In 1944 Lev D. Landau found a non-trivial solution to a stationary Navier-Stokes flow on R³, which was symmetric around some axis and fulfilled the condition, that the velocity decayed linearly
and the pressure quadratically in |x|, depending on a parameter -1 < d < 1 uniquely determined by the force acting on the fluid.
Landau was examinating this flow model when considering the following physical experiment. Imagine a thin pipe in a fluid without any boundary conditions. The fluid is driven by a small jet outflowing the pipe. If one looks at the limit case, where the diameter of the pipe shrinks to zero, there is no force acting on the fluid on R³. Nevertheless, if one wants to describe the physical behaviour of the fluid completely, one has to take into account the force acting at the origin, i.e. at the orifice of the pipe. Therefore, one
has to extend the solution in a suitable manner to the whole space R³. This can be done if one admits also weak solutions in the space of distributions.
When looking at the solutions found by Landau, it seems quite natural to ask, if there are also reasonable physical interpretations for the case |d| greater or equal to 1. Therefore we first derive Landau’s solution with the assumptions posed in the beginning, following an ansatz by Batchelor. From there it will be clear, that the
velocity is unbounded on a cone, if |d| > 1. As it turns out, such a modified Landau solution is no longer
a solution to a Navier-Stokes system - clearly not in the classical, but as well neither in the weak nor very
weak sense. If d = ±1, the velocity will be unbounded on the half line x1 = x2 = 0, ±x3 ≥ 0.
In this case, we have a bit more insight in the behaviour of the modified Landau solution, yet still no physically reasonable interpretation can be given in the whole space R³. In any case, one can consider modified Landau solutions to be solutions in certain subsets of R³, where they yield some interesting streamline plots, which are also provided in this thesis. Furthermore, we give an overview over known results concerning the Landau solution.
Contents
1. Introduction
2. Preliminaries
2.1. The Transport Theorem
2.2. Governing equations of fluid dynamics
2.3. Concepts of a solution
3. Analysis of the classical case
3.1. The classical Landau solution
3.2. Characterisation and approximation of solutions by the Landau solution
4. Analysis of the modified Landau solution
4.1. The critical Landau solution
4.2. The supercritical Landau solution
4.3. Summary and outlook
Research Objectives and Topics
This thesis examines the properties of the Landau solution to the steady Navier-Stokes equations, focusing on the derivation, mathematical characterization, and the analysis of singular behaviors when moving from classical to critical and supercritical cases.
- Mathematical derivation of the Landau solution using spherical coordinates.
- Analysis of flow symmetry, axisymmetry, and the physical interpretation of the Landau parameter.
- Investigation of critical and supercritical Landau solutions and their behavior as sinks.
- Distributional analysis of the divergence and pressure terms in the Navier-Stokes system.
- Evaluation of the validity of Landau solutions as physical models for fluid jets.
Excerpt from the Book
3.1 The classical Landau solution
Theorem 3.1.1. Let d ∈ (−1, 1). Then on R³\{0} a smooth solution (u, p) to the Navier-Stokes system (2.21) which is symmetric, but does not rotate about the x3-axis and fulfils the homogeneity conditions u(x)= 1/|x| u(x/|x|) and p(x)= 1/|x|² p(x/|x|) is given by (1.1). Such a solution is called a Landau solution. Furthermore, every smooth solution satisfying these conditions is a Landau solution.
Rather than checking all the conditions, we will derive the solution from the scratch to have a better insight in the physical meanings of the respective terms, which will help us understanding the more general situation later.
The main idea of our attempt to find a solution to the Navier-Stokes system is, that we reduce the problem to a solvable ordinary differential equation in only one variable. Even though the axisymmetry of the system might suggest cylindrical coordinates, it turns out that using spherical coordinates yields the fastest approach towards this, for then we can make use of our knowledge about the radial dependence of our solution. The proof provided here is due to [3] and [19].
Proof. Since the Navier-Stokes flow is assumed to be stationary and has, if one assumes in advance no rotation of the fluid around its axis of symmetry, only two degrees of freedom, the continuity equation is identically fulfilled by finding a stream function ψ as introduced in [3], such that the velocity components in spherical coordinates are described by u(r) = 1/(r² sin θ) ∂ψ/∂θ, u(θ) = − 1/(r sin θ) ∂ψ/∂r, u(φ) = 0.
Summary of Chapters
1. Introduction: Presents the historical context of Lev D. Landau's 1944 solution to the stationary Navier-Stokes equations and introduces the mathematical framework and notation used throughout the paper.
2. Preliminaries: Establishes the fundamental fluid dynamics background, including the Transport Theorem, governing equations of mass and momentum conservation, and formal definitions of solution concepts.
3. Analysis of the classical case: Derives the classical Landau solution and provides a rigorous analysis of its properties, including the physical interpretation of the Landau parameter and its relation to fluid force.
4. Analysis of the modified Landau solution: Investigates the behavior of Landau solutions beyond the classical case, specifically focusing on critical and supercritical solutions that exhibit singular behavior at the origin or along a cone.
Keywords
Navier-Stokes equations, Landau solution, fluid dynamics, spherical coordinates, stationary flow, axisymmetry, stream function, distributional solutions, critical Landau solution, supercritical Landau solution, momentum flux, continuity equation, singularity analysis.
Frequently Asked Questions
What is the core subject of this thesis?
The work focuses on the mathematical analysis of the Landau solution, which is a specific exact solution to the stationary Navier-Stokes equations representing a fluid flow driven by a point force.
What are the primary themes discussed?
The thesis covers the derivation of the Landau solution, the classification of classical versus singular (modified) solutions, and the physical interpretation of these flows in the context of fluid jets.
What is the main objective of the research?
The primary objective is to determine if and how Landau solutions, particularly modified versions, can be extended to the whole space R³ and whether they provide a valid physical interpretation for fluid flow problems.
Which scientific methods are utilized?
The author employs analytical methods from fluid dynamics, including the use of stream functions in spherical coordinates, distribution theory, and the study of ordinary differential equations derived from the Navier-Stokes system.
What is covered in the main body?
The main body systematically derives the classical Landau solution, analyzes its properties, and extends the investigation to critical and supercritical regimes to test the limits of these mathematical models.
Which keywords best characterize this work?
Key terms include Navier-Stokes equations, Landau solution, axisymmetry, fluid dynamics, distributional solutions, and point singularities.
How does the Landau parameter influence the flow?
The Landau parameter is uniquely related to the force at the origin; a higher magnitude of the parameter corresponds to a stronger, faster, and narrower fluid stream.
Why are critical Landau solutions considered to be sinks?
The critical Landau solution exhibits behavior where streamlines tend toward the origin or a critical half-line, and calculations of the divergence show that the flow acts as a sink, complicating the physical interpretation in the whole space.
- Citar trabajo
- Jonas Sauer (Autor), 2010, Analysis of the Landau Solution, Múnich, GRIN Verlag, https://www.grin.com/document/179272
