3/8 Stability Concepts (6s)

3/8 Stability Concepts (6s)

Prof. Václav Uruba, Praha, CZ

Częstochowa, 23.-25.4.2014

vaclav@uruba.eu

Osborn Reynolds

1842-1912

Reynolds experiment

Osborn Reynolds 1883 Uni Manchester

Lyapunov Stability Concept

• Aleksandr Michailovich Liapunov 1857-1918

Lyapunov Stability Concept

• Autonomous Dynamical System:

– Lyapunov Stable:

– Asymptotically Stable: LS +

– Exponentially Stable: AS +

Equilibrium:

 

x  f x ^{x t}  ^{ D} _ ⁿ f x e  ⁰

     

0, 0 : x 0 x_e x t x_e , t 0

     

          

   

0 : 0 _e lim _e 0

x x t x t x

 

       

     

, , 0 : x 0 x x t x x 0 x e ^t, t 0

     ^

        

Lyapunov Stability Concept

Stability in FD

stable indifferent

instable linear

stable instable nonlinear

stable Convective instability

Absolute instability instability

btw (b), (d)

Instability

• Convective I.

– Free Shear Layer – Boundary Layer

• Absolute I.

– Jet

– Wake

Hydro-Stability

• Closed Flow

• Open Flow

– Free (inflextion)

– Wall-bounded (no inf.)

Advection!!!

Transition type

• SOFT – continuous

• HARD – discontinuous

Competition, intermittency Circ. pipe

• Linear model

• Exponential growth

• Micro >>> Macro

• NONLINEAR >>> disturbance size ???

Stability in FM

N-S equations

• Kinetic theory

• Quantum physics

Exponential growth

• Exponential law

• Geometrical growth

• Malthus’ growth model

• Growth rate is proportional to the instantaneous size

•

Exponential growth FD

• Linear model

• Disturbance growth

• Final disturbance

• Initial disturbance

a ^{- gain}

Example:

   

1 ˆ exp1

u t^  u i t

1  

u t

r i i

    _i>0

1 1

ˆ 1

u  a u

10 10

1 1

10 ˆ 10 a

u ^ u



 

1

 

du dt L u 

Comparison of growth laws

2^x x³

50x

0 5 10 15 20 25 30 35 40 0E+00

1E+01 2E+01 3E+01 4E+01 5E+01 6E+01 7E+01 8E+01 9E+01 1E+02

Time

Multiplicator

Graph of the exponential growth

100

2^x

Number of cycles

Graph of the exponential growth

2^x

0 5 10 15 20 25 30 35 40

0E+00 1E+03 2E+03 3E+03 4E+03 5E+03 6E+03 7E+03 8E+03 9E+03 1E+04

Time

Multiplicator

10 000

Graph of the exponential growth

2^x

0 5 10 15 20 25 30 35 40

0E+00 1E+05 2E+05 3E+05 4E+05 5E+05 6E+05 7E+05 8E+05 9E+05 1E+06

Time

Multiplicator

1 000 000

Number of cycles

Graph of the exponential growth

2^x

0 5 10 15 20 25 30 35 40

0E+00 1E+07 2E+07 3E+07 4E+07 5E+07 6E+07 7E+07 8E+07 9E+07 1E+08

Time

Multiplicator

100 000 000

Graph of the exponential growth

2^x

0 5 10 15 20 25 30 35 40

0E+00 1E+09 2E+09 3E+09 4E+09 5E+09 6E+09 7E+09 8E+09 9E+09 1E+10

Time

Multiplicator

10 000 000 000

Number of cycles

Graph of the exponential growth

2^x

0 5 10 15 20 25 30 35 40

0E+00 1E+11 2E+11 3E+11 4E+11 5E+11 6E+11 7E+11 8E+11 9E+11 1E+12

Time

Multiplicator

1 000 000 000 000

Escherichia coli

• Rod shape

• Approx. 2 μm length, 0.5 μm dia

• Volume 0.6 (μm)³

• Weight 7 10^-7 μg

• 1.4 billions in 1 mg

Reproduction

• Division - mitosis

• One generation period 9.8 mins

Escherichia Coli population theory – math.

• t = 0 7 10^-7 mg

• 3h 20m (21) 1 mg

• 5h (31) 1 mg

• 6h 40m (41) 1 g

• 8h 15m (51) 1 kg

• 9h 50m (60) 1t = 10³ kg

• 11h 30m (70) 1000 t = 10⁶ kg

• 13h 10m (80) 1million t = 10⁹ kg

• 20h 30m (126) 7 10²² kg (Moon)

• 21h 40m (133) 6 10²⁴ kg (Earth)

• 24h (151) 2 10³⁰ kg (Sun)

Escherichia Coli population reality – nature

• 4 phs of pop.evol.

1. Preparation

2. Exponential growth 3. Stationary ph. –

saturation (run out resources, toxic wastes)

4. Death ph.

(resources

LINEAR

NONLINEAR

Messages

1. There is no too small perturbation 2. Linear model has limitations

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