Invariants of gas and fluid flows and exact solutions to boundary eigenvalue problems

气体和流体流动的不变量以及边界特征值问题的精确解

• 批准号: RGPIN-2022-03404
• 负责人: Bogoyavlenskij, Oleg
• 金额: $ 1.31万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758913
• 关键词: Invariants; gas; fluid; flows; exact

The project consists of three parts. Part A will be devoted to investigation of invariants of gas and fluid flows generalizing new invariants discovered in my paper CCV#7 for the isentropic gas dynamics in mushroom clouds appearing after the atomic and thermonuclear explosions in atmosphere. I intend: (a) To find material conservation laws for the axisymmetric gas dynamics in mushroom clouds with arbitrary equation of state; (b) To obtain invariants for dynamics of ideal incompressible fluid with vanishing swirl (atomic and thermonuclear explosions under water); (c) To derive invariants of axisymmetric and helically symmetric plasma flows in magnetohydrodynamics. The work on Part A is planed to do in 2022-2025 jointly with PhD1 student and in 2025-2027 jointly with MSc2 student. Part B will be committed to the study of chaotic dynamics of viscous fluid with applications to the turbulence problem. Based on our joint with *Post-Doctoral Fellow R. Kipka paper *CCV#1 I plan: (a) To investigate numerically different regimes of chaotic dynamics in more general non-symmetric exact solutions to the Navier - Stokes equations; (b) To derive new time-dependent exact solutions to the Navier - Stokes equations and viscous magnetohydrodynamics equations with chaotic streaklines and chaotic magnetic field lines. The work on Part B is planed to do in 2023-2026 jointly with my Post-Doctoral Fellow and in 2023-2025 jointly with MSc1 student. Part C will be dedicated to the study of exact solutions to the ideal fluid dynamics, viscous magnetohydrodynamics and Navier - Stokes equations that have a common property that fluid velocity V(x) satisfies Beltrami equation curl V(x) =gV(x). This amounts to the eigenvalue-boundary-value problem for the operator curl. Based on results of my work CCV#4 I plan: (a) To represent in elementary functions all series of eigenfields of operator curl inside a ball 0 < R < R1  and a spherical shell R2 < R < R1 with the non-penetration boundary condition. In work by Cantarella et al (2000) [18] the eigenfields were described by special functions (Bessel's and Legendre's); (b) To investigate the scale-invariant equations for the eigenvalues  gk.m; (c) To derive exact solutions to the Navier-Stokes equations with the no-slip boundary condition. The work on Part C is planed to do in 2025-2027 in collaboration with PhD2 student. Results of the research will be important to all specialists on mathematical physics and on gas and fluid mechanics. The anticipated outcomes and benefits for Canada are: (a) Young specialists in the advanced areas of science will be trained and will defend their Dissertations. Each PhD student and Post-Doctoral Fellow will publish 2 papers in USA Journal Physics of Fluids and in European Journal ZNA; each MSc student will publish at least 1 paper in the same Journals. (b) Important mathematical problems will be resolved and their practical applications will be developed.

该项目包括三个部分。A部分将致力于研究气体和流体流动的不变量，概括我在论文CCV#7中发现的新不变量，用于大气中原子和热核爆炸后出现的蘑菇云中的等熵气体动力学。我打算：（a）在任意状态方程的蘑菇云中找到轴对称气体动力学的物质守恒定律;（B）获得具有消失漩涡（水下原子和热核爆炸）的理想不可压缩流体的动力学不变量;（c）推导磁流体力学中轴对称和螺旋对称等离子体流的不变量。A部分的工作计划在2022-2025年与博士1学生共同完成，并在2025-2027年与硕士2学生共同完成。B部分将致力于研究粘性流体的混沌动力学及其在湍流问题中的应用。基于我们与 * 博士后研究员R。Kipka文件 *CCV#1我计划：（a）调查数值不同制度的混沌动力学更一般的非对称精确解的Navier-Stokes方程;（B）推导新的时间依赖精确解的Navier-Stokes方程和粘性磁流体力学方程与混沌条纹线和混沌磁力线。B部分的工作计划在2023-2026年与我的博士后研究员共同完成，并在2023-2025年与MSc 1学生共同完成。 C部分将致力于研究理想流体动力学、粘性磁流体动力学和Navier-Stokes方程的精确解，这些方程具有流体速度V（x）满足Beltrami方程curl V（x）=gV（x）的共同性质。这相当于算子旋度的特征值边值问题。基于我的工作CCV#4的结果，我计划：（a）用初等函数表示球0 < R < R1和球壳R2 < R < R1内具有非穿透边界条件的算子旋度的所有本征场系列。在Cantarella等人（2000）[18]的工作中，本征场由特殊函数（贝塞尔函数和勒让德函数）描述;（B）研究本征值gk.m的标度不变方程;（c）推导具有无滑移边界条件的Navier-Stokes方程的精确解。C部分的工作计划在2025-2027年与博士2学生合作完成。研究结果将是重要的所有专家对数学物理和气体和流体力学。对加拿大的预期成果和惠益是：（a）高级科学领域的年轻专家将得到培训，并将为他们的论文答辩。每个博士生和博士后研究员将在美国流体物理学杂志和欧洲ZNA杂志上发表2篇论文;每个硕士生将在相同的期刊上发表至少1篇论文。(b)重要的数学问题将得到解决，并将开发其实际应用。