Magnetic fields can alter fluid flow by either interacting with the microscopic magnetic moments of the fluid constituents or with electric currents in a conducting fluid. Control of the flow by magnetic fields is especially convenient, since influencing the flow by mechanical means relies on the interaction of a solid component in direct contact with the fluid. Fluids can be attracted or repelled by a magnetic field gradient depending on whether they are paramagnetic or diamagnetic. The paramagnetic properties depend on the spin configuration of the atoms, ions or molecules that constitute the fluid [1]. A fluid is diamagnetic when its constituents have no unpaired electrons. Magnetic fluids are most commonly encountered in the form of liquids, although paramagnetism also exists in the gaseous state with the prominent example of molecular oxygen [2]. The following sections give a concise overview of the key ideas behind the agglomeration of magnetic particles in magnetic fields, the influence of magnetic field gradients on fluid flows and magnetohydrodynamics.

1 Magnetic Liquids

Paramagnetism in the liquid state originates from solvated paramagnetic ions or ferrimagnetic/ferromagnetic particles in colloidal suspensions [3]. In all cases, the solvents or carrier liquids are diamagnetic. Fluids show no net magnetisation without exposure to an external magnetic field. This is due to Brownian motion that keeps the molecules from attaining any ordered state in which the exchange interaction between magnetic particles could have an effect. Brownian motion also ensures the homogeneity of the liquid by keeping the ions or particles from settling. However, the application of a magnetic field gradient to suspensions of magnetic particles can lead to their agglomeration in areas of high magnetic field intensity. The magnetically modified distribution N(H) of the density of suspended particles can be estimated from the Boltzmann factor \(\textrm{exp}(-E_\text {mag}/k_B\,T)\) with the magnetic energy \(E_\text {mag}\), Boltzmann constant \(k_B= 1.38\times 10^{-23}\)J K\(^{-1}\) and temperature T in the exponent. In a suspension made up of particles with volume V and magnetic susceptibility \(\chi \), the magnetic energy of the distribution of magnetic moments in an external field is given by \(E_\textrm{mag} = -\tfrac{1}{2} \mu _0 \chi H^2 V\). The dimensionless volume magnetic susceptibility \(\chi \) relates the applied magnetic field H (unit: A m\(^{-1}\)) to the induced magnetisation \(M = \chi H\) and \(\mu _0= 4 \pi \times 10^{-7}\) N A\(^{-2}\) is the vacuum permeability. The value of \(\chi \) determines the form of magnetism of a material. Paramagnetic materials have positive values for the susceptibility \(\chi > 0\), whereas diamagnetic substances have negative susceptibilities \(\chi < 0\). Thus, the expression for the Boltzmann distribution is simply:

$$\begin{aligned} N(H) = N(0) \, \text {exp}\left( \frac{\mu _0 \chi H^2\,V}{2k_B T}\right) , \end{aligned}$$
(2.1)

with the density of particles in absence of an external magnetic field N(0). Any magnetic interactions between the particles are neglected in this treatment. As soon as the magnetic energy exceeds the thermal energy (\(\tfrac{1}{2} \mu _0 \chi H^2 V \ge k_B T\)), particles are immobilised and gather in the area of highest magnetic field. Evidently, the agglomeration in a magnetic field is proportional to the volume of the suspended particles and it is possible to estimate the minimum particle size for their magnetic confinement. Rearranging for the diameter d of a spherical particle with volume \(V = \tfrac{4}{3} \pi r^3 = \tfrac{1}{6} \pi d^3\) leads to:

$$\begin{aligned} d \ge \left( \frac{12 k_B T}{\pi \mu _0 \chi H^2} \right) ^{\tfrac{1}{3}}. \end{aligned}$$
(2.2)

At room temperature, magnetic particles in a suspension with \(\chi = 1\) settle in a magnetic field of 0.2 T (\(H = 1.59 \times 10^5\) A m\(^{-1}\)) when their diameter exceeds 8 nm. The expression in Eq. 2.1 was first employed to explain the formation of magnetic powder patterns (Bitter patterns) that form above the domain walls on surfaces of ferromagnetic crystals [4]. The magnetic movement of individual particles lies at the heart of the phenomenon of magnetophoresis, which is particularly relevant for magnetic separation processes.

It is to be noted that the magnetic susceptibility is also temperature dependent itself for paramagnetic particles that follow Curie’s law \(\chi = \tfrac{C}{T}\) with the Curie constant C (unit: K). In practice, the amplification of the susceptibility by cooling is limited by the freezing point of the carrier liquid or solvent. Unlike the paramagnetic case, diamagnetic susceptibilities are temperature independent.

The agglomeration of the magnetic constituents of the liquid can be undesired as the heightened density of particles in the magnetic field compresses the fluid and hinders flow. For this reason, ideal ferrofluids comprise magnetic particles with a size below 10 nm [5] and are superparamagnetic. Solutions of paramagnetic transition metal or rare earth salts have far lower magnetic susceptibilities than ferrofluids and the thermal energy dwarfs the magnetic energy by at least two order of magnitude at room temperature.

Magnetic nanoparticles or solvated paramagnetic ions act like single domain microscopic magnets and the distribution law must be modified. The energy of a magnetic moment in a magnetic field is given by \(E= - \mu _0 \textbf{m} \cdot \textbf{H} = - \mu _0 m H \textrm{cos} \theta \). The distribution N(H) of single domain magnetic nanoparticles or ions in a magnetic field can be obtained by integrating the Boltzmann factor \(\textrm{exp}(-E_\textrm{mag} / E_\textrm{therm})\) = \(\textrm{exp}(\mu _0 m H \textrm{cos} \theta / k_BT)\) over all angles between \(0^\circ \) and \(180^\circ \) [4]:

$$\begin{aligned} N(H) &= N(0) \int \limits _{0}^{\pi } \textrm{sin}(\theta ) \textrm{exp}(\mu _0 m H \textrm{cos} \theta / k_BT) d \theta \end{aligned}$$
(2.3)
$$\begin{aligned} &= N(0) \left[ -\frac{\textrm{exp}(\frac{\mu _0 m H \textrm{cos} \theta }{k_BT})}{\mu _0 m H / k_BT} \right] ^\pi _0 \end{aligned}$$
(2.4)
$$\begin{aligned} &= N(0) \frac{\text {sinh}(\mu _0 m H/k_B T)}{ \mu _0 m H/k_B T}. \end{aligned}$$
(2.5)

Once again, the thermal energy must be larger than the magnetic energy to guarantee homogeneity in the magnetic field. A maximum value of the magnetic field can be estimated from \(k_B T \ge \mu _0 m H\). For a gadolinium ion with 7 \(\mu _B\) (Bohr magneton: \(\mu _B = 9.274 \times 10^{-24}\)J T\(^{-1}\)), this yields \(\mu _0 H = \tfrac{k_B T}{m}= \tfrac{4.11 ^{-21}}{7 \times 9.274 \times 10^{-24}}\textrm{T} \approx 63.4 \textrm{T}\). Continuous magnetic fields of such high intensities are not attainable, but pulsed field magnets can sustain such high magnetic fields for milliseconds. In ideal ferrofluids and paramagnetic salt solutions there is no danger of segregation of the magnetic nanoparticles or ions. However, the microscopic forces on the individual microscopic moments are passed on to the bulk fluid as a whole and the resulting force density can drive convection. An explanation of the effects of such external body forces on fluids can be found in fluid dynamics.

2 Fluid Dynamics

The Navier-Stokes equation describes the movement of a fluid that is exposed to force densities \(\textbf{F}\) (in N m\(^{-3}\)) and pressure gradients \(\nabla P\). It is also known as the momentum equation. The change of the fluid velocity \(\textbf{u}\) is introduced into the equation via its material derivative (\(\tfrac{\textrm{D}\textbf{u}}{\textrm{D}t} = \tfrac{\partial \textbf{u}}{\partial t} + (\textbf{u} \cdot \nabla ) \textbf{u}\)):

$$\begin{aligned} \rho \frac{\textrm{D}\textbf{u}}{\textrm{D}t} = - \nabla P + \eta \nabla ^2 \textbf{u} + \textbf{F}, \end{aligned}$$
(2.6)

with the density \(\rho \) (in kg m\(^{-3}\)) and dynamic viscosity \(\eta \) (unit: N s m\(^{-2}\)). The viscosity can be interpreted to cause momentum to diffuse along its gradient [6].

A dimensionless quantity central to fluid dynamics is the Reynolds number (Re), which is the ratio between inertia and viscous forces that appear in the Navier-Stokes equation (Eq. 2.6) as \(\rho (\textbf{u} \cdot \nabla ) \textbf{u}\) and \(\eta \nabla ^2 \textbf{u}\), respectively. The Reynolds number dictates whether fluid flow is laminar or turbulent and is defined as:

$$\begin{aligned} \textrm{Re} = \frac{\rho ul}{\eta } = \frac{ul}{\nu }, \end{aligned}$$
(2.7)

with the characteristic length scale of the fluid motion l, the fluid velocity u and the kinematic viscosity \(\nu \) (in m\(^2\) s\(^{-1}\)). Low Reynolds numbers indicate laminar flow that is dominated by viscous forces. Laminar flow is widespread in microfluidics due to the small diameter of the tubes in microfluidic circuits. Turbulent flow arises at high Reynolds numbers, at which inertial forces prevail.

Relevant for the following discussion of magnetic effects are the body forces, which enter into the equation as force densities. The most common force density on a fluid is due to gravity:

$$\begin{aligned} \textbf{F}_\textrm{g} = \rho \textbf{g}, \end{aligned}$$
(2.8)

where \(\textbf{g}\) is the gravitational acceleration (\(g=9.81\,\)m s\(^{-2}\)). The magnetic force density exerted on a fluid by a magnetic field gradient \(\nabla \textbf{H}\) (in A m\(^{-2}\)) is given by the Kelvin force:

$$\begin{aligned} \textbf{F}_{\textrm{K}} = \mu _0 (\textbf{M}\cdot \nabla ) \textbf{H}. \end{aligned}$$
(2.9)

For low susceptibility liquids (\(\chi \ll 1\)) such as paramagnetic salt solutions, an approximated form of the Kelvin force with the magnetic flux density \(\textbf{B} = \mu _0 (\textbf{H}+ \textbf{M})\) (unit: T) is often encountered:

$$\begin{aligned} \mathbf {F_{\nabla \textrm{B}}} = \frac{\chi }{\mu _0} (\textbf{B} \cdot \nabla ) \textbf{B}. \end{aligned}$$
(2.10)

The approximation \(\textbf{B} = \mu _0 (\textbf{H} +\textbf{M}) = \mu _0(\textbf{H} +\chi \textbf{H}) \approx \mu _0 \textbf{H}\) is used to obtain this expression, which is not applicable to ferrofluids with \(\chi \approx 1\).

2.1 Convection in a Magnetic Field Gradient

What effect do these force densities have upon fluids? Magnetic field gradients can deform free surfaces of liquids, but the discussion will be restricted to cases in which the fluid is completely confined by solid walls, as is the case in microfluidic systems. Furthermore, the fluid is assumed to be an incompressible liquid. This is a reasonable assumption for ideal ferrofluids and paramagnetic salt solutions. For incompressible fluids, the flow velocity is free of divergence:

$$\begin{aligned} \nabla \cdot \textbf{u} = 0. \end{aligned}$$
(2.11)

Whether a force density has any influence on the flow velocity of an enclosed liquid depends on the vorticity \(\pmb {\omega }\) (unit: s\(^{-1}\)), which quantifies the spinning motion of a fluid. The vorticity is defined as the curl of the flow velocity:

$$\begin{aligned} \pmb {\omega } = \nabla \times \textbf{u}. \end{aligned}$$
(2.12)

The curl in Eq. 2.12 evinces the three-dimensionality of the vorticity. Two-dimensional approximations can lead to misleading results and should generally be avoided. If a force density does not induce vorticity in an enclosed incompressible fluid, there is no noticeable effect. The force only presses the fluid against the solid wall, resulting in a pressure field that cancels it out. The density of an incompressible fluid is unperturbed by this and nothing happens. No irrotational flow is created. In contrast, internal flows can materialise in the fluid when vorticity is present. This is known as rotational flow. The Navier-Stokes equation can be transformed into the vorticity equation by application of the curl operator and division by \(\rho \) [7, 8]:

$$\begin{aligned} \frac{\textrm{D}\pmb {\omega }}{\textrm{D}t} = (\pmb {\omega } \cdot \nabla )\textbf{u} + \nu \nabla ^2 \pmb {\omega } + \frac{1}{\rho ^2} \nabla \rho \times \nabla {\text{ p }} + \nabla \times \left( \frac{\textbf{F}}{\rho } \right) , \end{aligned}$$
(2.13)

with the material derivative of the vorticity (\(\tfrac{\textrm{D}\pmb {\omega }}{\textrm{D}t} = \tfrac{\partial \textbf{u}}{\partial t} + \textbf{u} \cdot \nabla \pmb {\omega }\)). Here, the kinematic viscosity \(\nu = \tfrac{\eta }{\rho }\) (in m\(^2\) s\(^{-1}\)) has also been introduced. For compressible fluids, the term \(\pmb {\omega } (\nabla \cdot \textbf{u})\) must be added to the equation. The first term on the right \((\pmb {\omega } \cdot \nabla )\textbf{u}\) describes how gradients of the flow velocity influence the vorticity, whereas the second term (\(\nu \nabla ^2 \pmb {\omega }\)) acts as the diffusion of vorticity down its gradient due to viscosity. The third term is relevant for stratified fluids with \(\nabla \rho \ne 0\) in which the pressure field interacts with the density gradient. The curl of the force densities is the last term and this is where magnetic fields come into play. Hence, it is sufficient to analyse the curl of the body forces for incompressible fluids that are enclosed by solid walls.

The curl of the gravitational force density (Eq. 2.8) is given by:

$$\begin{aligned} \nabla \times \textbf{F}_\textrm{g} = \nabla \rho \times \textbf{g}. \end{aligned}$$
(2.14)

It follows that the gravitational force density is irrotational when the density gradient is parallel to the direction of gravity. Gravity induces flows in any system in which this is not the case. The effect of a magnetic field gradient can be analysed by means of the rotational component of the Kelvin force [3, 9]:

$$\begin{aligned} \nabla \times \textbf{F}_\textrm{K} = \mu _0 \nabla M \times \nabla H. \end{aligned}$$
(2.15)

This implies that there must be an inhomogeneity in the magnetisation, as well as in the applied magnetic field. Additionally, the gradients of both must be non-parallel. As the magnetisation of paramagnetic liquids is proportional to the concentration of their magnetic component, a gradient in the density usually accompanies the magnetisation gradient. Thus, an interplay of gravitational (Eq. 2.14) and magnetic force densities (Eq. 2.15) often develops in systems with an inhomogeneity. Large gradients in the magnetisation are located at the interface between magnetic and non-magnetic liquids. In the case of miscible liquids, such an interface is inevitably wiped out by molecular diffusion that establishes homogeneity and brings the system into thermodynamic equilibrium [10, 11]. On the other hand, the magnetic component in a system of immiscible liquids can be captured in a magnetic field gradient indefinitely [12].

The role of the concentration c (in mol m\(^{-3}\)) dependence of the magnetisation is easy to identify by inspecting the curl of the magnetic field gradient force (Eq. 2.10) when it is formulated with the molar magnetic susceptibility \(\chi _m\) (in m\(^3\) mol\(^{-1}\)):

$$\begin{aligned} \mathbf {F_{\nabla \textrm{B}}} = \frac{\chi _m \, c}{\mu _0} (\textbf{B} \cdot \nabla ) \textbf{B} = \frac{\chi _m \, c}{2 \mu _0} \nabla B^2. \end{aligned}$$
(2.16)

The identity \((\textbf{B} \cdot \nabla ) \textbf{B} = \tfrac{1}{2} \nabla (\textbf{B} \cdot \textbf{B}) - \textbf{B} \times (\nabla \times \textbf{B})\) with the approximation \(\nabla \times \textbf{B} = 0\) results in the right part of Eq. 2.16. This is valid for \(\textbf{B} \approx \mu _0 \textbf{H}\), when currents are absent and \(\nabla \times \textbf{H} = 0\). Applying the curl operator to Eq. 2.16 and using the identity \(\nabla \times (\psi \nabla \phi ) = \nabla \psi \times \nabla \phi \) leads to:

$$\begin{aligned} \nabla \times \textbf{F}_{\nabla \textrm{B}} = \frac{\chi _m}{2\mu _0} \nabla c \times \nabla B^2. \end{aligned}$$
(2.17)

At this point it is worth mentioning that there is another prevalent expression for the magnetic force density known as the Korteweg-Helmholtz force [3, 13]. It is defined in terms of gradients of the magnetic susceptibility or permeability:

$$\begin{aligned} \textbf{F}_{\textrm{H}} = - \frac{\textbf{H} \cdot \textbf{H}}{2} \mu _0 \nabla \chi = - \frac{1}{2} H^2 \mu _0 \nabla \chi . \end{aligned}$$
(2.18)

The approximation of the Korteweg-Helmholtz force (Eq. 2.18) for fluids with modest magnetic susceptibilities \(\chi \ll 1\) is called the concentration gradient force:

$$\begin{aligned} \textbf{F}_{c} = - \frac{1}{2} H^2 \mu _0 \chi _{m} \nabla c \approx - \frac{B^2}{2\mu _0} \chi _{m} \nabla c. \end{aligned}$$
(2.19)

The expressions for the Kelvin (Eq. 2.9) and the Korteweg-Helmholtz force (Eq. 2.18) differ by a gradient of the pressure and offer identical alternative descriptions for the motion of incompressible fluids in magnetic field gradients [3, 14, 15]. Both sets of equations describe the creation of vorticity in a magnetic field gradient in the same way. This is straightforward to verify by assuring oneself that the application of the curl operator to the Korteweg-Helmholtz force (Eq. 2.18) leads to the rotational component of the Kelvin force (Eq. 2.15). Likewise the curl of the concentration gradient force (Eq. 2.19) corresponds to that of the magnetic field gradient force (Eq. 2.17). The vector identity \(\nabla \times (\psi \nabla \phi ) = \nabla \psi \times \nabla \phi \) is needed to prove this.

If a fluid exhibits a free surface and is exposed to a magnetic field, the pressure at the interface can be modified by the magnetic force density and deform the surface. This is pertinent for the magnetic control of the wetting of a solid or the deformation of a magnetisable liquid drop [16]. Here, irrotational force densities contribute and potential flow ensues. The situation for incompressible fluids and irrotational flows is described by the time-dependent ferrohydrodynamic Bernoulli equation [5, 16]. However, rotational flows still dominate the flow field in an open beaker of magnetic fluid when the magnetic field is applied far from the surface of the liquid towards the bottom of the vessel. Such conditions are typical in magnetoelectrochemistry, where the electrodes are completely submerged in the electrolyte.

2.2 Magnetohydrodynamics

Magnetic fields can also interact with electrical currents in fluids and modify fluid flow through the Lorentz force. Magnetohydrodynamics (MHD) is the field of study that deals with the effect of the Lorentz force on the motion of electrically conducting fluids [17]. Electrodes can pass a current through electrolytic solutions, which can then interact with an externally applied magnetic field and drive flow. The Lorentz force density is defined as the cross product of the current density \(\textbf{j}\) (in A m\(^{-2}\)) and the magnetic flux density \(\textbf{B}\):

$$\begin{aligned} \textbf{F}_\textrm{L} = \textbf{j} \times \textbf{B}. \end{aligned}$$
(2.20)

The current and magnetic flux densities must be non-parallel for the generation of a Lorentz force component. An important dimensionless number quantity in MHD is the Hartmann number (Ha), which is the square root of the ratio between the Lorentz force and the viscous forces [17]:

$$\begin{aligned} \textrm{Ha} = Bl \sqrt{\frac{\sigma }{\eta }}, \end{aligned}$$
(2.21)

with the electrical conductivity \(\sigma \) (in \(\Omega ^{-1}\) m\(^{-1}\)), the magnetic flux density B, the dynamic viscosity \(\eta \) and the characteristic length scale l as in the definition of the Reynolds number (Eq. 2.7). The Hartmann number is proportional to the magnitude of the magnetic flux density B, but is diminished at small length scales. This means that the Lorentz force becomes negligible at the microscale in all but the highest conductivity liquids, regardless if the Lorentz force is curl-free or not. It is worth pointing out that the Kelvin force (Eq. 2.9) is more resilient to miniaturisation, as the smaller length scales increase the magnetic field gradients. A comparison between the magnitudes of the Kelvin and Lorentz force densities is given in Table 2.1 at the end of this chapter.

Electrolytic cells or vessels containing liquid metals are bounded by solid walls and the rotational component of the Lorentz force is necessary to introduce vorticity into the electrolyte solution. Utilising the vector identity for the curl of a cross product \(\nabla \times (\textbf{A} \times \textbf{B}) = (\textbf{B} \cdot \nabla ) \textbf{A} - (\textbf{A} \cdot \nabla ) \textbf{B} + \textbf{A}(\nabla \cdot \textbf{B}) - \textbf{B} (\nabla \cdot \textbf{A})\) leads to:

$$\begin{aligned} \nabla \times \textbf{F}_\textrm{L} = \nabla \times (\textbf{j} \times \textbf{B}) = (\textbf{B} \cdot \nabla ) \textbf{j} - (\mathbf {j \cdot \nabla }) \textbf{B} + \textbf{j} (\nabla \cdot \textbf{B}) - \textbf{B} (\nabla \cdot \textbf{j}). \end{aligned}$$
(2.22)

The last two terms drop out because of Gauss’s law for magnetism \(\nabla \cdot \textbf{B} = 0\) and charge conservation \(\nabla \cdot \textbf{j} = 0\), respectively. This leaves only the first two terms:

$$\begin{aligned} \nabla \times \textbf{F}_\textrm{L} = (\textbf{B} \cdot \nabla ) \textbf{j} - (\mathbf {j \cdot \nabla }) \textbf{B}. \end{aligned}$$
(2.23)

Accordingly, there must be an inhomogeneity in the electrical current or the external magnetic field in order for a rotational component of the Lorentz force to exist. This is different from the rotational component of the Kelvin force (Eq. 2.15), which also relies on a gradient in the magnetisation provided by concentration gradients (Eq. 2.17). A three-dimensional analysis of the situation in electrolytic cells is mandatory for reliable interpretations of experimental results [18]. When an electrical current flows through a paramagnetic electrolytic solution in a magnetic field gradient, both the Kelvin force and the Lorentz force are present. Which of these is more relevant for the motion of the electrolyte depends on the magnetic susceptibility and electrical conductivity of the liquid.

It is also possible to influence fluid flow with the Lorentz force and no input of electrical energy. A fluid of high electrical conductivity such as liquid metal can be moved through a magnetic field, which induces an electrical current proportional to \(\sigma (\textbf{u} \times \textbf{B})\) and the flow velocity \(\textbf{u}\). Liquid metals have a conductivity of around \(10^6 \,\Omega ^{-1}\,\textrm{m}^{-1}\), while that of sea water is approximately 5 \(\Omega ^{-1}\) m\(^{-1}\) for comparison. Induced currents in liquid metals moving at 1 m s\(^{-1}\) through an external field of 1 T magnitude are around 10 A cm\(^{-2}\). The appearance of these currents and the subsequent dissipation of kinetic energy finds application in metallurgical processes, where motion of the melt can be controlled by magnetic damping [17]. The magnitude and distribution of the magnetic damping force can be estimated with \(\sigma (\textbf{u} \times \textbf{B}) \times \textbf{B}\). In the case of a flowing electrolytic solution in an external magnetic field, the induced currents are negligible.

Another dimensionless number carrying the name of interaction parameter or Stuart number (\(\textrm{N}\)) serves to gauge if the Lorentz force due to currents induced by motion of a liquid in an external magnetic field have an effect upon the flow. The interaction parameter is defined as the ratio between the Lorentz force \(\textbf{j}\times \textbf{B}\), with \(|\textbf{j}| \sim \sigma u B\), and inertia \(\rho (\textbf{u} \cdot \nabla ) \textbf{u}\):

$$\begin{aligned} \textrm{N} = \frac{\sigma B^2 l }{\rho u} = \frac{\textrm{Ha}^2}{\textrm{Re}}. \end{aligned}$$
(2.24)

The interaction parameter equals the ratio of the squared Hartmann number (Eq. 2.21) and the Reynolds number (Eq. 2.7). All three dimensionless quantities are reduced when the characteristic length l reaches small values. Unlike the Hartmann number, the interaction parameter is proportional to the square of the magnetic flux density. The ratio given by Eq. 2.24 indicates that viscous forces dominate the Lorentz force at high flow velocities, even if the induced current is directly proportional to u.

It is possible to express the Lorentz force (Eq. 2.20) solely with the magnetic flux density \(\textbf{B}\) using Ampère’s lawFootnote 1 \(\nabla \times \textbf{B} = \mu _0 \textbf{j}\) [17, 19]:

$$\begin{aligned} \textbf{F}_\textrm{L} = \textbf{j} \times \textbf{B} = \frac{1}{\mu _0} (\mathbf {B \cdot \nabla }) \textbf{B} - \frac{\nabla \textbf{B}^2}{2\mu _0}. \end{aligned}$$
(2.25)

The first term is called the magnetic tension and the second is the gradient of the magnetic pressure. The former can have a non-zero rotational component \(\nabla \times (\tfrac{1}{\mu _0} (\mathbf {B \cdot \nabla }) \textbf{B}\)), whereas the latter has none. In any case, the existence of a gradient of the magnetic flux density \(\nabla B \ne 0\) is a prerequisite for the generation of vorticity in the liquid. On a final note, it is interesting that the magnetic tension and the approximation that is the magnetic field gradient force (Eq. 2.10) differ only by the factor of the magnetic susceptibility.

Table 2.1 Magnitudes of force densities acting on paramagnetic salt solutions (\(\chi = 10^{-3}\)), diamagnetic liquids (\(\chi = -10^{-5}\)) and ferrofluids (\(\chi = 1\)) with following parameters: \(B = 1\) T, \(\nabla B = 10\) T m\(^{-1}\), \(j = 10^3\) A m\(^{-2}\) and \(\Delta \rho = 10^2\) kg m\(^{-3}\)
Full size table

3 Summary

The present chapter gave an overview of the effect of magnetic fields on magnetic fluids and those with a non-negligible conductivity. Gradients in the magnetic field can affect fluid flow and even manipulate magnetic particles suspended in liquids. The magnetic particles must have sufficient size in order for a magnetic field to change their concentration. This is important for the phenomena of magnetorheology [20, 21], in which the viscosity of the fluid is magnetically modified, and magnetophoresis [21, 22], where particles migrate in a magnetic field. Magnetic fluids with particles permanently in suspension without sedimentation are incompressible. When an incompressible fluid is confined by solid walls, the rotational component of the Kelvin force can drive convection. For this to happen, there must be a gradient both in the magnetic field and the magnetisation. When conducting fluids move through a magnetic field, the Lorentz force can alter the fluid flow and lead to stirring or magnetic damping. Magnitudes of the different force densities present in fluids are summarised in Table 2.1. Whether or not the magnetic forces still have an effect upon scaling down to a microfluidic system depends on their strength with respect to the friction given by the viscous forces. In any case, maximisation of the magnetic susceptibility of the liquid, design of high magnetic field gradients and use of a highly conducting liquid for MHD flow will serve to exploit the magnetic field to greater effect.