ARNOLD STABILITY of TIME-OSCILLATING FLOWS Legacy of Vladimir - PowerPoint PPT Presentation

ARNOLD STABILITY of TIME-OSCILLATING FLOWS Legacy of Vladimir Arnold Fields Institute, November, 2014 Prof. V. A. Vladimirov University of York University of Cambridge Sultan Qaboos University Novosibirsk State University ... December 2, 2014 Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

I like this great photo of Vladimir Igorevich Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

SUMMARY: Slide 1 ◮ Oscillating flows appear in various applications : geophysics, coastal engineering, self-swimming, medicine, machinery ... One can say that oscillating flows are the most important in applied hydrodynamics ... ◮ The flow oscillations can be caused by various factors such as oscillating boundaries, surface waves, acoustic waves, MHD waves, etc. ◮ Our aim is to present asymptotic/averaging models for oscillating fluid flows with the use of the multi-scale (two-timing) method . Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

SUMMARY: Slide 2 ◮ We consider relatively ‘weak’ averaged flows , interacting with the flow oscillations. ◮ The distinctive property of the averaged flows is: they possess an additional advection with the drift velocity . ◮ All our consideration is Eulerian. The drift velocity is Lagrangian characteristic of a flow, however in our consideration it naturally appears in an Eulerian procedure. ◮ The relations to the Stokes drift , Langmuir circulations , acoustics, and MHD dynamo are discussed. Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

SUMMARY: Slide 3 ◮ Our models represent examples of Hamiltonian systems and interesting areas of exploiting of Arnold’s ideas in Hydrodynamics. ◮ The averaged equations and boundary conditions possess the energy-type integral , which allows us to consider some ‘energy-related’ results. ◮ We have derived a number of results such as the energy variational principle , the second variation of energy , and some Arnold-type stability criteria for averaged flows. Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Oscillating Flows in bio-applications: Slide 4A Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Oscillating Flows in med-applications: Slide 4B Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Oscillating Flows in turbine-applications: Slide 4C Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Drift motion brings impressive income: Slide 4D Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Langmuir Circulations in a lake: Slide 4E Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

3D Vortex Dynamics in Oscillating Domain Slide 5 A homogeneous inviscid incompressible fluid in a 3D domain Q with oscillating boundary ∂ Q . Velocity u † = u † ( x † , t † ), vorticity ω † ≡ ∇ † × u † ∂ ω † ∂ t † + [ ω † , u † ] † = 0 , ∇ † · u † = 0 where ‘dags’ mark dimensional variables, and square brackets stand for the commutator. The boundary condition at ∂ Q is dF † / dt † = 0 F † ( x † , t † ) = 0 at The characteristic scales of velocity, length, and two additional time-scales T † T † U † , L † , fast , slow Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Vortex Dynamics in 3D Oscillating Domain Slide 6 ∂Q ( t ) n 0 n ( t ) vortex flow u ( x , t ) Q ( t ) ∂Q 0 Oscillating flow domain Q ( t ) Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Scaling parameters: Slide 7 Two independent dimensionless parameters T † ≡ L † / U † T fast ≡ T † T slow ≡ T † fast / T † , slow / T † , where T fast – the given period of oscillations, the frequency of oscillations σ † ≡ 1 / T † σ ≡ T † / T † fast , fast The dimensionless independent variables x ≡ x † / L † , t ≡ t † / T † The dimensionless ‘fast time’ τ and ‘slow time’ s : S ≡ T † / T † τ ≡ t / T fast = σ t , s ≡ t / T slow ≡ St , slow Attention! T † slow is NOT given! It is unknown! Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Oscillating velocity Slide 8 We consider the oscillatory solutions u † = AU † u ( x , s , τ ) where τ -dependence is 2 π -periodic, s -dependence is general, A – the dimensionless amplitude of velocity. f f = f ( s ) + � f ( s , τ ) t Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Scaling Slide 9 Dimensionless variables and the chain rule give � ∂ � ∂τ + S ∂ ω + A σ [ ω , u ] = 0 σ ∂ s where s and τ are still mutually dependent variables. An auxiliary assumption: we operate with s and τ as mutually independent variables ; justification of it often can be given a posteriori by the estimation of the errors/residuals in the equation, rewritten back to the original variable t . Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Two independent small parameters Slide 10 ◮ In the two-timing method the basic small parameter is T slow / T fast = S /σ ◮ The term ∂ ω /∂τ must be dominating, in order to form an evolution equation. Hence, generally, we take two independent small parameters ε 1 , ε 2 as: ω τ + ε 1 ω s + ε 2 [ ω , u ] = 0; ε 1 ≡ S σ ≪ 1 , ε 2 ≡ A σ ≤ 1 • ε 1 is ratio of two characteristic time scales; • ε 2 is the ratio of amplitude over frequency. Note: the amplitude itself can be huge! ◮ Asymptotic solutions correspond to the limit ( ε 1 , ε 2 ) → (0 , 0). Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Distinguished Limits Slide 11 ◮ There are infinitely many asymptotic pathes ( ε 1 , ε 2 ) → (0 , 0). QUESTION: Is the number of different asymptotic solutions also infinite? ◮ We accept that the distinguished limit is given by such a path ( ε 1 , ε 2 ) → (0 , 0) that allows us to build a self-consistent asymptotic procedure, leading to the finite/valid solution in any approximation. ◮ ANSWER: By the method of trial and errors one can find that there are only two pathes, which allow to build such solutions: ε 1 = ε 2 ≡ ε : ω τ + ε ω s + ε [ ω , u ] = 0 2 ≡ ε 2 : ω τ + ε 2 ω s + ε [ ω , u ] = 0 ε 1 = ε 2 The second case leads to the Weak Vortex Dynamics (WVD). ◮ Any systematic procedure of finding all possible distinguished limits is unknown: it can be classified as an experimental mathematics (Arnold). This is why pure mathematicians do not like this research area. Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS

Notations Slide 12 Any function f = f ( x , s , τ ) in this paper is: • f = O (1) and all x -, s -, and τ -derivatives of f are also O (1). • f ( x , s , τ ) = f ( x , s , τ + 2 π ) • The averaging operation is � τ 0 +2 π � f � ≡ 1 f ( x , s , τ ) d τ, ∀ τ 0 2 π τ 0 • The tilde-functions (or purely oscillating functions) is such that � f ( x , s , τ ) = � � � f ( x , s , τ + 2 π ) , with f � = 0 , • The class of bar-functions is defined as f : f τ ≡ 0 , f ( x , s ) = � f ( x , s ) � • The tilde-integration keeps the result in the tilde-class: � τ � 2 π �� µ � f ( x , s , σ ) d σ − 1 f τ ≡ � � � f ( x , s , σ ) d σ d µ. 2 π 0 0 0 Prof. V. A. Vladimirov[2mm] University of York University of Cambridge Sultan Qaboos University Novosibirsk State University... ARNOLD STABILITY of [3mm] TIME-OSCILLATING FLOWS