In the description of the motion of magnetic liquids (ferrofluids) a principal issue is the acting stress. It is derived in different textbooks, including the well known book by Rosensweig [1], where the ponderomotive forces are derived by the energetical approach, and many others [2,3,4]. In this article a general approach that considers the conservation laws and is also valid in non-equilibrium situations is reviewed. It is based on an extension of an overview of the description of magnetizable media in [5].

One difficulty in describing magnetizable media is deciding what should be taken as the energy of the electromagnetic field and what as the energy of the medium. The problem is that in an applied field the energy of the medium depends on the field and a simple distinction between the two energies is not evident. Here we make our choice based on the Maxwell equations in continous media, which in a quasistationary approximation read

$$\begin{aligned} \nabla \times \vec {E}=-\frac{1}{c}\frac{\partial \vec {B}}{\partial t};~\nabla \times \vec {H}=\frac{4\pi }{c}\vec {j} \end{aligned}$$
(4.1)

as usually

$$\begin{aligned} \vec {B}=\vec {H}+4\pi \vec {M};~\nabla \cdot \vec {B}=0. \end{aligned}$$
(4.2)

Equation (4.1) give

$$\begin{aligned} \frac{1}{4\pi }\vec {H}\frac{\partial \vec {B}}{\partial t}+\vec {E}\cdot \vec {j}+\frac{c}{4\pi }\nabla \cdot [\vec {E}\times \vec {H}]=0, \end{aligned}$$
(4.3)

which using the identity \(\varrho \frac{d\tilde{a}}{dt}=\frac{\partial a}{\partial t}+\nabla \cdot (\varrho \tilde{a}\vec {v})\) (\(\tilde{a}=a/\varrho \) is the quantity per unit mass) may be rewritten as follows

$$\begin{aligned} \varrho \frac{d}{dt}\frac{\vec {H}^{2}}{8\pi \varrho }-\nabla \cdot (\frac{\vec {H}^{2}}{8\pi }\vec {v})+\vec {H}\varrho \frac{d\tilde{\vec {M}}}{dt}-\vec {H}\partial _{i}(v_{i}\vec {M})+\vec {E}\cdot \vec {j}+\frac{c}{4\pi }\nabla \cdot [\vec {E}\times \vec {H}]=0. \end{aligned}$$
(4.4)

Equation (4.4) shows that the volume density of the electromagnetic field energy may be identified as \(e_{f}=\vec {H}^{2}/(8\pi )\). Other terms in Eq.(4.4) may be identified as source terms for the internal and mechanical energy of the medium and as a flux of the electromagnetic energy.

For the total energy of the system the following local conservation law is valid (for an overview of energy conservation in systems interacting with electromagnetic field see [6])

$$\begin{aligned} \varrho \frac{d}{dt}\Bigl (\frac{\vec {v}^{2}}{2}+\tilde{e}_{f}+\tilde{e}\Bigr )=\partial _{k}(v_{i}(\sigma _{ik}+T_{ik}))-\nabla \cdot \vec {j}_{f}-\nabla \cdot \vec {j}_{q}, \end{aligned}$$
(4.5)

where the first term on the right side in Eq.(4.5) corresponds to the work done by the total stress \(\sigma _{ik}+T_{ik}\) (\(T_{ik}\) is the electromagnetic field contribution to the stress identified further), the second term describes the electromagnetic energy flux and the third describes the transfer of heat. Similar conservation laws are valid for momentum and mass

$$\begin{aligned} \varrho \frac{dv_{i}}{dt}=\partial _{k}(\sigma _{ik}+T_{ik}); \end{aligned}$$
(4.6)
$$\begin{aligned} \varrho \frac{d}{dt}\frac{1}{\varrho }=\nabla \cdot \vec {v}. \end{aligned}$$
(4.7)

It may be noted that (4.6) in the case of equilibrium magnetization and incompressible liquid reads

$$\begin{aligned} \varrho \frac{d\vec {v}}{dt}=-\nabla p+\eta \Delta \vec {v}+(\vec {M}\cdot \nabla )\vec {H}+\frac{1}{c}[\vec {j}\times \vec {H}]. \end{aligned}$$
(4.8)

We see that in term for the Lorenz force on current we have \(\vec {H}\) and not \(\vec {B}\) as was pointed out by A.Einstein et al. in [7]. In [8] it is shown that electromagnetic force in Eq.(4.8) is reduced to \(\frac{1}{c}[\vec {j}\times \vec {B}]\) if the condition \(\frac{\partial H_{i}}{\partial x_{j}}=0\) is valid. Taking \(\vec {j}_{f}=\frac{c}{4\pi }[\vec {E}'\times \vec {H}]\), where \(\vec {E}'=\vec {E}+\frac{1}{c}[\vec {v}\times \vec {B}]\) is the electric field strength in the reference frame of the material element, relation (4.4) may be put in the following form

$$\begin{aligned} \begin{aligned}& \varrho \frac{d}{dt}\frac{\vec {H}^{2}}{8\pi \varrho }+\vec {H}\varrho \frac{d\tilde{\vec {M}}}{dt}-\partial _{k}\Bigl (v_{i}\Bigl (\frac{H_{i}B_{k}}{4\pi }-\frac{\vec {H}^{2}}{8\pi }\Bigr )\Bigr )+v_{i}\vec {M}\partial _{i}(\vec {H}) \\ & \quad +\vec {E}\cdot \vec {j}+\frac{c}{4\pi }\nabla \cdot [\vec {E}'\times \vec {H}]=0.\end{aligned} \end{aligned}$$
(4.9)

The equation for the kinetic energy reads

$$\begin{aligned} \varrho \frac{d}{dt}\frac{\vec {v}^{2}}{2}=\partial _{k}(v_{i}\sigma _{ik})-\sigma _{ik}\frac{\partial v_{i}}{\partial x_{k}}+v_{i}\partial _{k}T_{ik}. \end{aligned}$$
(4.10)

Subtraction of \(\varrho \frac{d}{dt}\frac{\vec {H}^{2}}{8\pi \varrho }+\varrho \frac{d}{dt}\frac{\vec {v}^{2}}{2}\) given by Eqs. (4.9, 4.10) from Eq.(4.5) and identifying \(T_{ik}=\frac{H_{i}B_{k}}{4\pi }-\frac{H^{2}}{8\pi }\delta _{ik}\) allows us to obtain the source term for the internal energy of media \(\tilde{e}\) which reads

$$\begin{aligned} \varrho \frac{d\tilde{e}}{dt}=\varrho \vec {H}\cdot \frac{d\tilde{\vec {M}}}{dt}+v_{i}\vec {M}\partial _{i}\vec {H}+\sigma _{ik}\frac{\partial v_{i}}{\partial x_{k}}+\vec {E}\cdot \vec {j}-v_{i}\partial _{k}T_{ik}-\nabla \cdot \vec {j_{q}}. \end{aligned}$$
(4.11)

Using \(\partial _{i}H_{k}-\partial _{k}H_{i}=e_{ikj}\frac{4\pi }{c}j_{j}\) and

$$\begin{aligned} -v_{i}\partial _{k}T_{ik}+v_{i}M_{k}\partial _{i}H_{k}=\frac{1}{c}[\vec {v}\times \vec {B}]\cdot \vec {j} \end{aligned}$$

the equation for the internal energy reads

$$\begin{aligned} \varrho \frac{d\tilde{e}}{dt}=\varrho \vec {H}\cdot \frac{d\tilde{\vec {M}}}{dt}+\sigma _{ik}\frac{\partial v_{i}}{\partial x_{k}}+\vec {E}'\cdot \vec {j}-\nabla \cdot \vec {j_{q}}. \end{aligned}$$
(4.12)

Taking the stress tensor of the medium as \(\sigma _{ik}=-p\delta _{ik}+\tau _{ik}\), where \(\tau _{ik}\) is the viscous stress tensor, we have for the case when the magnetization is in thermal equilibrium the following expression

$$\begin{aligned} \varrho \frac{d\tilde{e}}{dt}=\varrho T\frac{d\tilde{S}}{dt} -p\varrho \frac{d}{dt}\frac{1}{\varrho }+\varrho \vec {H}\cdot \frac{d\tilde{\vec {M}}}{dt}, \end{aligned}$$
(4.13)

where for the specific entropy \(\tilde{S}\) we have

$$\begin{aligned} \varrho T\frac{d\tilde{S}}{dt}=\tau _{ik}\frac{\partial v_{i}}{\partial x_{k}}+\vec {E}'\cdot \vec {j}-\nabla \cdot \vec {j}_{q}. \end{aligned}$$
(4.14)

According to the relation (4.13) the internal energy is defined by the equation of state \(\tilde{e}=\tilde{e}(\tilde{S},\varrho ,\tilde{\vec {M}})\). In the case when the magnetization is in a non-equilibrium state the relation (4.13) is put in the following form [5]

$$\begin{aligned} \varrho \frac{d\tilde{e}}{dt}=\varrho T\frac{d\tilde{S}}{dt} -p\varrho \frac{d}{dt}\frac{1}{\varrho }+\varrho \vec {H}_{e}\cdot \frac{d\tilde{\vec {M}}}{dt}. \end{aligned}$$
(4.15)

In (4.15) instead of \(\vec {H}\) the relation for the internal energy contains the effective field \(\vec {H}_{e}\) according to which the magnetization is given by the equilibrium magnetization law \(\vec {M}=\vec {M}_{eq}(\vec {H}_{e})\). The effective field was introduced in [9] in order to describe a non-equilibrium state of the magnetization for diluted ferrofluids. As a result for the entropy production \(\sigma \) we obtain

$$\begin{aligned} \varrho \frac{d\tilde{S}}{dt}=-\nabla \cdot \vec {j}_{s}+\sigma , \end{aligned}$$
(4.16)

where

$$\begin{aligned} T\sigma =\vec {j}_{q}\cdot \nabla \frac{1}{T}+\tau _{ik}\frac{\partial v_{i}}{\partial x_{k}}+\vec {E}'\cdot \vec {j}-(\vec {H}_{e}-\vec {H})\rho \frac{d\tilde{\vec {M}}}{dt}. \end{aligned}$$
(4.17)

On the basis of the relation (4.17) for the entropy production the linear phenomenological laws for heat transfer, viscous stress, the charge transfer and magnetic relaxation may be formulated.

Let us consider in detail the phenomenology of the magnetic relaxation. The total stress tensor as it follows from the requirement of the angular momentum conservation is symmetric \(\sigma _{ik}+T_{ik}=\sigma _{ki}+T_{ki}\) (we are neglecting the internal angular momentum due to the spinning of particles). As a result the term \(\tau _{ik}\frac{\partial v_{i}}{\partial x_{k}}\) may be transformed as follows

$$\begin{aligned}\tau _{ik}\frac{\partial v_{i}}{\partial x_{k}}=\tau ^{s}_{ik}\frac{1}{2}\Bigl (\frac{\partial v_{i}}{\partial x_{k}}+\frac{\partial v_{k}}{\partial x_{i}}\Bigr )+\tau ^{a}_{ik}\frac{1}{2}\Bigl (\frac{\partial v_{i}}{\partial x_{k}}-\frac{\partial v_{k}}{\partial x_{i}}\Bigr ). \end{aligned}$$

Since \(\tau ^{a}_{ik}=-T^{a}_{ik}\) and \(T^{a}_{ik}=-\frac{1}{2}e_{ikl}[\vec {M}\times \vec {H}]_{l}\) we have

$$\begin{aligned}\tau ^{a}_{ik}\frac{1}{2}\Bigl (\frac{\partial v_{i}}{\partial x_{k}}-\frac{\partial v_{k}}{\partial x_{i}}\Bigr )=[\vec {M}\times \vec {H}]\cdot \vec {\Omega }_{0}, \end{aligned}$$

where \(\vec {\Omega }_{0}=\frac{1}{2}\nabla \times \vec {v}\) is the angular velocity of the local rotation of the fluid. According to this the entropy production may be put in the following form (\(\vec {M}\parallel \vec {H}_{e}\))

$$\begin{aligned} T\sigma =\vec {j}_{q}\cdot \nabla \frac{1}{T}+\tau _{ik}\frac{\partial v_{i}}{\partial x_{k}}+\vec {E}'\cdot \vec {j}-(\vec {H}_{e}-\vec {H})\Bigl (\rho \frac{d\tilde{\vec {M}}}{dt}-[\vec {\Omega }_{0}\times \vec {M}]\Bigr ). \end{aligned}$$
(4.18)

We see that as a thermodynamic flux for the magnetization relaxation there appears \(\rho \frac{d\tilde{\vec {M}}}{dt}-[\vec {\Omega }_{0}\times \vec {M}]\), which describes the magnetization relaxation in the reference frame of a rotating material element. Accounting for this term allows one to describe such effects as the increase of the effective viscosity of the ferrofluid in an applied field [10].

The general approach according to which the magnetic relaxation equation is formulated, in our opinion, resolves the issue of the correct form of this equation which was under dispute in the literature [11,12,13,14]. The kinetic coefficients for the magnetic relaxation are obtained considering the Brownian motion of magnetic dipoles in the effective field approximation [9] and read as follows

$$\begin{aligned} \Bigl (\rho \frac{d\tilde{\vec {M}}}{dt}-[\vec {\Omega }_{0}\times \vec {M}]\Bigr )_{\parallel ,\perp }=-\frac{1}{\gamma _{\parallel ,\perp }}\Bigl (\vec {H}_{e}-\vec {H}\Bigr )_{\parallel ,\perp }, \end{aligned}$$
(4.19)

where

$$\begin{aligned} \gamma ^{-1}_{\parallel }=\frac{nm^{2}}{\alpha }\frac{2L(\xi _{e})}{\xi _{e}};~\gamma ^{-1}_{\perp }=\frac{nm^{2}}{\alpha }\Bigl (1-\frac{L(\xi _{e}}{\xi _{e}}\Bigr ). \end{aligned}$$
(4.20)

Here m is the magnetic moment of a colloidal particle, n is their concentration, \(\alpha \) is the rotational drag coefficient per unit volume, \(\xi _{e}\) is the Langevin parameter of the effective field \(\xi _{e}=mH_{e}/k_{B}T\) and \(L(\xi _{e})\) is the Langevin function.

The magnetic relaxation Eq. (4.19) and the equation of motion of an incompressible ferrofluid (4.6) (the Lorentz force on the electric current is neglected) give the closed set of equations describing its motion in the case of the non-equilibrium magnetization

$$\begin{aligned} \varrho \frac{d\vec {v}}{dt}=-\nabla p+\eta \Delta \vec {v}+(\vec {M}\cdot \nabla )\vec {H}+\frac{1}{2}\nabla \times [\vec {M}\times \vec {H}]. \end{aligned}$$
(4.21)

We draw attention to the appearance of the volume force \(\frac{1}{2}\nabla \times [\vec {M}\times \vec {H}]\) in (4.21) due to the inhomogeneity of the volume torque \([\vec {M}\times \vec {H}]\). The set of equations (4.19,4.21) describes a broad variety of phenomena connected with the spinning of magnetic particles as edge flows of a ferrofluid with free boundaries under the action of a rotating field [15], the increase of its effective viscosity in an applied field [10] and others. The expression for the dissipative function allows one to estimate the effect of magnetic hyperthermia which is of great interest due to its applications in biomedicine [16].

In the case when the magnetization of the ferrofluid is in thermodynamic equilibrium \(\vec {M}=M_{eq}(H)\vec {H}/H\) the equation of motion includes the Kelvin force \(M\nabla H\) and describes a variety of phenomena, for example, the deformation of droplets in an applied field [17], labyrinthine instabilities [18, 19] and magnetic microconvection [20] among others.