scholarly journals The Brinkman-Fourier system with ideal gas equilibrium

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chun Liu ◽  
Jan-Eric Sulzbach
Keyword(s):  
Velocity Field ◽  
Weak Solutions ◽  
General Framework ◽  
Type Equation ◽  
Thermodynamic System ◽  
Ideal Gas ◽  
Mechanical Forces ◽  
Laws Of Thermodynamics ◽  
Classical Gas ◽  
Variational Approaches

<p style='text-indent:20px;'>In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistence between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existences of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.</p>

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

scholarly journals Ultrarelativistic Gas with Zero Chemical Potential

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 249
Author(s):  
Daniel Mata-Pacheco ◽  
Gonzalo Parga ◽  
Fernando Angulo-Brown
Keyword(s):  
Chemical Potential ◽  
Ideal Gas ◽  
Cavity Radiation ◽  
Classical Gas ◽  
Photon Gas ◽  
Spin Degeneracy ◽  
Classical Particles

In this work, we propose a set of conditions such that an ultrarelativistic classical gas can present a photon-like behavior. This is achieved by assigning a zero chemical potential to the ultrarelativistic ideal gas. The resulting behavior is similar to that of a Wien photon gas. It is found to be possible only for gases made of very lightweight particles such as neutrinos, as long as we treat them as classical particles, and it depends on the spin degeneracy factor. This procedure allows establishing an analogy between an evaporating gas and the cavity radiation.

Density expansion of the correlation function of a hard sphere gas

1979 ◽  
Vol 57 (3) ◽  
pp. 466-476 ◽  
Author(s):  
D. G. Blair ◽  
N. K. Pope ◽  
S. Ranganathan
Keyword(s):  
Correlation Function ◽  
Fourier Transforms ◽  
Hard Spheres ◽  
Kinetic Equations ◽  
Ideal Gas ◽  
Zero Density ◽  
Classical Gas ◽  
Density Expansions ◽  
The Moment ◽  
Derivatives Of

Using the grand canonical ensemble, the classical Van Hove correlation function G(r, t) is expanded in a power series in density. The zero density limit is the ideal gas result. We have derived, for a classical gas of hard spheres, exact expressions for [Formula: see text], the zero density derivative of the correlation function, and its Fourier transforms. These involve only two particle dynamics. The first two terms in the density expansions provide representation of the correlation functions for appropriate ranges of density and correlation function arguments. We also show that the same result can be obtained from generalized kinetic equations. To this order in density, the moment relations and the time derivatives of I(q, t) at t = 0+ are satisfied. Numerical results are compared with those of Mazenko, Wei, and Yip and with those of the Boltzmann equation and they show the expected behavior.

scholarly journals Holographic heat engines and static black holes: A general efficiency formula

2019 ◽  
Vol 28 (02) ◽  
pp. 1950030 ◽  
Author(s):  
Felipe Rosso
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Upper Bound ◽  
Heat Engine ◽  
Thermodynamic System ◽  
Ideal Gas ◽  
Heat Flows ◽  
Heat Engines ◽  
Static Black Hole

Starting from simple observations regarding heat flows for static black holes (or any thermodynamic system with [Formula: see text]), we get inequalities which restrict their change in energy and adiabatic curves in the [Formula: see text] plane. From these observations, we then derive an exact efficiency formula for virtually any holographic heat engine defined by a cycle in the [Formula: see text] plane, whose working substance is a static black hole. Moreover, we get an upper bound for its efficiency and show that for a certain class of black holes, this bound is universal and achieved by an “ideal gas” hole. Finally, we compute exact efficiencies for some particular and new engines.

Concentrating Bound States for Kirchhoff Type Problems in ℝ3 Involving Critical Sobolev Exponents

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Yi He ◽  
Gongbao Li ◽  
Shuangjie Peng
Keyword(s):  
Weak Solutions ◽  
Bound States ◽  
Type Equation ◽  
Positive Parameter ◽  
Kirchhoff Type ◽  
Critical Sobolev Exponents ◽  
Kirchhoff Type Equation ◽  
Small Positive Parameter ◽  
Kirchhoff Type Problems

AbstractWe study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth,where ε is a small positive parameter and a, b > 0 are constants, f ∈ C

scholarly journals EXISTENCE AND APPROXIMATION OF A (REGULARIZED) OLDROYD-B MODEL

2011 ◽  
Vol 21 (09) ◽  
pp. 1783-1837 ◽  
Author(s):  
JOHN W. BARRETT ◽  
SÉBASTIEN BOYAVAL
Keyword(s):  
Free Energy ◽  
Velocity Field ◽  
Weak Solutions ◽  
Stress Equation ◽  
Time Step ◽  
Polymeric Fluid ◽  
Dissipative Term ◽  
Advection Term ◽  
Energy Bound ◽  
Conformation Tensor

We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain [Formula: see text], d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler-type time discretization. This extends the results of Boyaval et al. (Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Math. Model. Numer. Anal.43 (2009) 523–561) on this free energy bound. There a piecewise constant approximation of the conformation tensor was necessary to treat the advection term in the stress equation, and a restriction on the time step, based on the initial data, was required to ensure that the approximation to the conformation tensor remained positive definite. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms in the system, similar to those introduced in Barrett and Süli (Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off, Math. Models Methods Appl. Sci.18 (2008) 935–971) for the microscopic–macroscopic finitely extensible nonlinear elastic model of a dilute polymeric fluid, we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, toward global-in-time weak solutions of this regularized Oldroyd-B system. Hence, we prove existence of global-in-time weak solutions to this regularized model. Moreover, in the case d = 2 carry out this convergence in the absence of cut-offs, but with a time step restriction dependent on the spatial discretization parameter, and hence show existence of a global-in-time weak solution to the Oldroyd-B system with an additional dissipative term in the stress equation.

scholarly journals On 𝓅 ( x ) -Kirchhoff-type equation involving 𝓅 ( x ) -biharmonic operator via genus theor

2020 ◽  
Vol 72 (6) ◽  
pp. 842-851
Author(s):  
S. Taarabti ◽  
Z. El Allali ◽  
K. Ben Haddouch
Keyword(s):  
Weak Solutions ◽  
Variational Approach ◽  
Type Equation ◽  
Biharmonic Operator ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Genus Theory ◽  
Kirchhoff Type Equation ◽  
Kirchhoff Type Problem

UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the 𝓅 ( x ) -Kirchhoff-type problem, u = Δ u = 0 o n ∂ Ω . By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for the 𝓅 ( x ) -Kirchhoff-type equation.

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