Fluid mechanics and some novel surface waves

Part I

Baile Zhang

Nanyang Technological University, Singapore

1

Ahmad Falahatpisheh and Arash Kheradvar, University of California, Irvine

Bathtub drain vortex

Bathtub Vortex

3Feynman Lectures, Vol. 2

Curl-free bathtub vortex

2v ds rv C

Circulation

0v

2

Cv

r

4Kundu and Cohen, Fluid Mechanics, 3rd edition

2v ds rv C

Circulation

2

Cv

r >R

Bathtub VortexCurl-free bathtub vortex

0v

5Physics Today 62, 38 (2009)/AIP

Aharonov-Bohm effect

2v ds rv C

Circulation

2

Cv

r >R

Bathtub Vortex vs AB EffectCurl-free bathtub vortex

0v

6

Eur. J. Phys. 1, 154-162 (1980)

M. V. Berry

Bathtub Vortex vs AB Effect

wave

Wavefront dislocations

7

r

ek i A

c

In the continuumrp i pr i

rk i kr i

For a charge in EM field

“Minimal coupling”

In a lattice

kr i + ?

‘Vector Potential’ in k-Space

8

In the continuumkr i

( ) ( ) ik rr dk k e ( ) ( ) ik rr r dkr k e

( )[ ]ik r

kdk k i e ( ) ik r

kdki k e

kr i

‘Vector Potential’ in k-Space

Kai Sun, Univ. of Michigan

‘Vector Potential’ in k-Space

9

In a lattice

,( ) ( ) ( )n n k

n

r dk k r , ,( ) ( ) ik r

n k n kr u r e

,( ) ( ) ( )( )ik r

n n k k

n

r r dk k u r i e

, ,[ ( ) ( ) ( ) ( )] ik r

k n n k n k n k

n

dk i k u r k i u r e

' ' '

, ,[ ( ) ( ) ( ) ( ) ( )] ik r

k n n k n k n k

n

dk i k u r k dr r r i u r e

' * ' '

, , , ,[ ( ) ( ) ( ) ( ) ( ) ( )] ik r

k n n k n m k m k k n k

n m

dk i k u r k dr u r u r i u r e

, ,[ ] ( ) ( )k m n n n k

n m

dk i k r

' * ' '

, , ,( ) ( )m k k n k m ni dr u r u r Define

kr i + ?

( )

Kai Sun, Univ. of Michigan

10

, ,( ) [ ] ( ) ( )k m n n n k

n m

r r dk i k r ' * ' '

, , ,( ) ( )m n m k k n ki dr u r u r where

If only a single band is considered

, ,( ) [ ] ( ) ( )k n n n n kr r dk i k r

Sok nr i

Berry connection

r

ek i A

c Vector potential

, ,| |n n k k n ki u u

‘Vector Potential’ in k-Space

Kai Sun, Univ. of Michigan

Vector Potential vs ‘Vector Potential’ in k-Space

11

Berry connection

n

Vector potential

A

Magnetic field

B A

Berry curvature

n nk

Magnetic monopole

2M

cB ds n

e

Chern number

2BZ

dk C

12

Difficulty: It’s not a periodic system

Bloch theorem not applicable

A Lattice of Vortices?

13

1v r 1r rAt

0v

2r r

A Lattice of Confined Vortices

PRL 114, 114301 (2015)

At

14Kundu and Cohen, Fluid Mechanics, 3rd edition

22

1 12 1 1 22

1 2 2

1

1 ( / )

R Rv r

R R R r

When2 0

22

1 11 12

1 2 2

1

1 ( / )

R Rv r

R R R r

Circular Couette Flow

Analytic solution from Navier-Stokes equations

15

1tv v v p

Euler’s equation:

0v v v

1r r0( )tp v

0 02

1 1( ) ( ) 0t tv v

c

22 2

2

1 1( , ) ln ln

4 2V x y

c

Schrodinger-type

where

J. Acoust. Soc. Am. 87 (6), 2292 (1990).

2[( ) ( , )] 0effiA V x y

0n

0

2

( , )eff

v x yA

c

At0p p p

Master equation

Let

Continuity:

( ) 0t v

Acoustic Waves with ‘Magnetic Flux’

16

Haldane model, PRL 61, 2015-2018 (1988)

Complex next-nearest-neighbor (NNN) hopping

The first theoretical model of topological insulator beyond quantum Hall effect

Haldane Model with Staggered Magnetic Field

17

A

B A

Haldane Model with Staggered Magnetic Field

A-B reciprocal

A-A nonreciprocal

18

Acoustic Band Structure

Chern number

| |nk nkn kiA

1( )

2nn x y kBZ

C dk dk A

19

Topology and Topological Insulator

=

=

= =

Hole num.: 0 Hole num.: 1 Chern num.: 0 Chern num.: 1

=

20

Trivial insulator

C=0

C≠0Topological insulator

insulating

insulating

conductive

Topology and Topological Insulator

21

Topological Edge State

20X1 supercell

A strip

22

Source

One-way edge state

Topological Edge State

PRL 114, 114301 (2015)

23

Source

Defect 1: a cavity

Topological Edge State

PRL 114, 114301 (2015)

24

Source

Defect 2: a step

Topological Edge State

PRL 114, 114301 (2015)

25

Source

Defect 3: sharp corners

Topological Edge State

PRL 114, 114301 (2015)

26

Thanks!