MY PUBLICATIONS I 1

Jitender Singh and Renu Bajaj, Couette flow in ferrofluids with magnetic field. J. Magn. Magn. Mat. 294 (2005), 53-62.

2

Jitender Singh and Renu Bajaj, Stability of nonaxisymmetric ferrofluid flow in rotating cylinders with magnetic field. Int. J. Math. Math. Sci. 23 (2005) 3727-3737.

3

Jitender Singh and Renu Bajaj, Stability of ferrofluid flow in rotating porous cylinders with radial flow. Magnetohydrodynamics. 42, (2006), 41-56.

4

Jitender Singh and Renu Bajaj, Nonaxisymmetric modes of Couette-Taylor instability in ferrofluids with radial flow. Magnetohydrodynamics. 42, (2006), 57-68.

5

Jitender Singh and Renu Bajaj, Parametric Modulation in the Taylor-Couette Ferrofluid flow. Fluid Dyn. Res.(IOP). 40, (2008), 737-752.

6

Jitender Singh and Renu Bajaj, Thermal Modulation in Rayleigh-Benard Convection. The ANZIAM J. 50, 2, (2008), 231-245.

7

Jitender Singh, Defining Power sums of n and ϕ(n) Integers. Int. Jour. Num. Th. 5, (1), (2009), 41-53.

8

Jitender Singh and Renu Bajaj, Temperature Modulation in Ferrofluid Convection. Phys. Fluids (AIP). 21, 6, (2009), 0641051-12.

9

Jitender Singh and Renu Bajaj, Convective instability in a Ferrofluid layer with temperature modulated rigid boundaries. (2010). (Under Review)

10 Jitender Singh, On Energy-Relaxation for Transient Convection in Ferrofluids (2010). (Under Review)

© SINGH & HINES (GNDU, ISEL)

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Suspended Sediments on Onset of Alternate River-Bars

J. Singh1 , E. Hines2 1 Department

of Mathematics, Guru Nanak Dev University, Amritsar Email: [email protected] 2 School of Engineering, University of Warwick Email: [email protected]

July–2010

© SINGH & HINES (GNDU, ISEL)

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Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Introduction

Introduction

River flow provides a fascinating phenomenon where an interaction between fluid and its container determines the shape of latter, the cause being noncohesive character of the river bed.

$ O(sediment-size) −→ rippled beds * ale c o-s icr m Flow-Bottom  macro-scale - O(flow-depth) −→ dunes/antidunes me HH gaInteraction sca l HH e j O(channel-width) −→ river-bars H & % '

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

3 / 17

Finally, patterns mayIntroduction be recognized in various characteristics of the channel boundary. The instability of the bed interface affects bed elevation (bed forms). Similarly, the straight alignment of fluvial channels is typically unstable to planform perturbations arising from a complex interaction between outer-bank erosion (a mechanism whereby the floodplain loses sediment to A BRIEF DESCRIPTION the channel) andRev. inner-bank deposition (a mechanism of floodplain reconstruction). As a result, Seminara, Fluvial Sedimentary Patterns. Annu. Fluid Mech. 2010. 42:43-66 planform patterns develop, either building up channel sinuosity (meandering) (Figure 2a) or

a

b

2081

Figure 1 (a) Erosional pattern: landscape near Orland, California, with rhythmic sequences of valleys spaced roughly 100 m apart. Figure courtesy of J. Kirchner. (b) Depositional pattern: the Wax Lake fluvial delta, Louisiana. Red lines show predictions (Parker & Sequeiros 2006) of its progradation in time. Figure courtesy of G. Parker. 44

Seminara

© SINGH & HINES (GNDU, ISEL)

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July–2010

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RV400-FL42-03

ARI

13 November 2009

11:49

Introduction

A BRIEF DESCRIPTION Seminara, Fluvial Sedimentary Patterns. Annu. Rev. Fluid Mech. 2010. 42:43-66

a

b

Figure 2 (a) Forced (steady) bar at the inner bend of a meander very close to neck cutoff, a process occurring when two meander branches merge, cutting the meander loop. Historical bend development can be traced by the sequence of so-called scroll bars showing up in the flood plain adjacent to the inner bend. (b) Multiple row (migrating) bars in the Waikariri River, a braiding river of New Zealand (courtesy of B. Federici).

network of curved channelsRiver–Bars (braiding) (Figure 2b). Finally, so-called ©forming SINGHan & interconnected HINES (GNDU, ISEL)

July–2010

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Introduction

A BRIEF DESCRIPTION

a

b

2 1 0 2

n 1 0 2 1 0

se only.

0

©

50

100

150

way to like un size tha stream As opp riffle p migrate which Froude into a step is a bould 200 betwee

downst result, of the (a) 5Alternate Bars in Lab.: W. Bertoldi (b)3.4OoiView River, Japan, Figure of theFrom OoiIkeda(2001) River, Japan, Figure 3.6 sho showing sorting of gravel and sand on of a sed (a) Alternate bars observed in the laboratory. Figure courtesy bars. From Ikeda (2001). of W. Bertoldi. (b) Num SINGH & HINES (GNDU, ISEL) River–Bars July–2010 6 / 17 the development of alternate bars show the convective nature of bar instability. Persis

Introduction

PRELIMINARY CONSIDERATIONS

A BRIEF DESCRIPTION z w

y

V

x

U0

U0

U

H0

H

D0*

D*

2B*

2B*

Initially Flat Bed Surface Cohesionless erodible bed

δ

(c)

1

Cohesionless erodible bed

δ

(d)

A turbulent flow in a rectangular channel of width 2B∗ , and flow depth D∗ over an initially flat erodible cohesionless sediment bed.

2

River banks are assumed to be fixed and non-erodible.

3

Both bed-load transport and suspended load transport are considered.

4

Non-homogeneity of the sediment is taken into considerations.

5

Variable size distribution of grains is also considered.

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

7 / 17

Introduction

PRELIMINARY CONSIDERATIONS

A BRIEF DESCRIPTION z w

y

V

x

U0

U0

U

H0

H

D0*

D*

2B*

2B*

Initially Flat Bed Surface Cohesionless erodible bed

δ

(e)

1

Cohesionless erodible bed

δ

(f)

A turbulent flow in a rectangular channel of width 2B∗ , and flow depth D∗ over an initially flat erodible cohesionless sediment bed.

2

River banks are assumed to be fixed and non-erodible.

3

Both bed-load transport and suspended load transport are considered.

4

Non-homogeneity of the sediment is taken into considerations.

5

Variable size distribution of grains is also considered.

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

7 / 17

Introduction

PRELIMINARY CONSIDERATIONS

A BRIEF DESCRIPTION z w

y

V

x

U0

U0

U

H0

H

D0*

D*

2B*

2B*

Initially Flat Bed Surface Cohesionless erodible bed

δ

(g) 1

Cohesionless erodible bed

δ

(h)

A turbulent flow in a rectangular channel of width 2B∗ , and flow depth D∗ over an initially flat erodible cohesionless sediment bed.

2

River banks are assumed to be fixed and non-erodible.

3

Both bed-load transport and suspended load transport are considered.

4

Non-homogeneity of the sediment is taken into considerations.

5

Variable size distribution of grains is also considered.

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

7 / 17

Introduction

PRELIMINARY CONSIDERATIONS

A BRIEF DESCRIPTION z w

y

V

x

U0

U0

U

H0

H

D0*

D*

2B*

2B*

Initially Flat Bed Surface Cohesionless erodible bed

δ

(i) 1

Cohesionless erodible bed

δ

(j)

A turbulent flow in a rectangular channel of width 2B∗ , and flow depth D∗ over an initially flat erodible cohesionless sediment bed.

2

River banks are assumed to be fixed and non-erodible.

3

Both bed-load transport and suspended load transport are considered.

4

Non-homogeneity of the sediment is taken into considerations.

5

Variable size distribution of grains is also considered.

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

7 / 17

Introduction

PRELIMINARY CONSIDERATIONS

A BRIEF DESCRIPTION z w

y

V

x

U0

U0

U

H0

H

D0*

D*

2B*

2B*

Initially Flat Bed Surface Cohesionless erodible bed

δ

(k)

1

Cohesionless erodible bed

δ

(l)

A turbulent flow in a rectangular channel of width 2B∗ , and flow depth D∗ over an initially flat erodible cohesionless sediment bed.

2

River banks are assumed to be fixed and non-erodible.

3

Both bed-load transport and suspended load transport are considered.

4

Non-homogeneity of the sediment is taken into considerations.

5

Variable size distribution of grains is also considered.

© SINGH & HINES (GNDU, ISEL)

River–Bars

July–2010

7 / 17

Introduction

PRELIMINARY CONSIDERATIONS

The Governing Equations

f

U

∂U ∂H τx ∂U +V =− −β ∂x ∂y ∂x D

(1)

U

τy ∂V ∂V ∂H +V =− −β ∂x ∂y ∂y D

(2)

∂ ∂ (UD) + (VD) = 0 ∂x ∂y

(3)

∂ 2 ∂f ∂ ∂ (Fr H − D) + La + {f (Qx + qx )} + {f (Qy + qy )} = 0 ∂t ∂x ∂x ∂y

(4)

V = Qy = qy = 0 for y = ±1.

(5)

© SINGH & HINES (GNDU, ISEL)

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Introduction

PRELIMINARY CONSIDERATIONS

Abbreviations and Definitions

Dimensionless scales: Velocity Field Scale: Flow depth scale: Free surface elevation: Time scale: Horizontal distance scale: Bottom shear stress scale:

© SINGH & HINES (GNDU, ISEL)

U0∗ D∗0 D∗0 ∗ 3/2 (1 − p)D∗0 B∗ /Q∗0 , Q∗0 := (ρs /ρ − 1)1/2 dg0 B∗ ρU0∗ 2

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Introduction

PRELIMINARY CONSIDERATIONS

Abbreviations I

(U, V) pD Fr (= U0∗ / g(1 − p)D∗0 ) β √ (τx , τy ) = (U, V) U 2 + V 2 C C = {6 + 2.5 ln(D/ks )}−2 dg (= 2−φg ) ks (= 2dσ ) La (Qx , Qy ) Φ

© SINGH & HINES (GNDU, ISEL)

:= Velocity Field along (x, y) direction := Flow depth := Froude Number := B∗ /D∗0 := Bottom shear stresses := Friction coefficient := dimensionless grain size := grain roughness height := Active layer thickness := Bed-load vector := |(Qx , Qy )| River–Bars

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Introduction

PRELIMINARY CONSIDERATIONS

Abbreviations II

(qx , qy ) p f f0 (φ) Zdg0 ∞

φg := f φdφ Z ∞ −∞ σ 2 := (φ − φg )2 fdφ

:= suspended load vector := Porosity of the sediment := p.d.f. for the grain size distribution := initial p.d.f. (d = 2−φ ) := 2−φg0 := geometric mean phi size Z ∞ := variance of φ; f (φ, x, y, t)dφ = 1

−∞

dσ (= dg 2σ ) Φ ∗ ) Θg = τ ∗ /(ρ∆gdg0 G(ζ)

© SINGH & HINES (GNDU, ISEL)

−∞

:= Coarse grain size 3/2 := Θg G(ζ) := Shields Parameter (Bed Shear Stress) := Transport Capacity fn (Empirically found)

River–Bars

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Introduction

PRELIMINARY CONSIDERATIONS

Basic State and Analysis I

Suspended Load Transport: Neglect the vertical variation of eddy viscosity . C¯:= Sediment concentration, w := Settling velocity. ∂C ∂C =w + ∇2 C ∂t ∂z

Diffusion Equation : The Basic State:

(U, V, D, H, Qx , Qy , τx , τy , f , qx , qy ) := (1, 0, 1, H0 , Φ0 , 0, C0 , 0, f0 , qx0 , 0) Z z ¯ 0 C¯dz qx0 := U a

C¯ = C¯a Einstein (1950) : © SINGH & HINES (GNDU, ISEL)



D−z a D z−a qx0



w 0.4U 0 f

n wz o = C¯0 exp −      30D 0 ¯ = 11.6Uf Ca I1 log + I2 2σ0 +1 dg0 River–Bars

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Introduction

PRELIMINARY CONSIDERATIONS

Perturbations I (U, V, D, H, Qx , Qy , τx , τy , f , qx , qy , σ, φg ) = (1, 0, 1, H0 , Φ0 , 0, C0 , 0, f0 , qx0 , 0, σ0 , φg0 ) +h(U1 , V1 , D1 , H1 , Qx1 , Qy1 , τx1 , τy1 , f1 , qx1 , qy1 , σ1 , φg1 ) + o(h2 ) 2βC0 U1 +

∂U1 ∂H1 + + β(CD − 1)C0 D1 + βC0 tσ (σ1 − φg1 ) = 0 ∂x ∂x βC0 V1 +

∂V1 ∂H1 + =0 ∂x ∂y

∂V1 ∂D1 ∂U1 + + =0 ∂x ∂y ∂x   ∂U1 ∂ ∂ ∂D1 ∂V1 2Φ0 qη f0 + f0 − Φ0 R (Fr2 H1 − D1 ) + f0 Φ0 CD qη + f0 Φ0 + ∂x ∂t ∂y ∂x ∂y     ∂φg1 ∂qy1 ∂σ1 ∂ ∂ ∂qx1 f0 Φ0 qφ + f0 Φ0 qσ + La0 + (Φ0 + qs0 ) f1 + f0 + =0 ∂x ∂x ∂t ∂x ∂x ∂y (6) © SINGH & HINES (GNDU, ISEL)

River–Bars

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Introduction

PRELIMINARY CONSIDERATIONS

Normal Mode Solution I

(U1 , D1 , H1 ) = (U1 , D1 , H1 )(t) sin(yπ/2) exp {ιαx}, V1 = V1 (t) cos(yπ/2) exp {ιαx} (f1 , σ1 , φg1 ) = (f1 , σ1 , φg1 ) sin(yπ/2) exp {ιαx} Fr2 H1 − D1 = η1 The System of ODEs: df1 dη1 + La0 = Γ1 η1 + Γ2 f1 + Γ3 φg1 + Γ4 σ1 (7) ∂t dt Z +∞ Z +∞ Z +∞ Z +∞ dη1 = η1 Γ1 dφ+ Γ2 f1 dφ+φg1 Γ3 dφ+σ1 Γ4 dφ (8) dt −∞ −∞ −∞ −∞ Z +∞ φg1 = φf1 dφ (9) f0

−∞ © SINGH & HINES (GNDU, ISEL)

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Introduction

PRELIMINARY CONSIDERATIONS

Normal Mode Solution II

1 σ1 = 2σ0

Z

+∞

−∞

(φ − φg0 )2 f1 dφ

(10)

Sediment-Concentration C = C¯ + hC1 + o(h2 ). ∂C1 ∂C1 ∂C1 + U0 =w + ∇2 C1 ∂t ∂x ∂z C1 = ϕ(z, t) sin(yπ/2) exp {ιαs} 

∂C1 + wC1 = 0 at z = 1 ∂z

Second boundary condition for C1 ?

© SINGH & HINES (GNDU, ISEL)

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Introduction

PRELIMINARY CONSIDERATIONS

OUTCOME I

An instability analysis of the basic state so obtained would be carried out and possible effect of the suspended sediment transport and other parameters on the formation of Alternate Bars in straight alluvial channels, will be obtained in terms of the control parameter β. The analysis would involve (a) Intelligent system approach (Professor Hines) and (b) Mathematical approach (Dr. Jitender Singh).

© SINGH & HINES (GNDU, ISEL)

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Introduction

REFERENCES

REFERENCES

1 S. Lanzoni and M. Tubino, Grain sorting and bar instability, J. Fluid Mech., 393,(1999) 149-174. 2 G. Seminara and M. Tubino, Sand bars in tidal channels. Part 1. Free bars, J. Fluid Mech., 440,(2001) 49-74. 3 J. N. Hunt, The turbulent transport of suspended sediment in open channels, Proc. Royal Soc. Lond., 224,1158, (1954), 322-335. 4 J. A. Zyserman and J. Freds∅e, Data analysis of bed concentration of suspended sediment, J. Hyd. Engg., 120,9, (1994), 1021-1042. 5 J. Freds∅e, Meandering and braiding of rivers, J. Fluid Mech., 84,4, (1978) 609-624. 6 G. Seminara, Fluvial Sedimentary Patterns. Annu. Rev. Fluid Mech. 2010. 42:43-66. 7 G. Parker, Surface-based bedload transport relation for gravel rivers. Jour. Hyd. Res. 1990. 28: 4, 417 - 436. © SINGH & HINES (GNDU, ISEL)

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