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TitleConvection in Ferro-Nanofluids: Experiments and Theory: Physical Mechanisms, Flow Patterns, and Heat Transfer
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Total Pages279
Table of Contents
1 Ferrofluids: Composition and Physical Processes
	1.1 Brief History and Composition of Ferrofluids
	1.2 Physical Processes Taking Place in Ferrofluids
	1.3 Physical Properties of Ferrofluids
2 Governing Equations
	2.1 Simplifying Physical Assumptions and Basic Equations
	2.2 Nondimensionalisation and Governing Parameters
3 Infinite Vertical Layer
	3.1 Introduction
	3.2 Problem Definition and Basic Flow Solutions
	3.3 Flow Patterns in a Normal Magnetic Field
		3.3.1 Linearised Equations for Infinitesimal Perturbations
		3.3.2 Stability Results for an Equivalent Two-Dimensional Problem
		3.3.3 Perturbation Energy Balance
		3.3.4 Three-Dimensional Results
		3.3.5 Symmetry-Breaking Effects of Nonuniform Fluid Magnetisation
		3.3.6 Variation of Stability Characteristics and Summary of Results for Convection in Normal Field
	3.4 Flow Patterns in an Oblique Magnetic Field
		3.4.1 Linearised Perturbation Equations in Zero Gravity
		3.4.2 Flow Stability Characteristics in Zero Gravity
		3.4.3 Perturbation Energy Balance in Zero Gravity
		3.4.4 Perturbation Fields in Zero Gravity
		3.4.5 Linearised Perturbation Equations in Non-zeroGravity
		3.4.6 Wave-Like Instabilities in an Oblique Field in Non-zero Gravity
		3.4.7 Stability Diagrams for an Equivalent Two-Dimensional Problem in an Oblique Field and Non-zero Gravity Field Inclined in the Plane Containing the Main Periodicity Direction (γ"0365γ=0) Arbitrary Field Orientation
	3.5 Weakly Nonlinear Consideration of Thermomagnetic Convection
		3.5.1 Amplitude Expansion
		3.5.2 Linearised Disturbances
		3.5.3 Mean Flow Correction and Second Harmonic
		3.5.4 Fundamental Harmonic Distortion and Landau Equation
		3.5.5 Numerical Results and Their Physical Interpretation
		3.5.6 Conclusions
4 Experimental Methodology
	4.1 Properties of Ferrofluids
	4.2 Requirements for Experimental Setup
	4.3 Experimental Chamber Design
	4.4 Interpretation of Thermal Field Visualisations
		4.4.1 Edge Effects in Magnetic Fluid Flows
		4.4.2 Convection Patterns
	4.5 Heat Flux Measurements
	4.6 Spherical Configuration
	4.7 Experiments in Magnetic Field
	4.8 Evaluation of Governing Parameters
5 Thermogravitational Convection
	5.1 Introduction
	5.2 Horizontal Layer
		5.2.1 Temporal Behaviour of Convection Flows
		5.2.2 Spatial Patterns
	5.3 Vertical Layer: The Influence of Sedimentation
		5.3.1 Fully Stratified Fluid
		5.3.2 Partially Stratified Fluid
	5.4 Inclined Layer
	5.5 Sphere
	5.6 Conclusions
6 Thermomagnetic Convection
	6.1 Magnetic Control of Magneto-Polarisable Media
	6.2 Horizontal Layer
		6.2.1 Historical Overview and the Current State of Knowledge
		6.2.2 Convection and Heat Transfer
		6.2.3 Convection in a Horizontal Layer Placed in a Magnetic Field Parallel to the Layer
	6.3 Vertical Layer
		6.3.1 Problem Overview
		6.3.2 Thermomagnetic Convection Patterns
		6.3.3 Heat Transfer Characteristics
		6.3.4 Influence of Fluid Stratification
		6.3.5 Other Factors Influencing Experimental FlowPatterns
	6.4 Inclined Layer
		6.4.1 Convection in an Inclined Layer Placed in a Normal Magnetic Field
		6.4.2 Convection in an Inclined Layer Placed in a Magnetic Field Parallel to the Layer
	6.5 Sphere
		6.5.1 Problem Overview
		6.5.2 Thermomagnetic Convection in a Sphere Heated from Top
7 Concluding Remarks
Appendix A  Brief Summary of the Used Numerical Approximation
Appendix B  Copyright Permissions
Document Text Contents
Page 1

Advances in Mechanics and Mathematics 40

Aleksandra A. Bozhko
Sergey A. Suslov

Convection in
Experiments and
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

Page 139


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].






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


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


∼ 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




aggregates, 3, 12, 17

amplitude expansion, 92


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


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


dipole-dipole interaction, 2

double diffusion, 129, 147


ferrofluid, 26

ferrohydrodynamics, 2

ferromagnetic fluid, 16

ferromagnetics, 1


gliding dislocation, 149

gravitational convection, 5

gravitational sedimentation, 12, 139, 153,
154, 168, 178, 179, 184, 188, 210, 213,


Helmholtz coils, 124, 167

hysteresis, 152, 153, 174


induction approximation, 6

inductionless approximation, 6, 244


Kelvin force, 4, 13, 29, 73


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


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,


Page 279


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

nanofluid, 2
nanoparticle, 1
Navier-Stokes equations, 11
Newtonian fluid, 8, 13, 152
Nusselt number, 119

oscillatory instability, 23

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

quasi-harmonic oscillations, 154, 159

rotational viscosity, 8, 106

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


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,

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


Van der Waals force, 2

viscous time, 12


wavelet analysis, 134, 156, 158

wavelet coefficients, 159

wavelet transform, 156, 159

weakly nonlinear analysis, 92

Weiss Mean Field model, 17


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

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