Isentropic Equations Research Articles

Energy equality in the isentropic compressible Navier-Stokes equations allowing vacuum

It is well-known that a Leray-Hopf weak solution in L4(0,T;L4(T3)) for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This further allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations, which is an improvement of corresponding results obtained by Yu in [31, Arch. Ration. Mech. Anal., 225 (2017)] and answers a question posed by Liang in [18, Proc. R. Soc. Edinb., Sect. A, Math. (2020)].

Design and test a converging and de Laval nozzle using additive manufacturing

The advent of additive manufacturing technology has facilitated the design and fabrication of parts and models in both academia and aerospace industry. Compressible flow in the nozzles is not a new research topic; however, the accuracy of the experimental results obtained from the nozzles using additive manufacturing has not been assessed comprehensively. Surface roughness and strength of 3D-printed nozzles are two major concerns when they are applied to compressible flows. In this paper, a converging and a de Laval nozzle fabricated by additive manufacturing using ABS filament are designed and tested. Surface roughness inside the converging nozzle is quantified using a nondestructive method. In general, the experimental results compare well with the analytical solutions from isentropic equations for the converging nozzle and the numerical simulations conducted in ANASYS Fluent for the de Laval nozzle. 3D-printed nozzles can be employed to quickly demonstrate and verify novel ideas and concepts in the pedagogy and research at large Reynolds numbers.

On regular solutions to compressible radiation hydrodynamic equations with far field vacuum

Abstract The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ \rho in a power law ρ δ {\rho }^{\delta } (with 0 < δ < 1 0\lt \delta \lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

Vanishing Magnetic Field Limits of Solutions to the Isentropic Chaplygin Gas Magnetogasdynamics Equations

Vanishing Magnetic Field Limits of Solutions to the Isentropic Chaplygin Gas Magnetogasdynamics Equations

Thermodynamics of Hot Neutron Stars and Universal Relations

Over the last few years, the detection of gravitational waves from binary neutron star systems has rekindled our hopes for a deeper understanding of the unknown nature of ultradense matter. In particular, gravitational wave constraints on the tidal deformability of a neutron star can be translated into constraints on several neutron star properties using a set of universal relations. Apart from binary neutron star mergers, supernova explosions are also important candidates for the detection of multimessenger signals. Such observations may allow us to impose significant constraints on the binding energy of neutron stars. The purpose of the present study is twofold. Firstly, we investigate the agreement of finite temperature equations of state with established universal relations. Secondly, we examine the possible existence of a universal relation between the binding energy and the dimensionless tidal deformability, which are the bulk properties connected to the most promising sources for multimessenger signals. We find that hot equations of state are not always compatible with accepted universal relations. Therefore, the use of such expressions for probing general relativity or imposing constraints on the structure of neutron stars would be inconclusive (when thermal effects are present). Additionally, we show that the binding energy and the dimensionless tidal deformability exhibit a universal trend at least for moderate neutron star masses. The latter allows us to set bounds on the binding energy of a 1.4 M⊙ neutron star using data from the GW170817 event. Finally, we provide a relation between the compactness, the binding energy and the dimensionless tidal deformability of a neutron star that is accurate for cold and hot isentropic equations of state.

Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish

This paper is concerned with the Riemann problem for the isentropic Chaplygin gas magnetogasdynamics equations and the formation of $$\delta $$ -shocks and vacuum states as pressure and magnetic field vanish. Firstly, the Riemann problem of the isentropic magnetogasdynamics equations for Chaplygin gas is solved analytically. Secondly, it is rigorously proved that, as both the pressure and the magnetic field vanish, the Riemann solution containing two shocks tends to a $$\delta $$ -shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -measure which form the $$\delta $$ -shock; while the Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the transport equations if we do not consider the virtual velocity in the vacuum region, the intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

Asymptotic stability of planar rarefaction waves under periodic perturbations for 3-d Navier-Stokes equations

In this paper, we study a Cauchy problem for the 3-d compressible isentropic Navier-Stokes equations, in which the initial data is a 3-d periodic perturbation around a planar rarefaction wave. We prove that the solution of the Cauchy problem exists globally in time and tends to the background rarefaction wave in the L∞(R3) space as t→+∞. The result reveals that even though the initial perturbation has infinite oscillations at the far field and is not integrable along any direction of space, the planar rarefaction wave is nonlinearly stable for the 3-d N-S equations. The key point is to construct a suitable ansatz (ρ˜,u˜) to carry the same oscillations as those of the solution (ρ,u) at the far field in the normal direction of the rarefaction wave, so that the difference (ρ−ρ˜,u−u˜) belongs to some Sobolev space and the energy method is available.

An enhanced, rational model to study acoustic vaporization dynamics of a bubble encapsulated within a nonlinearly elastic shell

Acoustic droplet vaporization (ADV) is a new approach to generate vapor bubbles that have potentially broad medical applications. ADV-generated bubbles can be used as contrast agents in acoustic imaging, as drug carriers to deliver drugs to particular targets, and also in embolotherapy, thermal therapy, and histotripsy. However, despite much progress, ADV dynamics have still not been well understood and properly modeled. In this paper, we present a theoretical study of ultrasound-induced evaporation of a droplet encapsulated by a shell. The main emphasis of this theoretical study is on a proper description of the supercritical state occurring after bubble collapse. For this purpose, an isentropic equation of state for a van der Waals gas is used to describe the bubble behavior in the supercritical state. Sensitivity of the vaporization process is investigated for different acoustic and geometrical parameters and mechanical properties of the shell. Results show that the value of the minimum pressure required for direct vaporization (without any oscillatory behavior) depends on shell elasticity and initial size of the droplet, especially at high frequencies (greater than 2[MHz]). Moreover, it has been shown that applying an acoustic wave with proper phase such that thermal equilibrium of the bubble temperature with the surrounding liquid is attained, results in direct vaporization at lower acoustic pressure.

Linear stability of viscous shock wave for 1-D compressible Navier-Stokes equations with Maxwell’s law

In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.

Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

<p style='text-indent:20px;'>In this paper, we prove the global existence of the strong solutions to the vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with small initial data and axial symmetry, where the solutions are independent of the axial variable and the angular variable. The solutions capture the precise physical behavior that the sound speed is <inline-formula><tex-math id="M1">\begin{document}$ C^{1/2} $\end{document}</tex-math></inline-formula>-Hölder continuous across the vacuum boundary provided that the adiabatic exponent <inline-formula><tex-math id="M2">\begin{document}$ \gamma\in(1, 2) $\end{document}</tex-math></inline-formula>. The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the obstacles by constructing some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and establishing the uniform-in-time weighted energy estimates of solutions by delicate analysis, in which the balance of pressure and self-gravitation, and the dissipation of velocity are crucial.</p>

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

Existence and nonlinear stability of steady-states to outflow problem for the full two-phase flow

The outflow problem for the viscous full two-phase flow model in a half line is investigated in the present paper. The existence, uniqueness and nonlinear stability of the steady-state are shown respectively corresponding to the supersonic, sonic or subsonic state at far field. This is different from the outflow problem for the isentropic Navier-Stokes equations, where there is no steady-state for the subsonic state. Furthermore, we obtain either exponential time decay rates for the supersonic state or algebraic time decay rates for supersonic and sonic states in weighted Sobolev spaces.

Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.

Prediction of Interlobe Leakages in Screw Compressors

The clearance gaps in twin-screw compressors are critical for their performance and reliable operation, as the leakage flows through these clearances influence the volumetric and adiabatic efficiencies. The amount of leakage flows depends on the clearance size and shape as well as various geometric and operating parameters. Usually, the isentropic nozzle equations along with appropriate flow coefficients are used for more accurate estimation of the leakage flow rates through the clearance gaps. Hence, a proper understanding of the flow coefficients and their relationship with the dimensionless parameters (such as pressure ratio, Reynolds number, and aspect ratios) is critical for an accurate prediction of the leakage flows. In the present study, considering the interlobe clearance gap in screw compressor in terms of rectangular openings, the interlobe leakage flow rates are estimated for various opening sizes and pressure conditions using isentropic nozzle equations and an iterative method. The flow coefficients are determined by comparing the experimental values obtained using a specialized test rig and the flow rates obtained from the analytical methods. The dimensionless parameters are varied to see their individual effect on the leakage mass flow rates and on the flow coefficients. The mean deviation from the experimental results when using an analytical iterative procedure (-8.5%) is substantially lower in comparison to the mean deviation (+26.8%) using the isentropic nozzle equations. The study validates that the iterative method can be preferred (for an interlobe leakage flow rate prediction) over the isentropic nozzle equation method.

Experimental Investigations on Leakages in Positive Displacement Machines

The clearance gaps in positive displacement machines such as compressors, pumps, expanders, and turbines are critical for their performance and reliability. The leakage flow through these clearances influences the volumetric and adiabatic efficiencies of the machines. The extent of the leakage flow depends on the size and shape of clearance paths and pressure differences across these paths. Usually, the mass flow through the gaps is estimated using the isentropic nozzle equation with the flow coefficients applied to correct for the real flow conditions. However, the flow coefficients applied generally do not take into account the shape and size of these leakage paths. For that reason, a proper understanding of the relationship between flow coefficients and shape parameters is crucial for an accurate prediction of leakage flows. The present study investigates the influence of the various dimensionless parameters such as Reynolds number, Mach number, and pressure ratio on the flow coefficients for circular and rectangular clearance shapes. The flow coefficients are determined by comparing the experimental values obtained in an experimental test rig and the flow rates obtained from the isentropic nozzle equation. It is observed that in the case of circular clearances, the mean deviation of the experimental leakage results (in comparison to the analytical results using isotropic nozzle equations) is +9.1%, which is significantly lower than the mean deviation (+20.5%) in the case of rectangular clearance leakages. The study indicates that the isentropic nozzle equation method is more suitable for predicting the leakages through the circular clearances and needs modifications for consideration of the rectangular clearances. Using regression analysis, empirical correlations are developed to predict the flow coefficient in terms of Reynolds number, Mach number, pressure ratio, aspect ratio, and β ratio, which are found to match within ±6.4 percent of the numerical results for the rectangular clearance and within the range of -3.6 percent to +5.1 percent of the numerical result for the circular clearance. The empirical relationships presented in this study can be extended to evaluate the flow coefficients in a positive displacement machine.

Initial boundary value problems for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms

We consider the initial boundary value problem for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms. This problem is related to the motion of the compressible viscous flow surrounding a rotating obstacle. We show the global existence and uniqueness of strong solutions with initial data close to some approximate solution, provided that the friction coefficient α attached to the Navier-slip boundary condition is small. The large time behavior of the solutions is also studied.

Well-posedness of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power (ρδ with 0<δ<1), based on some analysis of the nonlinear structure of this system, we identify a class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one cannot obtain any global regular solution whose L∞ norm of u decays to zero as time t goes to infinity.

Analytical Solution to 1D Compressible Navier-Stokes Equations

There exist complex behavior of the solution to the 1D compressible Navier-Stokes equations in half space. We find an interesting phenomenon on the solution to 1D compressible isentropic Navier-Stokes equations with constant viscosity coefficient on x , t ∈ 0 , + ∞ × R + , that is, the solutions to the initial boundary value problem to 1D compressible Navier-Stokes equations in half space can be transformed to the solution to the Riccati differential equation under some suitable conditions.

Large time behavior of solutions to a two phase fluid model in [formula omitted

We establish the global existence and large time behavior of a unique classical solution for a two phase fluid model consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations via a drag force, provided the initial data is sufficiently small in H4(R3)∩L1(R3). The main tools employed in the analysis are the spectrum analysis and energy method. Our results show that the classical solution converges to a given equilibrium state at algebra decay rate.

Thermodynamical Description of Hot, Rapidly Rotating Neutron Stars, Protoneutron Stars, and Neutron Star Merger Remnants

Abstract The prediction of the equation of state of hot, dense nuclear matter is one of the most complicated and interesting problems in nuclear astrophysics. At the same time, knowledge of it is the basic ingredient for some of the most interesting studies. In the present work, we concentrate our study on the construction of the equation of state of hot, dense nuclear matter, related mainly to the interior of the neutron star. We employ a theoretical nuclear model, which includes momentum-dependent interaction among the nucleons, along with state-of-the-art microscopic calculations. Thermal effects are introduced in a self-consistent way, and a set of isothermal and isentropic equations of state are predicted. The predicted equations of state are used in order to acquire and extend the knowledge of the thermal effect on both nonrotating and rapidly rotating with the Kepler frequency neutron stars. The simultaneous study of thermal and rotation effects provides useful information on some of the most important quantities, including the mass (gravitational and baryon) and radius, the Kepler frequency and Kerr parameter, the moment of inertia, etc. These quantities are directly related to studies of protoneutron stars and mainly the hot and rapidly rotating remnant of a binary neutron star merger. Data from the late observations of binary neutron star mergers and the present study may offer useful tools for investigation and help in providing possible constraints on the equation of state of nuclear matter.