##### Document Text Contents

Page 1

Advances in Mechanics and Mathematics 40

Aleksandra A. Bozhko

Sergey A. Suslov

Convection in

Ferro-Nanofluids:

Experiments and

Theory

Physical Mechanisms, Flow Patterns,

and Heat Transfer

Page 2

Advances in Mechanics and Mathematics

Volume 40

Series Editors

David Gao, Federation University Australia

Tudor Ratiu, Shanghai Jiao Tong University

Advisory Board

Antony Bloch, University of Michigan

John Gough, Aberystwyth University

Darryl D. Holm, Imperial College London

Peter Olver, University of Minnesota

Juan-Pablo Ortega, University of St. Gallen

Genevieve Raugel, CNRS and University Paris-Sud

Jan Philip Solovej, University of Copenhagen

Michael Zgurovsky, Igor Sikorsky Kyiv Polytechnic Institute

Jun Zhang, University of Michigan

Enrique Zuazua, Universidad Autónoma de Madrid and DeustoTech

Kenneth C. Land, Duke University

More information about this series at http://www.springer.com/series/5613

http://www.springer.com/series/5613

Page 139

GR18.eps

5.1 Introduction 129

were observed to decay giving rise to pure conduction state. Thus, spatio-

temporal chaos in the vicinity of convection onset can exist in the form of

various irregular structures and defects. Such flow structures arise due to

the density gradients caused by several competing mechanisms of a thermal

and concentrational nature. In particular, convection patterns could result

from double diffusion in binary mixtures. In this case, if the thermodiffu-

sion coefficient is negative [4, 79] and a mixture is heated from below, the

Fig. 5.1 Schematic dia-

gram of the density gradi-

ents ∇ρT , ∇ρTD and ∇ρGS

caused by thermal expan-

sion, thermal diffusion and

gravitational sedimenta-

tion, respectively, arising in

a ferromagnetic nanofluid

heated from below [136].

GS∇T∇ TD∇

Cold

Hot

suppressing

driving

r r r

concentrational density gradient ∇ρTD is directed downwards and has a sta-

bilising influence, while thermal expansion of fluid leads to the occurrence

of the destabilising upward density gradient ∇ρT . The onset of convection is

then dictated by the relative strength of these two opposing effects. In con-

trast to binary fluids, the thermodiffusion coefficient in magnetic colloids is

positive1, and both density gradients due to thermal expansion and due to

thermodiffusion have a destabilising effect; see Figure 5.1. However, in mag-

netic fluids these two upward density gradients are opposed by a stabilising

negative density gradient ∇ρGS arising due to gravitational sedimentation of

solid particles and their aggregates [84, 100]. Moreover, ferrofluids based on

organic carrier fluids (kerosene, transformer oil) contain molecules of different

masses and sizes as well as molecules of a surfactant (up to 10% by volume).

Such different molecules are also subject to thermodiffusive redistribution

and can lead to liquid phase stratification. In addition, organic carrier liquids

forming the base of ferrofluids can contain contaminants and chemical species

that also form insoluble sediment that leads to the fluid density stratification.

Fluid mixing caused by fluid motion on the other hand tends to destroy such

density gradients resulting in irregular intermittent convection in magnetic

nanofluids [38, 102, 136, 200].

A large number of various de-homogenising mechanisms present in realistic

non-isothermal ferro-nanofluids and complexity of their analytical description

makes it virtually impossible to model them accurately and consistently at

1 The intensity of thermodiffusion is proportional to the Soret coefficient St = DT /D,

where DT and D are the coefficients of thermodiffusion and Brownian diffusion, respec-

tively. It is positive for magnetic fluids in the absence of magnetic field and has the value

of approximately 0.1K−1 [23, 75, 162], which is several orders of magnitude larger than

for binary mixtures.

Page 140

GR42.eps

130 5 Thermogravitational Convection

present. For this reason the many theoretical studies of convection in fer-

rofluids opted to neglect the non-homogeneity of the fluid treating it as a

hypothetical single-component medium [12, 88, 89, 209, 222, e.g.]. A stan-

dard justification given to such an approach is based on estimating the char-

acteristic time tD = h

2/D over which substantial changes of fluid density

due to diffusion of species occur. For typical experimental and industrial se-

tups, this time is of order of a week. However, this time corresponds to the

variation of concentration across a fluid layer of thickness h by a factor of e,

which is much larger than that required to initiate macroscopic convection.

Indeed, the density gradients due to the thermal expansion and the variation

of species concentration C are β∇T and βc∇C, respectively, where the typical

values of thermal expansion coefficient β ∼ 10−3 K−1, temperature gradient

∇T ∼ 1K/cm and solutal expansion coefficient βc = 1ρ

∂ρ

∂C

∼ 1 [23, 100, 212].

Thus, the solutal density gradient comparable with thermal gradient required

to induce convection occurs when the concentration gradient is of the order

of 10−4−10−3 cm−1. Therefore, the occurrence of solutal convection in mag-

netic colloids cannot always be ruled out. Yet its numerical modelling remains

virtually impossible because it requires the knowledge of various transport

properties such as the thermodiffusion coefficient and the rotational viscosity.

Their values for practical ferrofluids with moderate and large concentrations

of a solid phase are not always known because they depend on the fluid stor-

age and use conditions and can vary even for the same sample of fluid. Thus, a

direct evaluation of the governing nondimensional parameters corresponding

to experimental conditions remains impossible, which limits the opportunity

of a meaningful comparison of experimental and theoretical results. Since

accounting of all acting transport mechanisms by a single theoretical model

is not always possible, experimental investigation of flow phenomena in non-

isothermal magnetic colloids remains an important investigation avenue of

their prospective application that will be discussed in detail next.

5.2 Horizontal Layer

5.2.1 Temporal Behaviour of Convection Flows

The main distinction of behaviour of colloidal fluids from their single-

component counterparts contained in a layer bounded by horizontal plates

is that colloid remains an “active medium” even if it is isothermal and re-

mains at rest at a macroscopic scale. Microscopically, the continuous action

of gravity on the contained solid particles and their aggregates leads to their

barometric redistribution, which breaks fluid’s homogeneity. If the fluid is

kept in non-isothermal conditions, the effects of gravitational sedimentation

can be enhanced by thermodiffusion. The overall degree of inhomogeneity

depends on the time fluid remains at macroscopic rest. Therefore, the be-

haviour of a colloid becomes dependent on a pre-history of experiment. We

discuss this feature of ferrofluid behaviour in detail below.

Page 278

GR29.eps

Index

A

aggregates, 3, 12, 17

amplitude expansion, 92

B

bifurcation, supercritical or subcritical, 91

binary mixture, 129, 184

Bingham fluid, 13

blinking state, 128, 147, 193

Boussinesq approximation, 11

Brownian motion, 2, 4, 8

Brownian relaxation, 8

buoyancy, 16, 44

C

carrier fluid, 17

climbing dislocation, 193, 194

colloid, 1

confined state, 128, 191

cross-roll instability, 134, 174, 189

Curie coefficient, 17

Curie effect, 4, 17

D

dipole-dipole interaction, 2

double diffusion, 129, 147

F

ferrofluid, 26

ferrohydrodynamics, 2

ferromagnetic fluid, 16

ferromagnetics, 1

G

gliding dislocation, 149

gravitational convection, 5

gravitational sedimentation, 12, 139, 153,

154, 168, 178, 179, 184, 188, 210, 213,

214

H

Helmholtz coils, 124, 167

hysteresis, 152, 153, 174

I

induction approximation, 6

inductionless approximation, 6, 244

K

Kelvin force, 4, 13, 29, 73

L

Landau constant, 97

Langevin magnetisation law, 16, 17

Langevin’s parameter, 17

linearised perturbation energy balance

analysis, 42

localised state, 128, 145, 184

longitudinal rolls, 144

Lorentz force, 4

M

magnetic buoyancy, 30

magnetic field lines, 29, 70

magnetic moment, 1

magnetic pressure, 28, 101

magnetic relaxation, 3

© Springer International Publishing AG, part of Springer Nature 2018

A. A. Bozhko, S. A. Suslov, Convection in Ferro-Nanofluids: Experiments

and Theory, Advances in Mechanics and Mathematics 40,

https://doi.org/10.1007/978-3-319-94427-2

271

https://doi.org/10.1007/978-3-319-94427-2

Page 279

GR42.eps

272 Index

magnetic succeptibility, 9

magnetic susceptibility, 12, 16, 166

magnetic viscosity, 108

magnetisation, 22, 28, 73

magnetisation of saturation, 17, 106

magnetisation relaxation time, 12

magneto-concentrational convection, 7

magneto-polarisable media, 165

magnetoconvection, 16, 38, 40, 60, 68, 166,

215, 217

magnetophoresis, 12, 168

magnetorheological fluids, 1

magnetoviscosity, 8, 178

magnetoviscous effect, 8, 168, 179, 244

Maxwell’s equations, 11

Modified Mean Field model, 17

momentum diffusion time, 151

multiple timescales, 92

N

nanofluid, 2

nanoparticle, 1

Navier-Stokes equations, 11

Newtonian fluid, 8, 13, 152

Nusselt number, 119

O

oscillatory instability, 23

P

paramagnetic fluid, 16, 26, 68, 70

paramagnetics, 1

particle diffusion time, 151

pinning effect, 149, 193

ponderomotive force, 13, 45, 46, 165

pyromagnetic coefficient, 16

pyromagnetic coefficient, 14, 26, 167

Q

quasi-harmonic oscillations, 154, 159

R

rotational viscosity, 8, 106

S

saturated hydrocarbones, 161

Schmidt-Milverton method, 112, 118, 122

self-induced oscillations, 152

shear instability, 22

shear rate, 12

sliding dislocation, 194

Soret coefficient, 9, 168, 178, 188

Soret effect, 7, 9

spiral defect chaos, 128, 136, 189

Squire’s transformation, 32

steric repulsion, 3

subcritical transition, 152

superparamagnetics, 2

T

target chaos, 128

thermal conductivity, 8

thermal diffusion time, 151

thermal energy equation, 11

thermal waves, 23

thermodiffusion, 7–9, 129, 130, 150, 154,

164, 168, 184, 188

thermodiffusion coefficient, 129

thermogravitational convection, 35, 42, 49,

214

thermogravitational waves, 37, 40, 48

thermomagnetic convection, 4, 22, 91, 174,

217, 220

thermomagnetic instability, 30

thermomagnetic waves, 37, 41, 48

thermomagnetically sensitive fluids, 26

thermophoresis, 12, 168

V

Van der Waals force, 2

viscous time, 12

W

wavelet analysis, 134, 156, 158

wavelet coefficients, 159

wavelet transform, 156, 159

weakly nonlinear analysis, 92

Weiss Mean Field model, 17

Z

zipper state, 128, 138, 144, 174, 231

Advances in Mechanics and Mathematics 40

Aleksandra A. Bozhko

Sergey A. Suslov

Convection in

Ferro-Nanofluids:

Experiments and

Theory

Physical Mechanisms, Flow Patterns,

and Heat Transfer

Page 2

Advances in Mechanics and Mathematics

Volume 40

Series Editors

David Gao, Federation University Australia

Tudor Ratiu, Shanghai Jiao Tong University

Advisory Board

Antony Bloch, University of Michigan

John Gough, Aberystwyth University

Darryl D. Holm, Imperial College London

Peter Olver, University of Minnesota

Juan-Pablo Ortega, University of St. Gallen

Genevieve Raugel, CNRS and University Paris-Sud

Jan Philip Solovej, University of Copenhagen

Michael Zgurovsky, Igor Sikorsky Kyiv Polytechnic Institute

Jun Zhang, University of Michigan

Enrique Zuazua, Universidad Autónoma de Madrid and DeustoTech

Kenneth C. Land, Duke University

More information about this series at http://www.springer.com/series/5613

http://www.springer.com/series/5613

Page 139

GR18.eps

5.1 Introduction 129

were observed to decay giving rise to pure conduction state. Thus, spatio-

temporal chaos in the vicinity of convection onset can exist in the form of

various irregular structures and defects. Such flow structures arise due to

the density gradients caused by several competing mechanisms of a thermal

and concentrational nature. In particular, convection patterns could result

from double diffusion in binary mixtures. In this case, if the thermodiffu-

sion coefficient is negative [4, 79] and a mixture is heated from below, the

Fig. 5.1 Schematic dia-

gram of the density gradi-

ents ∇ρT , ∇ρTD and ∇ρGS

caused by thermal expan-

sion, thermal diffusion and

gravitational sedimenta-

tion, respectively, arising in

a ferromagnetic nanofluid

heated from below [136].

GS∇T∇ TD∇

Cold

Hot

suppressing

driving

r r r

concentrational density gradient ∇ρTD is directed downwards and has a sta-

bilising influence, while thermal expansion of fluid leads to the occurrence

of the destabilising upward density gradient ∇ρT . The onset of convection is

then dictated by the relative strength of these two opposing effects. In con-

trast to binary fluids, the thermodiffusion coefficient in magnetic colloids is

positive1, and both density gradients due to thermal expansion and due to

thermodiffusion have a destabilising effect; see Figure 5.1. However, in mag-

netic fluids these two upward density gradients are opposed by a stabilising

negative density gradient ∇ρGS arising due to gravitational sedimentation of

solid particles and their aggregates [84, 100]. Moreover, ferrofluids based on

organic carrier fluids (kerosene, transformer oil) contain molecules of different

masses and sizes as well as molecules of a surfactant (up to 10% by volume).

Such different molecules are also subject to thermodiffusive redistribution

and can lead to liquid phase stratification. In addition, organic carrier liquids

forming the base of ferrofluids can contain contaminants and chemical species

that also form insoluble sediment that leads to the fluid density stratification.

Fluid mixing caused by fluid motion on the other hand tends to destroy such

density gradients resulting in irregular intermittent convection in magnetic

nanofluids [38, 102, 136, 200].

A large number of various de-homogenising mechanisms present in realistic

non-isothermal ferro-nanofluids and complexity of their analytical description

makes it virtually impossible to model them accurately and consistently at

1 The intensity of thermodiffusion is proportional to the Soret coefficient St = DT /D,

where DT and D are the coefficients of thermodiffusion and Brownian diffusion, respec-

tively. It is positive for magnetic fluids in the absence of magnetic field and has the value

of approximately 0.1K−1 [23, 75, 162], which is several orders of magnitude larger than

for binary mixtures.

Page 140

GR42.eps

130 5 Thermogravitational Convection

present. For this reason the many theoretical studies of convection in fer-

rofluids opted to neglect the non-homogeneity of the fluid treating it as a

hypothetical single-component medium [12, 88, 89, 209, 222, e.g.]. A stan-

dard justification given to such an approach is based on estimating the char-

acteristic time tD = h

2/D over which substantial changes of fluid density

due to diffusion of species occur. For typical experimental and industrial se-

tups, this time is of order of a week. However, this time corresponds to the

variation of concentration across a fluid layer of thickness h by a factor of e,

which is much larger than that required to initiate macroscopic convection.

Indeed, the density gradients due to the thermal expansion and the variation

of species concentration C are β∇T and βc∇C, respectively, where the typical

values of thermal expansion coefficient β ∼ 10−3 K−1, temperature gradient

∇T ∼ 1K/cm and solutal expansion coefficient βc = 1ρ

∂ρ

∂C

∼ 1 [23, 100, 212].

Thus, the solutal density gradient comparable with thermal gradient required

to induce convection occurs when the concentration gradient is of the order

of 10−4−10−3 cm−1. Therefore, the occurrence of solutal convection in mag-

netic colloids cannot always be ruled out. Yet its numerical modelling remains

virtually impossible because it requires the knowledge of various transport

properties such as the thermodiffusion coefficient and the rotational viscosity.

Their values for practical ferrofluids with moderate and large concentrations

of a solid phase are not always known because they depend on the fluid stor-

age and use conditions and can vary even for the same sample of fluid. Thus, a

direct evaluation of the governing nondimensional parameters corresponding

to experimental conditions remains impossible, which limits the opportunity

of a meaningful comparison of experimental and theoretical results. Since

accounting of all acting transport mechanisms by a single theoretical model

is not always possible, experimental investigation of flow phenomena in non-

isothermal magnetic colloids remains an important investigation avenue of

their prospective application that will be discussed in detail next.

5.2 Horizontal Layer

5.2.1 Temporal Behaviour of Convection Flows

The main distinction of behaviour of colloidal fluids from their single-

component counterparts contained in a layer bounded by horizontal plates

is that colloid remains an “active medium” even if it is isothermal and re-

mains at rest at a macroscopic scale. Microscopically, the continuous action

of gravity on the contained solid particles and their aggregates leads to their

barometric redistribution, which breaks fluid’s homogeneity. If the fluid is

kept in non-isothermal conditions, the effects of gravitational sedimentation

can be enhanced by thermodiffusion. The overall degree of inhomogeneity

depends on the time fluid remains at macroscopic rest. Therefore, the be-

haviour of a colloid becomes dependent on a pre-history of experiment. We

discuss this feature of ferrofluid behaviour in detail below.

Page 278

GR29.eps

Index

A

aggregates, 3, 12, 17

amplitude expansion, 92

B

bifurcation, supercritical or subcritical, 91

binary mixture, 129, 184

Bingham fluid, 13

blinking state, 128, 147, 193

Boussinesq approximation, 11

Brownian motion, 2, 4, 8

Brownian relaxation, 8

buoyancy, 16, 44

C

carrier fluid, 17

climbing dislocation, 193, 194

colloid, 1

confined state, 128, 191

cross-roll instability, 134, 174, 189

Curie coefficient, 17

Curie effect, 4, 17

D

dipole-dipole interaction, 2

double diffusion, 129, 147

F

ferrofluid, 26

ferrohydrodynamics, 2

ferromagnetic fluid, 16

ferromagnetics, 1

G

gliding dislocation, 149

gravitational convection, 5

gravitational sedimentation, 12, 139, 153,

154, 168, 178, 179, 184, 188, 210, 213,

214

H

Helmholtz coils, 124, 167

hysteresis, 152, 153, 174

I

induction approximation, 6

inductionless approximation, 6, 244

K

Kelvin force, 4, 13, 29, 73

L

Landau constant, 97

Langevin magnetisation law, 16, 17

Langevin’s parameter, 17

linearised perturbation energy balance

analysis, 42

localised state, 128, 145, 184

longitudinal rolls, 144

Lorentz force, 4

M

magnetic buoyancy, 30

magnetic field lines, 29, 70

magnetic moment, 1

magnetic pressure, 28, 101

magnetic relaxation, 3

© Springer International Publishing AG, part of Springer Nature 2018

A. A. Bozhko, S. A. Suslov, Convection in Ferro-Nanofluids: Experiments

and Theory, Advances in Mechanics and Mathematics 40,

https://doi.org/10.1007/978-3-319-94427-2

271

https://doi.org/10.1007/978-3-319-94427-2

Page 279

GR42.eps

272 Index

magnetic succeptibility, 9

magnetic susceptibility, 12, 16, 166

magnetic viscosity, 108

magnetisation, 22, 28, 73

magnetisation of saturation, 17, 106

magnetisation relaxation time, 12

magneto-concentrational convection, 7

magneto-polarisable media, 165

magnetoconvection, 16, 38, 40, 60, 68, 166,

215, 217

magnetophoresis, 12, 168

magnetorheological fluids, 1

magnetoviscosity, 8, 178

magnetoviscous effect, 8, 168, 179, 244

Maxwell’s equations, 11

Modified Mean Field model, 17

momentum diffusion time, 151

multiple timescales, 92

N

nanofluid, 2

nanoparticle, 1

Navier-Stokes equations, 11

Newtonian fluid, 8, 13, 152

Nusselt number, 119

O

oscillatory instability, 23

P

paramagnetic fluid, 16, 26, 68, 70

paramagnetics, 1

particle diffusion time, 151

pinning effect, 149, 193

ponderomotive force, 13, 45, 46, 165

pyromagnetic coefficient, 16

pyromagnetic coefficient, 14, 26, 167

Q

quasi-harmonic oscillations, 154, 159

R

rotational viscosity, 8, 106

S

saturated hydrocarbones, 161

Schmidt-Milverton method, 112, 118, 122

self-induced oscillations, 152

shear instability, 22

shear rate, 12

sliding dislocation, 194

Soret coefficient, 9, 168, 178, 188

Soret effect, 7, 9

spiral defect chaos, 128, 136, 189

Squire’s transformation, 32

steric repulsion, 3

subcritical transition, 152

superparamagnetics, 2

T

target chaos, 128

thermal conductivity, 8

thermal diffusion time, 151

thermal energy equation, 11

thermal waves, 23

thermodiffusion, 7–9, 129, 130, 150, 154,

164, 168, 184, 188

thermodiffusion coefficient, 129

thermogravitational convection, 35, 42, 49,

214

thermogravitational waves, 37, 40, 48

thermomagnetic convection, 4, 22, 91, 174,

217, 220

thermomagnetic instability, 30

thermomagnetic waves, 37, 41, 48

thermomagnetically sensitive fluids, 26

thermophoresis, 12, 168

V

Van der Waals force, 2

viscous time, 12

W

wavelet analysis, 134, 156, 158

wavelet coefficients, 159

wavelet transform, 156, 159

weakly nonlinear analysis, 92

Weiss Mean Field model, 17

Z

zipper state, 128, 138, 144, 174, 231