Analysis of the Landau Solution

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Analysis of the Landau Solution

Vorgelegte Bachelor-Thesis von Jonas Sauer

1. Gutachten: Prof. Dr. Reinhard Farwig 2. Gutachten: Prof. Dr. Dr. h.c. Hans-Dieter Alber

Tag der Einreichung:

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Very special thanks to Prof. Dr. Farwig from the Technische Universität Darmstadt, who sparked my interest in the topic of this present paper and who provided a lot of helpful corrections and hints. I also want to thank Prof. RNDr. Josef Málek, CSc., DSc. from Univerzita Karlova v Praze for patronising me during my stay in Prague.

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1 Introduction

In 1944 Lev D. Landau [15] found a non-trivial solution to a stationary Navier-Stokes flow on ³ \{0}, which was symmetric around some axis and fulfilled the condition, that the velocity decayed linearly and the pressure quadratically in |x|. If the axis of symmetry is the x 3 -axis, the solution is of the form

where d ∈ (−1, 1). We note here, that Squire [19] found this solution independently in 1951. 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 ³ \{0}. 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 ³ . This can be done if one admits also weak solutions in the space of distributions which satisfy

where b is a vector and δ is the Dirac distribution.

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| ≥ 1. Therefore we first derive Landau’s solution with the assumptions posed in the beginning, following an ansatz by Batchelor [3]. 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 {x ∈ ³ : x 1 = x 2 = 0, ±x 3 ≥ 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 ³ . In any case, one can consider modified Landau solutions to be solutions in certain subsets of ³ , 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.

In this paper we will use some common notation. Ω usually denotes an open subset of ³ . For the space of k-times continuously differentiable functions from Ω to n we write C k (Ω, n ). Furthermore we set C k (Ω, ) = C k (Ω) and analogously for the other spaces introduced here. We will also not distinguish

³ and C k (Ω). The space of infinitely often differentiable functions with compact sup-

between

port, i.e. the test functions, will be denoted by C ∞

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(Ω) is the space of distributions. L p (Ω) is the Lebesgue space of p-integrable functions and W k,p (Ω) is the Sobolev space of k-times weakly differentiable p-integrable functions, whose derivatives are also p- integrable.The subscript loc indicates, that the functions need to be p-integrable only on every compact subset of Ω. A ball around the origin with radius R will be denoted by B R . Due to the concepts dealt with in this paper, we will also use some notation from fluid dynamics. The tensor product of two vectors a and b is the 3 × 3-matrix a ⊗ b := (a i b j ) i j . The Cauchy stress tensor will be denoted by T = −p + ∇u +(∇u) T . Note that the momentum flux density tensor is given by u ⊗ u − T . The reader should note, that throughout this paper we will work without explicitly taking the dimension of physical quantities into account. Furthermore we set for the density of the fluid ρ = 1. Except for the solution u and the position vector x, vectors in ³ and vector-valued functions will be denoted by a bold face letter.

When speaking about distributions, the angled brackets denote the action of a distribution on a test

f , ϕ means, that the distribution f is acting on the test function ϕ. Unless explicitly said function, i.e.

otherwise, the symbol δ will always represent the Dirac distribution.

In the appendix the reader may find some basic theorems used in this paper. We usually refrain from referring explicitly to the theorems provided in the appendix, as they are mostly well-known and only meant for the convenience of the reader.

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2 Preliminaries

As we will be concerned with different kinds of solutions to a Navier-Stokes system, we first have to introduce the setup we are working with.

Following the usual procedures in fluid dynamics, we will work with quantities such as the velocity u or the pressure p in order to describe certain flows. These quantities are functions of space and time, that is their domain is

:= {(x, t) ∈ Ω t × (0, T )} ⊂ ⁴ , 0 < T < ∞ (2.1)

where Ω t is the (open) domain occupied by the fluid at time t. We will assume, that is open. A sufficient condition for this is, that Ω t changes continuously in time. In fact, we will be only considering the situation, where Ω t = Ω will be constant. Nevertheless, in this section we will state the main results with explicitly taking the change in time into account.

2.1 The Transport Theorem

Our mathematical model will treat fluids as a continuum, and our fundamental hypothesis is, that exactly one fluid particle passes through any point x ∈ Ω at any time t ∈ (0, T ).

That gives rise to some results laid out in this section. The first is the Transport Theorem, which is of utmost importance. Let us consider a system of fluid particles in a bounded domain (t) ⊂ Ω t at a time t. It is well known, that for given velocity u ∈ C ¹ ( ), the initial value problem

∂ ϕ has a unique maximal solution x(t) = ϕ(X , t 0 ; t), where ϕ has continuous first derivatives and both

∂ t 0

∂ ϕ have continuous first spatial derivatives ([14], theorem 10.1.1, 11.1.5, 13.1.1). If we fix t 0 , ϕ and

∂ t

defines the change of the domain (t) with time

(t) = {ϕ(X , t 0 ; t) : X ∈ ∈ (t 0 )}. (2.3)

We are now ready to state the assertion of the Transport Theorem.

Transport Theorem. Let t 0 ∈ (0, T ) and (t 0 ) be a bounded domain with (t 0 ) ⊂ Ω t 0 . Let ϕ : (t 0 ) → (t) be a continuously differentiable and bijective map (defining the change of (t 0 ) with time) with continuous and bounded Jacobian determinant J, which satisfies J(X , t) > 0 for all X ∈ ∈ (t 0 ) and for all t in an interval (t 1 , t 2 ) ∈ (0, T ) containing t 0 . Let F : → have continuous and bounded first derivatives on the set {(x, t) : t ∈ (t 1 , t 2 ), x ∈ ∈ (t)}.

Then we have for all t ∈ (t 1 , t 2 ) that there exists the finite derivative

∂ F

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Proof. Observe, that there exists an interval (t 1 , t 2 ) containing t 0 , such that ϕ and J satisfy the conditions of the theorem, see for example [10]. Then, by the Change of Variable Formula, we get

By a straight forward calculation which can be found in [9], we see that ∂ J (X , t) = J(X , t) div u(X , t).

∂ t

If one calls the integrand appearing in the right integral F (X , t), we see that for all t it holds that F is measurable as a function of X and that it has finite derivatives for almost every X ∈ ∈ (t 0 ). Furthermore, the time derivatives of F are bounded by an integrable majorizing function g and there exists t ∗ ∈ (t 1 , t 2 )

such that F (X , t ∗ ) is integrable. We therefore get by the theorem on differentiation of an integral with
respect to one parameter ⎛ ⎛ ⎞ ⎞

(2.10)

Please observe, that we omitted the arguments here and that we used the well-known relation ∇F · u + F div u = div (Fu).

Using this theorem, we can derive the mathematical formulation of the fundamental physical laws of conservation, such as the conservation of mass or the conservation of momentum, in differential form, known as the governing equations of fluid dynamics.

2.2 Governing equations of fluid dynamics

Let ρ : → (0, ∞) denote the density of the fluid. Then by the conservation law of mass and by the Transport Theorem, we get ∂ ρ

for all control volumes (t) ⊂ (t) ⊂ Ω t . By the continuity of the integrand we thus conclude

This equation is known as the continuity equation. In the steady-state situation, that is, if all time derivatives vanish, this simplifies to div (ρu) = 0. Throughout this paper we will only be concerned with the case, where the density is constant also with respect to the spatial components, i.e. with incompressible flows. Obviously, in that case the conservation law of mass is turned into the statement

div u = 0. (2.13)

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So we see, that there is no flux of mass across any closed surface for incompressible fluids, or in other words: At any time and for any control volume, the inward flow equals the outward flow. In the same manner one can examine the conservation law of momentum. Recall that the force is defined as the time derivative of the momentum. Thus, for the total force F acting on a domain (t) ∈ Ω t , we have the relation

But on the other hand, for Newtonian fluids we can decompose the total force F into the volume force and the surface force by virtue of its density f and the Cauchy stress tensor T

Combining these two expressions for the total force and using Green’s theorem will yield the equations of motion

∇u + (∇u) T .

Observe, that we have the two relations

∂ u

∇u + (∇u) T = Δu + ∇div u, ∂ ρu

+ div (ρu ⊗ u) = ρ + (u · ∇)u . div (2.18)

∂ t ∂ t

We use here the convention

This leads in the stationary case, i.e. div u = 0 to the Navier-Stokes equation

−Δu + ∇p + u · ∇u = f . (2.20)

Usually we refer to the system of equations

div u = 0,

(2.21)

−Δu + ∇p + u · ∇u = f ,

as the Navier-Stokes system with force f .

2.3 Concepts of a solution

In the classical work frame, we are usually interested in solutions to partial differential equations, which are at least as smooth as the degree of the equation. This concept of a solution turns out to be not suitable for certain applications. There we have to use a more general definition of a solution to a partial differential equation, which will allow us to give a physical interpretation to certain equations. First let us state the classical definition.

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Definition 2.3.1. Let n, m ∈ and let Ω ⊂ n . Then we call u : Ω → m a classical solution on Ω to the partial differential equation

ξ, {D α ψ(ξ) : |α| ≤ k} = 0,

(2.22) F

where ξ is an independent variable, if u ∈ C k (Ω) and for all x ∈ Ω we have F (x, {D α u(x) : |α| ≤ k}) = 0.

As already mentioned, this concept of a solution to partial differential equations is not always an appropriate approach. There are various concepts of solutions, but we will state here two definitions, which we will be needing when analysing the behaviour of the Landau solution.

Definition 2.3.2. Let n ∈ and let Ω ⊂ n and let L be a linear differential operator. Then we call u ∈ (Ω) a weak solution in the sense of distributions or short weak solution on Ω to the partial differential equation

Lψ = 0, (2.23)

if for all test functions ϕ ∈ C ∞ 0 (Ω) it holds true that

= 0.

Lu, ϕ (2.24)

This definition extends in an obvious way to solutions to partial differential equations in multiple functions.

Observe, that for distributions we have the definition

function f is in L 1

loc

Especially for the case of the Navier-Stokes system, we may consider the concept of a very weak solution

in the following sense. For this definition we have to recall, that if Ω ⊂ ³ is bounded, then L ¹ (Ω) ⊃

loc L ² loc (Ω).

Definition 2.3.3. Let f ∈ ∈ (Ω), where Ω is a finite region in ³ . Then u : Ω → ³ is a very weak solution on Ω to the Navier-Stokes system (2.21) with force f , if u ∈ L ² loc (Ω) and it holds true that u

is weakly divergence-free in the sense of distributions as defined above and for all divergence-free test

functions ϕ ∈ C ∞ 0 (Ω, ³ ) we have

Note that due the special properties of ϕ, namely div ϕ = 0, we do not have to take the pressure p of the original Navier-Stokes equation into account.

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3 Analysis of the classical case