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تسجيل مستخدم جديدتشكيل التجمعات الكثافية في الهيدروديناميكا المثالية للتبريد الحر لغازات غير مرونة: عائلة من الحلول الدقيقة
Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: a family of exact solutions
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نحن نستخدم الديناميكا الهيدروديناميكية الحبيبية لاستكشاف مشكلة التجمع المثالية للجسيمات في غاز حبيبي متخلص يبرد. نحن ننظر إلى الحركات الهيدروديناميكية الكبيرة الحجم حيث يمكن تجاهل السطوح والتدفئة ويصل إلى معادلات الغاز المثالي مع عبارة إضافية تصف فقدان الطاقة الجمعية بسبب الاصطدامات غير المتصلة. نحن نستخدم الإحداثيات اللاجانية ونستنتج عائلة واسعة من الحلول التحليلية غير المستقرة الدقيقة التي تعتمد فقط على متغير مساحي واحد. هذه الحلول تظهر نوعا جديدا من الخطأ، حيث ينتشر الغاز بسرعة في وقت محدد عند البدء من الظروف الناتجة الناعمة. تشير الانتشارات الغزيرة إلى تشكيل تجمعات مجتمعة من الجسيمات. باتجاه الوقت الذي يصل إلى $t_c$، يعرض الحد الأقصى للغاز قانونا بوور المثل $(t_c-t)^{-2}$. ينتشر التسارع بوور المثل $- (t_c-t)^{-1}$ بينما يبقى السرعة متصلة وتطور حاكما (بدلا من نقطة قطبية مفصلة) في الخطأ. تذهب درجة حرارة الغاز إلى الصفر في الخطأ، ويتبع الخطأ السيناريو الأيزوباري: يبقى ضغط الغاز متوازنا وتقريبا متحدثا في المكان وثابتا في الوقت بالقرب من الخطأ. حلا آخرا دقيقا يظهر أن الانتشار الغزير، من نفس النوع، يمكن أن يتواجد مع الصدمة العادية، حيث يكون الحقول الهيدروديناميكية مفصلة ولكن محدودة في الخطأ. نحن نؤكد ثبات الحلول الدقيقة بالنسبة إلى التثبيطات الأحادية الصغيرة عن طريق حل معادلات الهيدروديناميكا المثالية بشكل رقمي. بالإضافة إلى ذلك، تظهر الحلول الرقمية أن الخصائص المحلية للانتشار الغزير يمكن أن تكون عامة بشكل دون تأثير الظروف الأولية والحدود.
We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact non-stationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time $t_c$ is approached, the maximum density exhibits a power law $sim (t_c-t)^{-2}$. The velocity gradient blows up as $sim - (t_c-t)^{-1}$ while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas temperature vanishes at the singularity, and the singularity follows the isobaric scenario: the gas pressure remains finite and approximately uniform in space and constant in time close to the singularity. An additional exact solution shows that the density blowup, of the same type, may coexist with an ordinary shock, at which the hydrodynamic fields are discontinuous but finite. We confirm stability of the exact solutions with respect to small one-dimensional perturbations by solving the ideal hydrodynamic equations numerically. Furthermore, numerical solutions show that the local features of the density blowup hold universally, independently of details of the initial and boundary conditions.
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