do they differ from classical shock conditions?
• RQ3 How is mass, momentum, and energy transported from the surrounding medium onto the δ-shock front?
• RQ4 What is the mechanism of energy transfer from kinetic to internal energy during δ-shock propagation?
• RQ5 How do the total mass, momentum, and energy of the system behave over time, and what conservation laws govern the δ-shock front?

Key findings

• The total mass, momentum, and energy of the system are conserved globally, even as the δ-shock front accumulates mass and energy.
• The δ-shock front acts as a sink for mass and energy, with both quantities increasing over time due to transport from the surrounding medium.
• The total kinetic energy of the system is fully converted into internal energy during δ-shock propagation, as the velocity field collapses onto the front.
• The Rankine-Hugoniot conditions for δ-shocks are derived using integral identities and involve the normal velocity and curvature of the front.
• The surface transport theorem enables the derivation of evolution equations for conserved quantities on the moving δ-shock front.
• The framework supports modeling of pressureless continua such as dusty gases, two-phase flows, and granular gases with solid particles or droplets.