scholarly journals Causal dissipation for the relativistic dynamics of ideal gases

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160729 ◽  
Author(s):  
Heinrich Freistühler ◽  
Blake Temple
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Second Order ◽  
Order System ◽  
Navier Stokes ◽  
Unique Element ◽  
Relativistic Euler Equations ◽  
Navier Stokes Equations ◽  
First Order ◽  
Ideal Gases

We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.

A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

2017 ◽  
Vol 22 (2) ◽  
pp. 375-392 ◽  
Author(s):  
Huiqiang Yue ◽  
Jian Cheng ◽  
Tiegang Liu
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Equivalent Form ◽  
Second Order ◽  
Heat Fluxes ◽  
Order System ◽  
Navier Stokes ◽  
Optimal Order ◽  
Navier Stokes Equations ◽  
Optimal Order Of Convergence

AbstractIn this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

scholarly journals A multilayer shallow model for dry granular flows with the -rheology: application to granular collapse on erodible beds

2016 ◽  
Vol 798 ◽  
pp. 643-681 ◽  
Author(s):  
E. D. Fernández-Nieto ◽  
J. Garres-Díaz ◽  
A. Mangeney ◽  
G. Narbona-Reina
Keyword(s):  
Transient Flow ◽  
Stokes Equations ◽  
Low Cost ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Runout Distance ◽  
Multilayer Model ◽  
First Order ◽  
Constant Friction ◽  
Erodible Beds

In this work we present a multilayer shallow model to approximate the Navier–Stokes equations with the ${\it\mu}(I)$-rheology through an asymptotic analysis. The main advantages of this approximation are (i) the low cost associated with the numerical treatment of the free surface of the modelled flows, (ii) the exact conservation of mass and (iii) the ability to compute two-dimensional profiles of the velocities in the directions along and normal to the slope. The derivation of the model follows Fernández-Nieto et al. (J. Comput. Phys., vol. 60, 2014, pp. 408–437) and introduces a dimensional analysis based on the shallow flow hypothesis. The proposed first-order multilayer model fully satisfies a dissipative energy equation. A comparison with steady uniform Bagnold flow – with and without the sidewall friction effect – and laboratory experiments with a non-constant normal profile of the downslope velocity demonstrates the accuracy of the numerical model. Finally, by comparing the numerical results with experimental data on granular collapses, we show that the proposed multilayer model with the ${\it\mu}(I)$-rheology qualitatively reproduces the effect of the erodible bed on granular flow dynamics and deposits, such as the increase of runout distance with increasing thickness of the erodible bed. We show that the use of a constant friction coefficient in the multilayer model leads to the opposite behaviour. This multilayer model captures the strong change in shape of the velocity profile (from S-shaped to Bagnold-like) observed during the different phases of the highly transient flow, including the presence of static and flowing zones within the granular column.

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