• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.529-540
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• 2008

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system, $${u_t}-{\triangle}_{m,p}u=u^{{\alpha}_1}\;{\int}_{\Omega}\;{\upsilon}^{{\beta}_1}\;(x,\;t)dx,\;{\upsilon}_t-{\triangle}_{n,p}{\upsilon}={\upsilon}^{{\alpha}_2}\;{\int}_{\Omega}\;u^{{\beta}_2}\;(x,{\;}t)dx,$$ with homogeneous Dirichlet boundary condition. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depends on the initial data and the relations of the parameters in the system.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.10 no.2
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• pp.227-237
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• 1998

A numerical simulation has been carried out for the automotive air conditioning system. The purpose of this simulation is to present the methods for simulating car air conditioning components, systems and cool-down performance by computerized mathematical model and to analyze the performance of A/C system. In analyzing the heat exchanger(evaporator and condenser), the finite volume model which has a merit in predicting the temperature field in detail because it can consider partial variation of thermal property and heat transfer coefficient is used. In analyzing the compressor, the polytropic approach which regards the actual compression process as a reversible polytropic process is employed. In analyzing vehicle passenger compartment, the thermal network is employed to simulate the car cool down process. This A/C system program can be used for analyzing a component performance when a component is alternated or designed and for analyzing the engine cooling system when A/C system is operated.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1109-1129
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• 2021

This paper considers an anisotropic polytropic infiltration equation with a source term $$u_t={\sum\limits_{i=1}^{N}}{\frac{{\partial}}{{\partial}x_i}}\(a_1(x){\mid}u{\mid}^{{\alpha}_i}{\mid}u_{x_i}{\mid}^{p_i-2}u_{x_i}\)+f(x,t,u)$$, where p_i > 1, α_i > 0, a_i(x) ≥ 0. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}{\in}W_{loc}^{1,p_i}(\Omega)$, the stability of weak solutions is proved without boundary value condition by the weak characteristic function method. One of the essential characteristics of an anisotropic equation different from an isotropic equation is found originally.

• Journal of The Korean Astronomical Society
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• v.53 no.2
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• pp.49-57
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• 2020

We present numerical simulations of decaying hydrodynamic turbulence initially driven by solenoidal (divergence-free) and compressive (curl-free) drivings. Most previous numerical studies for decaying turbulence assume an isothermal equation of state (EOS). Here we use a polytropic EOS, P ∝ ρ^γ, with polytropic exponent γ ranging from 0.7 to 5/3. We mainly aim at determining the effects of γ and driving schemes on the decay law of turbulence energy, E ∝ t^-α. We additionally study probability density function (PDF) of gas density and skewness of the distribution in polytropic turbulence driven by compressive driving. Our findings are as follows. First of all, we find that even if γ does not strongly change the decay law, the driving schemes weakly change the relation; in our all simulations, turbulence decays with α ≈ 1, but compressive driving yields smaller α than solenoidal driving at the same sonic Mach number. Second, we calculate compressive and solenoidal velocity components separately and compare their decay rates in turbulence initially driven by compressive driving. We find that the former decays much faster so that it ends up having a smaller fraction than the latter. Third, the density PDF of compressively driven turbulence with γ > 1 deviates from log-normal distribution: it has a power-law tail at low density as in the case of solenoidally driven turbulence. However, as it decays, the density PDF becomes approximately log-normal. We discuss why decay rates of compressive and solenoidal velocity components are different in compressively driven turbulence and astrophysical implication of our findings.

• Water for future
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• v.28 no.5
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• pp.141-150
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• 1995

An air chamber is designed to keep the pressure from exceeding a predetermined value, or to prevent low pressures and column separation. Therefore, it can be used to protect against rapid transients in a pipe system following abrupt pump stoppage. In this research, an air chamber was applied to a hypothetical pipe system to analyze attenuation effect of pressure head for different air volumes, locations, chamber areas, coefficients of orifice loss and polytropic exponents. With an increase of air volume, the maximum pressure head at pump site is decreased and the minimum pressure head is increased. For different locations and areas of the chamber, the attenuation effects do not show much difference. Also, as the orifice loss coefficient increases, the maximum pressure head is decreased. For different polytropic exponents, isothermal process shows lower maximum pressure head than that of the adiabatic process.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1159-1173
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• 2009

This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents $q_1,\;q_2\;{\in}\;(0,+{\infty})$) with $q_1\;{<}\;q_2$. In other words, when q belongs to different intervals (0, $q_1),\;(q_1,\;q_2),\;(q_2,+{\infty}$), the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval (0, $q_2$]. However, when q ${\in}\;(q_2,+{\infty}$), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval ($q_1,+{\infty}$), while for q ${\in}\;(0,\;q_1$), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = $q_1$ is concerned, the other parameter ${\lambda}$ will play an important role. In other words, when $\lambda$ belongs to different interval (0, ${\lambda}_1$) or (${\lambda}_1$,+${\infty}$), where ${\lambda}_1$ is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

• The Bulletin of The Korean Astronomical Society
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• v.36 no.2
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• pp.111.2-111.2
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• 2011

While many astrophysical disks are vertically stratified and obey a polytropic equation of state, most studies on gravitational instability (GI) of flattened systems consider isothermal, razor-thin disks by taking vertical averages of disk properties. We investigate local GI of rotating pressure-confined polytropic disks with resolved vertical stratification by performing linear stability analysis. We find that the GI of vertically-stratified disks is in general a combination of conventional razor-thin Jeans modes and incompressible modes. The incompressible modes that dominate in the limit of the maximal disk compression require surface distortion and are an unstable version of terrestrial water waves. Disks with a steeper equation of state are found to be more Jeans unstable because they tend to have a smaller vertical scale height as well as a steeper temperature gradient corresponding to lower pressure support. GI depends more sensitively on the vertical temperature than density distribution. The density-weighted, harmonic mean, rather than the simple mean, of the adiabatic sound speed well describes the dispersion relation of horizontal modes, and thus is appropriate in the expression for Toomre Q stability parameter of razor-thin disks. We generalize Q into vertically-stratified disks, and discuss astrophysical application of our work.

• The Bulletin of The Korean Astronomical Society
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• v.7
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• pp.6.1-6.1
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• 1982

• The Bulletin of The Korean Astronomical Society
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• v.33 no.2
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• pp.64.1-64.1
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• 2008

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.27 no.4
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• pp.201-206
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• 2015

The MVR was theoretically modeled by performing the polytropic process, and the polytropic coefficient was estimated by using the performance curve provided by the manufacturers. The TVR was investigated by applying the conservation equations to the movement of fluids inside the TVR. The size of the nozzle and diffuser was determined. Theoretical MVR and TVR modeling was verified by comparing the results of the model with the available design data. Besides, the effects of multi-staging of the MVR on power consumption, and the effects of suction and primary pressure on the sizing of TVR were investigated.