## Abstract

Thermomagnetic convection is based on the use of external magnetic fields to better control heat transfer fluxes in ferrofluids, finding important applications in engineering and many related areas. The improvement of such methods relies on fundamentally understanding the flow of ferrofluids under temperature gradients and external magnetic fields. However, the underlying physics of this phenomenon is very complex and not yet well characterized. The problem we analyze in this paper consists of a ferrofluid confined in a square cavity heated from the top and subjected to an external magnetic field applied in the horizontal direction. Differently from earlier investigations, this scenario leads to a clear competition between stabilizing gravitational forces and destabilizing magnetic forces; the unbalance between them drives the flow and dictates the mechanisms of heat transfer in the system. The problem is described by the equations of conservation of mass, momentum, and energy; both gravitational and magnetic effects are accounted for through the definition of the body force following the well-known Boussinesq approximation to express the ferrofluid’s density and magnetization as functions of temperature. The resulting set of fully coupled, nonlinear equations of the model is solved with a fully implicit finite element method. The results show that thermomagnetic convection increases convective heat transfer fluxes in the flow, but its net effects essentially depend on a complex balance among viscous, magnetic, and gravitational forces which determines the pattern of recirculation regions inside the cavity. There are two critical values associated with the external field’s intensity: the first marks the onset of thermomagnetic convection when the destabilizing magnetic effects become comparable to the stabilizing gravitational ones and the second corresponds to the external field’s strength that suffices to make the effects of gravity nearly negligible in the flow dynamics. The latter regime is dictated by a balance between viscous and magnetic forces only, and the corresponding numerical predictions agree notoriously well with a scaling analysis which suggests that \({\overline{\mathrm{Nu}}} \sim \mathrm{Ra}_\mathrm{m}^{1/4}\), where \({\overline{\mathrm{Nu}}}\) is the average Nusselt number at the isothermal walls and \(\mathrm{Ra}_\mathrm{m}\) is the magnetic Rayleigh number, a dimensionless parameter associated with the intensity of the external field.

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## Notes

- 1.
In general, the magnetic susceptibility is written as \(\chi = \chi (T,H)\), where \(H = ||{\varvec{H}}||\) is the intensity of the local magnetic field. Indeed, from the superparamagnetic theory for dilute ferrofluids [33], we have that \({\varvec{M}}= M_\mathrm{s} \mathcal {L}(\xi ) {\hat{{\varvec{H}}}}\), where \(M_\mathrm{s}\) is the saturation magnetization (that is, the maximum magnetization that can be achieved by the ferrofluid), \(\mathcal {L}(\xi ) = \coth (\xi ) - \xi ^{-1}\) is the Langevin function, and \({\hat{{\varvec{H}}}} = {\varvec{H}}/H\) is the unit vector along the direction of \({\varvec{H}}\). Here, \(\xi = \mu _0 mH/k_\mathrm{B} T\) is a dimensionless parameter that represents the ratio of magnetic-to-thermal energy, where

*m*is the average intensity of the magnetic dipole of the suspended particles (which depends on their size and material) and \(k_\mathrm{B}\) is the Boltzmann constant. If the magnetic energy is much smaller than the thermal energy, which is the case considered herein, it follows that \(\xi \) is very small and the Langevin function can be approximated as \(\mathcal {L}(\xi ) \approx \xi /3\). Under this assumption, we have that \({\varvec{M}}= \chi (T){\varvec{H}}\) with \(\chi (T) = M_\mathrm{s} \mu _0 m/3k_\mathrm{B}T\). - 2.
In general, the local magnetic field is written as \({\varvec{H}}= {\varvec{H}}_\mathrm{e} + {\varvec{H}}_s\), where \({\varvec{H}}_s\) is the stray field (or demagnetizing field) that arises because of the interaction between the magnetization of the ferrofluid with the external magnetic field. However, if we assume that the saturation magnetization is much smaller than the external magnetic field, \({\varvec{H}}_s\) becomes negligible compared to \({\varvec{H}}_\mathrm{e}\) and hence \({\varvec{H}}\approx {\varvec{H}}_\mathrm{e}\). This approximation can be seen as a one-way coupling: the external magnetic field induces the ferrofluid flow, but the magnetic field created by the ferrofluid does not affect the external magnetic field in the flow domain.

- 3.
The reference power-law curve used here is \({\overline{\hbox {Nu}}} = \exp [A\ln (\hbox {Ra}_\mathrm{m}) + B]\) with \(A \approx 0.2612\) and \(B \approx -1.5722\), obtained by fitting the results for \(\hbox {Ra} = 0\) in the range \(10^8 \le \hbox {Ra}_\mathrm{m} \le 10^9\).

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## Funding

was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (BR) (Grant No. 303835/2018-4).

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Lucas H. P. Cunha, Ivan R. Siqueira, and Arthur A. R. Campos have contributed equally to this work.

Communicated by Oleg Zikanov.

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Cunha, L.H.P., Siqueira, I.R., Campos, A.A.R. *et al.* A numerical study on heat transfer of a ferrofluid flow in a square cavity under simultaneous gravitational and magnetic convection.
*Theor. Comput. Fluid Dyn.* (2020). https://doi.org/10.1007/s00162-020-00515-1

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### Keywords

- Ferrofluids
- Thermomagnetic convection
- Convective heat transfer
- Nusselt number
- Finite element method