Numerical analysis of self-driven temperature-sensitive magnetic fluid heat flow around two side-by-side heating cylinders under a solenoidal magnetic field

Abstract

A temperature-sensitive magnetic fluid (TSMF) is a functional fluid that enables heat transfer due to the temperature difference and magnetic field. Using this property, we can increase the driving force while promoting heat transfer by increasing the heat transfer area, in the same manner as a heat sink. However, when multiple cylinders are juxtaposed, the interference between the cylinders will greatly affect the heat transfer. In the present study, we used the hybrid lattice Boltzmann method to investigate the flow and heat transfer characteristics of a TSMF around two side-by-side heating circular cylinders under a solenoidal magnetic field. As the magnetic field increases, the heat transfer effect increases. Moreover, when the distance between the cylinders is close enough, the heat flow becomes unstable. Under the same magnetic field conditions, the unstable flow can promote heat transfer.

Keywords

Temperature-sensitive magnetic fluid heat transfer flow around cylinder numerical simulation

1. Introduction

A temperature-sensitive magnetic fluid (TSMF) has temperature-sensitive magnetization. The magnetization changes significantly with temperature because ferromagnetic fine particles with a low Curie temperature are stably dispersed in the solvent [1]. Based on this characteristic, a long-distance heat transfer device has been reported [2]. Since this fluid is driven by a magnetic field and a temperature difference, the resulting flow phenomenon is very complex, especially because the temperature difference changes the force, which leads to a change in heat flow. The heat flow around a single cylinder has been analyzed [3], and both the temperature gradient and the magnetic field can effectively enhance heat transfer in a flow-driven TSMF. In the present study, we introduce complex shapes in the heating section to increase the heat transfer area, which can improve the heat transfer capability of a self-driven magnetic fluid. When complex shapes (such as typical heat sinks) are used in the heating section, the interference of the heating sections with each other can cause the heat flow to become very complicated. The present study focuses on a relatively simple model of heat flow around a TSMF around side-by-side heating cylinders. This model was chosen because the flow around paired cylinders has been the subject of numerous studies [4–6]. It has been reported that when the distance between the cylinders is small, the flow between the cylinders is prone to interference.

Since the self-driven TSMF is very different from the above studies in terms of driving method, research on two side-by-side heating cylinders under a magnetic field is necessary. In the present study, we use the hybrid lattice Boltzmann method [7,8] to analyze the thermal flow in a self-driven TSMF around two side-by-side cylinders and focus on heat flow around a cylinder and the interference between two cylinders.

2. Numerical simulation

2.1. Simulation method

The motions of the self-driven TSMF are dominated mainly by the magnetic body force, which is defined as $\begin{eqnarray}\displaystyle \boldsymbol{F}_{\boldsymbol{m}}={\mu}_{0}(\boldsymbol{M}\cdot {\nabla})\boldsymbol{H}, & & \displaystyle\end{eqnarray}$ (1) where μ₀ is the magnetic permeability in a vacuum, H is the magnetic field intensity, and M is the magnetization.

The motion of a fluid is analyzed using the lattice Boltzmann method [7] as a continuum by considering a hypothetical particle distribution function and solving the time evolution equation for the distribution function. In the lattice Boltzmann method [7], the fictive particles perform stream and collision processes over the lattice mesh. Motions of the fictive particles are determined by the time development of the distribution function f_i at each lattice point as the following equation [7]. $\begin{eqnarray}\displaystyle f_{i}(\boldsymbol{r}+\boldsymbol{c}_{i}{\Delta}t,t+{\Delta}t)=f_{i}(\boldsymbol{r},t)-\frac{1}{{\tau}_{f}}\{f_{i}(\boldsymbol{r},t)-f_{i}^{eq}(\boldsymbol{r},t)\}+{\Delta}t\boldsymbol{F}, & & \displaystyle\end{eqnarray}$ (2) where τ_f is the relaxation time, Δt is the time step, $f_{i}^{eq}$ is the local equilibrium distribution function, F is the external force, c _i is the lattice velocity in the i-direction. In the D2Q9 model we used which shown in Fig. 1, i = 0 ∼8. The relationship between time interval Δt and lattice Δ r is set as cΔt∕Δ r = 1

Fig. 1.

D2Q9 model.

Macroscopic fluid density 𝜌 and flow velocity u are calculated using the following equations: $\begin{eqnarray}\displaystyle {\rho}=\mathop{\sum }_{i}f_{i}, & & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \boldsymbol{u}=\mathop{\sum }_{i}f_{i}\boldsymbol{c}_{i}/{\rho}. & & \displaystyle\end{eqnarray}$ (4) To represent heated cylinders, the immersed boundary method was introduced in our simulations to describe the particle and to consider interactions between the cylinders and the fluid. The interaction between the fluid and the cylinders, and the influence of the magnetic body force on the fluid are acting on the fluid through the external force term of Eq. (1).

Due to the high Prandtl number for the TSMF in the present study, it is difficult to analyze the temperature field using the Boltzmann method. Thus, we use the finite difference method to calculate the magnetic body force and energy equations. The dimensionless forms of the equation of continuity, the equation of motion, and the energy equation [9] are defined as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \boldsymbol{u}=0,\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}\left[\frac{\partial u}{\partial t}+(\boldsymbol{u}\cdot {\nabla})\boldsymbol{u}\right]=-{\nabla}p+Pr{\nabla}^{2}\boldsymbol{u}+Rm\cdot Pr(\boldsymbol{M}\cdot {\nabla})\boldsymbol{H},\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial T^{\ast }}{\partial t^{\ast }}+(\boldsymbol{u}\cdot {\nabla})T={\nabla}^{2}T,\end{eqnarray}$ (7) where 𝜌 is the density, u is the velocity vector, p is the pressure, and T is the temperature. Since our model is two-dimensional, Eq. (6) is based on the Rosensweig equation [1] and the effect of gravity is ignored.

The magnetic effect parameter (Rm) and the Prandtl number (Pr) are defined as: $\begin{eqnarray}\displaystyle Rm\text{} & =\text{} & \displaystyle \frac{{\mu}_{0}{\chi}_{0}{H_{max}}^{2}D^{2}}{{\eta}{\alpha}},\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle Pr\text{} & =\text{} & \displaystyle {\eta}/{\alpha}{\rho},\end{eqnarray}$ (9) where D is the cylinder diameter, 𝛼 is the thermal diffusivity, 𝜂 is the viscosity, ΔT is the difference between the temperatures of the heated plate and the inlet, 𝜒₀ is the magnetic susceptibility, and H_max is the maximum magnetic field intensity.

2.2. Analytical model

Figure 2 shows the analytical model used in the present study. The red circles indicate the heating cylinders, which are set to the heating temperature of the heater. Since the cylinder is set to the right of the centerline, according to the principle of self-driving, the main flow direction will be to the right. We consider that the TSMF flows out from the outlet after being cooled by the cooling section and then flows in again at the inlet. The left boundary can be regarded as the inlet, and the right border is the outlet. Bounce-back and adiabatic conditions were used for the upper and lower walls, and the outflow boundary condition was used for the right wall. The left wall was used to rectify the boundary condition.

Fig. 2.

The numerical model and boundary condition of this study.

The rectified boundary condition is based on the Zou-He boundary condition [10]. Since the driving force in the present study comes entirely from the vicinity of the heated cylinder, the fluid at the inlet and outlet is not changed by external forces. Therefore, the inlet flow can be considered as the outlet flow after rectification into a Poiseuille flow. At the left boundary, we first set up the uniform flow using the Zou-He boundary condition. Due to the possibility of generating a velocity v_out in the y-direction at the outlet and a Poiseuille flow v = v_in = 0 at the inlet, we define the inlet velocity as $u=∼\overline{u}_{in}=∼\sqrt{\overline{u}_{out}^{2}+\overline{v}_{out}^{2}}$ and the density as ${\rho}=∼\bar{{\rho}}_{in}=∼\bar{{\rho}}_{out}$ . Then, we calculate the Poiseuille flow from the average velocity $\bar{u}_{in}$ and bring the Poiseuille flow to the Zou-He boundary.

Figure 3 shows the applied two-dimensional magnetic field distributions. In this analysis, a solenoid magnetic field is assumed and calculated using the Biot-Savart law. The numerical conditions were set as shown in Table 1.

Fig. 3.

Distribution of magnetic field.

Table 1

Numerical conditions

Mesh size	4,000 × 600
Prandtl number, Pr	134
Magnetic effect parameter, Rm	6.8, 27.2, 61.2, 108.8 ×10⁷
Ratio of distance between cylinders to diameter, L∕D	1.5, 1.75, 2, 2.25, 2.5

Fig. 4.

Time change of driving force.

Fig. 5.

Temperature distribution around cylinders (x = 0 to 5) when Rm = 108.8 ×10⁷.

Fig. 6.

Temperature distribution for D∕L = 2.5.

Fig. 7.

Nusselt number distribution for all conditions.

3. Simulation results and discussion

The present study is based on the self-driven TSMF. The self-driving principle [9] uses the difference in magnetization caused by temperature changes to generate a difference in the offset magnetic body force, which is defined in Eq. (1), thereby driving the flow of the TSMF. Therefore, we define the driving force as $\begin{eqnarray}\displaystyle \boldsymbol{F}_{\text{driven}}=\mathop{\sum }_{i,j}\sqrt{\boldsymbol{F }_{mi,j}^{2}}. & & \displaystyle\end{eqnarray}$ (10) Figure 4 shows the time change of the driving force. For Rm = 108.8 ×10⁷, the driving force results, except those for L∕D = 2.5, all show instability. Figure 5 compares the time changes of the temperature field around the heating cylinders for L∕D = 1.5, 2.0, and 2.5. We found that the wake of the cylinders becomes stable as the distance between the cylinders decreases. We believe that this is due to indirect interactions between the cylinders, because when the distance between cylinders is smaller, the closeness of the cylinder wakes causes the heat flow to become unstable. The flow along the centerline is accelerated by squeezing of the two cylinders, which results in a velocity difference between the upper and lower edges of the cylinders, resulting in unstable flow.

Figure 6 shows the effect of Rm on the temperature distribution. As Rm becomes larger (magnetic field becomes stronger), the heated area behind the cylinder increases, which means that heat escape is more difficult, and this heat also increases the driving force.

In the present study, the average Nusselt number around the cylinders was used to evaluate the heat transfer effect and was calculated by converting Cartesian coordinates into cylindrical coordinates. The average Nusselt number Nu and local Nusselt number Nu (θ) are given by $\begin{eqnarray}\displaystyle Nu=\int _{-{\pi}}^{{\pi}}Nu({\theta})d{\theta}, & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle Nu({\theta}^{\ast })=-\left[\frac{\partial T}{\partial r}\right]_{r=0}, & & \displaystyle\end{eqnarray}$ (12) where r is the radius of the cylindrical coordinate system after coordinate transformation, and θ is the angle. The results are shown in Fig. 7. For all L∕D values, the heat transfer effect tended to increase as Rm increased. This indicates that the magnetic field increases the driving force and thus increases the heat transfer effect. Except for L∕D = 2.5, Nu changes approximately linearly with Rm. However, at Rm = 108.8 ×10⁷, the heat transfer effect increases as the distance between the cylinders decreases. We believe that the instability of the flow can promote heat transfer.

4. Conclusion

The behavior of the self-driven TSMF around two side-by-side cylinders was investigated using the hybrid lattice Boltzmann method. The increase of the magnetic field (Rm) can enhance the heat transfer, and when the magnetic field is strong enough and the distance between the cylinders is small, the heat flow wake produced by the cylinders becomes unstable and can promote heat transfer.

Footnotes

Acknowledgements

The present study was supported by JST SPRING, Grant Number JPMJSP2112.

References

Rosensweig

R.E.

, Ferrohydrodynamics, Cambridge University Press, Cambridge, 1985.

Iwamoto

Nakasumi

Ido

and Yamaguchi

, Long-distance heat transport based on temperature-dependent magnetization of magnetic fluids, Int. J. Appl. Electromagn. Mech.62 (2020), 711–724.

Iwamoto

Nakasumi

Ido

and Niu

X.D.

, Heat transfer of temperature-sensitive magnetic fluids around single heating pipe, Int. J. Appl. Electromagn. Mech.64 (2020), 1039–1046.

Sumner , Fluid be haviour of side-by-side circular cylinders in steady cross-flow, Journal of Fluids and Structures13 (1999), 309–338.

Zhiwen

W.A.N.G.

and Hui

, Flow-visualization of a two side-by-side cylinder wake, Journal of Flow Visualization and Image Processing9(2&3) (2002).

Mahbub

Alam Md

Moriya

and Sakamoto

, Aerodynamic characteristics of two side-by-side circular cylinders and application of wavelet analysis on the switching phenomenon, Journal of Fluids and Structures18.3-4 (2003), 325–346.

Mohamad

A.A.

, Lattice Boltzmann Method, Springer-Verlag, London limited, 2011.

Feng

and Michaelides

, The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comp. Phys.195 (2004), 602–628.

Rong

Z.W.

Iwamoto

and Ido

, Thermal flow analysis of self-driven temperature-sensitive magnetic fluid between partially heated parallel plates, J. Mag. Mag. Mat.552 (2022), 169079.

10.

Zou

Q.S.

and He

X.Y.

, On pressure and velocity flow boundary conditions and bounce back for the lattice Boltzmann BGK model, eprint, arXiv: preprint comp-gas, 1996.