Heat Transfer Study of the Ferrofluid Flow in a Vertical Annular Cylindrical Duct under the Influence of a Transverse Magnetic Field

Abstract

We studied the laminar fully developed ferrofluid flow and heat transfer phenomena of an otherwise magnetic fluid into a vertical annular duct of circular cross-section and uniform temperatures on walls which were subjected to a transverse external magnetic field. A computational algorithm was used, which coupled the continuity, momentum, energy, magnetization and Maxwell’s equations, accompanied by the appropriate conditions, using the continuity–vorticity–pressure (C.V.P.) method and a non-uniform grid. The results were obtained for different values of field strength and particles’ volumetric concentration, wherein the effects of the magnetic field on the ferrofluid flow and the temperature are revealed. It is shown that the axial velocity distribution is highly affected by the field strength and the volumetric concentration, the axial pressure gradient depends almost linearly on the field strength, while the heat transfer significantly increases due to the generated secondary flow.

1. Introduction

The examination of the characteristics of ferrofluids has generated much interest in past decades due to the many important technological applications [1,2,3,4]. The dispersion of a magnetizable mineral, such as an iron oxide powder, in a liquid solution, creates a fluid with unique properties such as magnetoviscosity [5,6,7] or negative viscosity [8], in the presence of a magnetic field. These properties can be exploited to create specialized engineering systems targeted for applications such as magnetic dampers [8] and shaft seals [9].

The numerical works that deal with ferrofluids usually focus on simple geometries, such as parallel plates and circular tubes, to facilitate the understanding of the complex phenomena that arise in these flows. The different timescales of the various physical mechanisms, i.e., the Brownian diffuse timescale, the hydrodynamic (convective) timescale and the timescale associated with applied magnetic field, are responsible for the non-equilibrium magnetization dynamics of the flow problem [10]. This variation in the relaxation time creates a misalignment between the local magnetization and the local magnetic field, leading to enhanced effective viscosity [11]. The interaction between the magnetic forces and the pressure gradient, apart from affecting the flow properties, induces changes in the flow pattern which may be utilized to control flow separation [12].

The numerical studies of ferrofluid flow focus also on geometries which are close to the actual engineering applications. The properties that are examined in this context are the pressure drop and heat transfer [13,14,15], while special attention is given to the velocity distributions that arise due to the interaction of the pressure gradients and the magnetic forces [16]. The flow and heat transfer characteristics of magnetic nanofluids have been compiled in a review paper [17] that discusses the properties of magnetic nano-fluids and their effect on natural convection, forced convection, and boiling. Specifically for thermal applications, annular pipes and double pipes are common configurations in heat exchangers, and their study is always interesting either in the case of ferrofluid flow or in neighboring scientific fields such as magnetohydrodynamics [18,19,20]. The utilization of different pipe shapes (e.g., sinusoidal or wavy [21]) and the inclusion of porous zones [22] in the flow passage can have a positive effect in heat transfer, although there is a penalty in pressure drop. Square ducts have also been reported in the bibliography [23], where flow takes place under the influence of transverse or axial magnetic fields. The different types of magnetic fields also play an important role in the flow properties and some of the configurations that have been examined are different types of magnets, such as quadrupole magnets [24], or current-carrying wires and coils [14,25]. An experimental study [26] showed that the use of ferrofluid flow in a circular pipe under the effect of a constant magnetic field can increase the heat transfer by up to 7.19%. Alternating magnetic fields are also important because they can enhance the heat transfer properties by up to 13.9% [13], when the frequency is optimum. Finally, the type of ferrofluid is an important element in the different configurations, with the most common being Fe₃O₄–water ferrofluids [27,28]. However, the range of applications is not restricted to engineering; there have been reports on the use of blood as ferrofluid [29,30].

In this context, the present work examines the ferrofluid flow of a magnetite–water solution in a vertical annular pipe. The magnetic field is constant and acts along the horizontal direction. This type of flow is important for cooling units and heat exchangers, and it can be employed in various engineering applications. The numerical model uses the formulation derived by Hatzikonstantinou and Vafeas [7] and the solution procedure is based on a validated in-house code. The main parameters that are studied are the magnetic field strength and the concentration of magnetic particles, which play an important role in the pressure drop, the velocity profiles and the heat transfer characteristics of the flow. To the best of our knowledge, this is the first work that examines the effects of these parameters in a forced convection problem with the inclusion of buoyancy effects. The results highlight the interaction between the gravitational, viscous, and magnetic forces, and provide useful information concerning the heat transfer augmentation that can be achieved.

2. Mathematical Model

We formulate our problem concerning the straight circular cylindrical annular duct, depicted in Figure 1, given the inner $R_i$ and the outer $R_o$ radii of the coaxial cylinders, wherein a conveniently adjusted Cartesian reference system $\left(x,y,z\right)$ is sketched. The inner and outer cylinders are maintained at uniform temperatures, assuming that the inner wall temperature $T_i$ is lower than the outer wall temperature $T_o$.

As for the developed buoyancy forces, the Boussinesq approximation is used, while the effect of viscous dissipation is neglected. On the other hand, the non-conductive ferrofluid is constituted by a stable suspension of solid spherical particles of radius $r_p$ and density ${\rho }_p$ with volumetric concentration $\varphi =\left(4/3\right)\pi r_p^{3}n$, where $n$ stands for the number of particles per unit volume. They control the variation of the viscosity and the pressure of the ferrofluid, provided that an external magnetic field $H_a$ is applied, as demonstrated in Figure 1. The best fitted circular cylindrical coordinate system $\left(r,\theta ,z\right)$ with $r\in \left(R_i,R_o\right)$, $\theta \in \left[0,2\pi \right)$ and $z\in \left(-\infty ,+\infty \right)$ is implied, which is connected to the Cartesian coordinates via $\left(x,y,z\right)=\left(r\cos \theta ,r\sin \theta ,z\right)$.

In terms of the gradient $\nabla$ and Laplacian $\mathsf{\Delta }$ differential operators, the governing equations that describe the physical phenomenon of the particular ferrohydrodynamic flow within the annular duct with heat alterations, interconnect the velocity $v$, the angular velocity $\mathsf{\Omega }=\nabla \times v$, the total pressure $P$ that incorporates the gravitational force, the magnetization $M$, the magnetic field $H$, which is the summation of the applied and the induced field, the latter being neglected, the magnetic induction $B$ and the temperature $\mathbf{T}$. Each implicated physical field is then a function of the current position vector $r=x\widehat{\mathit{x}}+y\widehat{\mathit{y}}+z\widehat{\mathit{z}}$, written in view of the Cartesian basis $\widehat{\mathit{x}}$, $\widehat{\mathit{y}}$, $\widehat{\mathit{z}}$ and the time variable $t$, omitting this dependence thereafter for notational clarity. Therein, supposing an incompressible Newtonian magnetic fluid with constant mass density $\rho$ and with constant dynamic viscosity $\eta$, we introduce the following set of dimensional equations, which are comprised by the continuity equation:

$$\nabla \cdot v=0,$$

(1)

the momentum equation:

$$\rho \frac{Dv}{Dt}\equiv \rho \left[\frac{\partial v}{\partial t}+\left(v\cdot \nabla \right)v\right]=-\nabla P+\eta \mathsf{\Delta }v+{\mu }_0\left(M\cdot \nabla \right)H+\frac{{\mu }_0}{2}\nabla \times \left(M\times H\right)-\rho g\beta \left(T-T_i\right)\widehat{\mathit{z}},$$

(2)

where ${\mu }_0$ is the magnetic permeability of vacuum, $g$ is the gravitational acceleration and $\beta$ is the coefficient of thermal expansion, the energy equation:

$$\frac{D\mathbf{T}}{Dt}\equiv \left[\frac{\partial \mathbf{T}}{\partial t}+\left(v\cdot \nabla \right)\mathbf{T}\right]=\frac{k}{\rho c_p}\mathsf{\Delta }\mathbf{T},$$

(3)

where $k$ is the thermal conductivity and $c_p$ is the specific heat, and the Maxwell’s equations:

$$\nabla \cdot B=0\mathrm{and}\nabla \times H=0,$$

(4)

where for a linear, homogeneous and isotropic fluid, it holds that $B={\mu }_0\left(H+M\right)$. Those are supplemented by the magnetization equation:

$$\frac{DM}{Dt}\equiv \frac{\partial M}{\partial t}+\left(v\cdot \nabla \right)M=\frac{1}{2}\mathsf{\Omega }\times M+\frac{{\mu }_0{\tau }_S}{I}\left(M\times H\right)\times M-\frac{1}{{\tau }_B}\left(M-M_0\frac{H}{H}\right)$$

(5)

where $M_0$ denotes the equilibrium magnetization field, rendered by the Langevin function:

$$M_0=nmL\left(\xi \right)\mathrm{with}L\left(\xi \right)=\coth \xi -\frac{1}{\xi },\mathrm{where}\xi =\frac{{\mu }_{0}mH}{KT},$$

(6)

wherein $m$ is the magnetic moment of a single particle, whose magnetization $M_p$ is related to the saturation magnetization $M_s$ via the relationship $M_s=\varphi M_p=nm$. Otherwise, ${\tau }_S=r_p^2{\rho }_p/15{\eta }_0$ is the relaxation time of particle rotation (${\eta }_0$ corresponding to the rotational viscosity), $I=8\pi r_p^5{\rho }_{p}n/15$ is the sum of moments of inertia of the spherical particles per unit volume and ${\tau }_B=4\pi \eta r_p^3/KT$ is the relaxation time of the Brownian rotation ($K$ being the Boltzmann’s constant). However, due to the approximate homogeneity of the magnetic field and the very small magnetization inertia of the magnetic fluid, resulting in the almost immediate orientation of the magnetization leads to $DM/Dt\cong 0$, so that partial differential Equation (5) becomes algebraic and is simplified to the analytical formula:

$$M=\frac{M_0}{H}{\left[1+{\left(\frac{\Omega {\tau }_B}{2f\left(H,\Omega \right)}\right)}^2\right]}^{-1}\left\{H+\frac{{\tau }_B}{2f\left(H,\Omega \right)}\mathsf{\Omega }\times H+{\left(\frac{{\tau }_B}{2f\left(H,\Omega \right)}\right)}^2\left(H\cdot \mathsf{\Omega }\right)\mathsf{\Omega }\right\}$$

(7)

where $H=\left|H\right|$ and $\Omega =\left|\mathsf{\Omega }\right|$. Using the dimensionless function:

$$g\left(H\right)=1+\frac{{\mu }_0{\tau }_S{\tau }_B}{I}H{M}_0,$$

(8)

it is readily obtained:

$$f\left(H,\Omega \right)=\frac{g\left(H\right)}{3}\left\{1+R\left(H,\Omega \right)+\frac{1}{R\left(H,\Omega \right)}\left[1-3{\left(\frac{\Omega {\tau }_B}{2g\left(H\right)}\right)}^2\right]\right\}$$

(9)

with:

$$R\left(H,\Omega \right)={\left\{Q_{\phi }\left(H,\Omega \right)+\sqrt{Q_{\phi }^2\left(H,\Omega \right)-{\left[1-3{\left(\frac{\Omega {\tau }_B}{2g\left(H\right)}\right)}^2\right]}^3}\right\}}^{1/3},$$

(10)

in which, bearing in mind that $\phi$ is the angle between the magnetic field $H$ and the vorticity $\mathsf{\Omega }$, which is $\mathsf{\Omega }\cdot H=\Omega H\cos \phi$, then:

$$Q_{\phi }\left(H,\Omega \right)=1+{\left(\frac{3\Omega {\tau }_B}{2g\left(H\right)}\right)}^2\left[1-\frac{3\left(g\left(H\right)-1\right)}{2g\left(H\right)}{\sin }^2\phi \right],$$

(11)

providing adequate information to compute Equation (7). In summary, we are obliged to solve the continuity Equation (1), the momentum law (Equation (2)), the energy Equation (3), Maxwell’s relationships (Equation (4)) and, instead of Equation (5), the analytical form for the magnetization of the ferrofluid given by Equation (7), taking into account Equations (6) and (8)–(11). Those are accompanied by the non-slip boundary conditions for the velocity field, i.e., $v=0$ and the imposition of the standard temperatures $T_i$ and $T_o$ on both the walls of the annular duct for $r=R_i,R_o$, as well as implying the excitation through the magnetic field $H_a$. Additionally, the flow is considered fully developed in the $z$-direction, while the magnetization on the boundaries can be recovered directly from the compact Equation (7).

The numerical implementation of the aforementioned boundary value problem requires the production of the corresponding dimensionless forms for all the involved equations. Following that, we define the dimensionless variables:

$$\begin{array}{l}\tilde{r}=\frac{r}{R_o},\tilde{t}=\frac{Ut}{R_o},\tilde{R}=\frac{R_i}{R_o},\tilde{\nabla }=R_o\nabla ,\tilde{\mathsf{\Delta }}=R_o^2\mathsf{\Delta }\\ \tilde{v}=\frac{v}{U},\tilde{P}=\frac{P}{\rho U^2},\tilde{T}=\frac{T-T_i}{T_o-T_i},\tilde{\mathsf{\Omega }}=\frac{R_o\mathsf{\Omega }}{U}\\ \tilde{H}=\frac{H}{H_a},\tilde{M}=\frac{M}{M_{s,\max }},{\tilde{M}}_0=\frac{M_0}{M_{s,\max }}=\varphi L\left(\xi \right),\end{array}$$

(12)

where $U$ is the mean velocity of the ferrofluid, $H_a=\left|H_a\right|$ and $M_{s,\max }$ are the saturation magnetization for $\varphi =1$ and $H=H_a$, i.e., $M_{s,\max }=3m/4\pi r_p^3$, because $M_s=nm$, given that $\varphi =\left(4/3\right)\pi r_p^{3}n$. By virtue of Equation (12), direct substitution to Equations (1)–(5) (or Equation (7) as the analytical counterpart of Equation (5) with Equations (8)–(11)) provides us with the dimensionless relationships of the continuity equation:

$$\tilde{\nabla }\cdot \tilde{v}=0,$$

(13)

the momentum equation:

$$\frac{\partial \tilde{v}}{\partial \tilde{t}}+\left(\tilde{v}\cdot \tilde{\nabla }\right)\tilde{v}=-\tilde{\nabla }\tilde{P}+\frac{1}{\mathrm{Re}}\tilde{\mathsf{\Delta }}\tilde{v}+\frac{{\mathrm{R}}_{\mathrm{M}}}{{\mathrm{Re}}^2}\left[\left(\tilde{M}\cdot \tilde{\nabla }\right)\tilde{H}+\frac{1}{2}\tilde{\nabla }\times \left(\tilde{M}\times \tilde{H}\right)\right]-\frac{\mathrm{Gr}}{{\mathrm{Re}}^2}\mathrm{T}˜\widehat{z},$$

(14)

the energy equation:

$$\frac{\partial \tilde{\mathbf{T}}}{\partial \tilde{t}}+\left(\tilde{v}\cdot \tilde{\nabla }\right)\tilde{\mathbf{T}}=\frac{1}{\mathrm{Re}\mathrm{Pr}}\mathsf{\Delta }\tilde{\mathbf{T}},$$

(15)

the Maxwell’s equations:

$$\tilde{\nabla }\cdot \tilde{B}=0\mathrm{and}\tilde{\nabla }\times \tilde{H}=0$$

(16)

and, additionally, either the differential form of the magnetization equation:

$$\frac{\partial \tilde{M}}{\partial \tilde{t}}+\left(\tilde{v}\cdot \tilde{\nabla }\right)\tilde{M}=\frac{1}{2}\left(\tilde{\mathsf{\Omega }}\times \tilde{M}\right)+\frac{{\mathrm{R}}_{\mathrm{M}}}{6\varphi \mathrm{Re}}\left(\tilde{M}\times \tilde{H}\right)\times \tilde{M}-\frac{{\mathrm{R}}_{\mathrm{H}}}{\mathrm{Re}}\left(\tilde{M}-\varphi \frac{\tilde{H}}{\tilde{H}}L\left(\xi \right)\right)$$

(17)

or its analytical expression:

$$\tilde{M}=\varphi L\left(\xi \right)\frac{1}{\tilde{H}\left(1+{\left(N\tilde{\Omega }\right)}^2\right)}\left[\tilde{H}+N\left(\tilde{\mathsf{\Omega }}\times \tilde{H}\right)+N^2\left(\tilde{\mathsf{\Omega }}\cdot \tilde{H}\right)\tilde{\mathsf{\Omega }}\right]$$

(18)

Above are written in terms of the Reynolds, Grashof and Prandtl numbers:

$$\mathrm{Re}=\frac{\rho U{R}_o}{\eta },\mathrm{Gr}=\frac{g\beta \left(T_o-T_i\right){\rho }^2{R}_o^3}{{\eta }^2}\mathrm{and}\mathrm{Pr}=\frac{c_p\eta }{k},$$

(19)

respectively, the dimensionless characteristic numbers:

$${\mathrm{R}}_{\mathrm{M}}=\frac{{\mu }_0\rho R_o^2{M}_{s,\max }{H}_a}{{\eta }^2}\mathrm{and}{\mathrm{R}}_{\mathrm{H}}=\frac{\rho R_o^2}{\eta {\tau }_B},$$

(20)

as well as the known quantity:

$$N=\frac{\mathrm{Re}}{2{\mathrm{R}}_{\mathrm{H}}}{\left[\frac{q}{3}\left(1+R+\frac{1-3{\left(\tilde{\Omega }p\right)}^2}{R}\right)\right]}^{-1}\mathrm{with}p=\frac{\mathrm{Re}}{2{\mathrm{R}}_{\mathrm{H}}g}\mathrm{and}q=1+\frac{{\mathrm{R}}_{\mathrm{M}}}{6{\mathrm{R}}_{\mathrm{H}}}\tilde{H}L\left(\xi \right),$$

(21)

where

$$R={\left[Q+\sqrt{Q^2-{\left[1-3{\left(\tilde{\Omega }p\right)}^2\right]}^3}\right]}^{\frac{1}{3}}\mathrm{with}Q=1+9{\left(\tilde{\Omega }p\right)}^2\left[1-\frac{3\left(q-1\right)}{2q}{\left(\frac{\left|\tilde{\mathsf{\Omega }}\times \tilde{\mathbf{H}}\right|}{\tilde{\Omega }\tilde{H}}\right)}^2\right],$$

(22)

concluding our dimensionless analysis. The cylindrical geometry is implied and $\tilde{\mathbf{H}}\cong H_a$ as for the physical assumption, although, because $\tilde{r}=r/R_o$, the imposed boundary conditions read $\tilde{v}=0$ for $\tilde{r}=\tilde{R}$ and $\tilde{r}=1$, as well as $\tilde{T}=0$ for $\tilde{r}=\tilde{R}$ and $\tilde{T}=1$ for $\tilde{r}=1$. On the other hand, because $\tilde{z}=z/R_o$, the fully developed flow in the $\tilde{z}$-direction is secured by the fact that are all the components of the fields are independent of the axial coordinate $\tilde{z}$. Although the pressure field $\tilde{P}$ is a linear function of $z$, all the axial derivatives are neglected, except for the axial pressure gradient ${\tilde{P}}_{c,z}\equiv d{\tilde{P}}_a\left(\tilde{z}\right)/d\tilde{z}$, which is constant. Finally, $\tilde{M}$ on $\tilde{r}=\tilde{R}$ and $\tilde{r}=1$ is obtained via the analytical relationship in Equation (18), while the corresponding differential Equation (17) for the magnetization will be used for validation. In conclusion, a very important quantity is the stream function $\psi$ ($\tilde{\psi }=\psi /U{R}_o$ as for its dimensionless form), which is connected to the velocity components $v_r$ and $v_{\theta }$ in cylindrical coordinates through the relationships:

$$v_r=\frac{1}{r}\frac{\partial \psi }{\partial \theta }\mathrm{and}{v}_{\theta }=-\frac{\partial \psi }{\partial r}\mathrm{or}{\tilde{v}}_r=\frac{1}{\tilde{r}}\frac{\partial \tilde{\psi }}{\partial \theta }\mathrm{and}{\tilde{v}}_{\theta }=-\frac{\partial \tilde{\psi }}{\partial \tilde{r}},$$

(23)

wherein trivial integration techniques give the stream function in terms of the velocity field.

Heat transfer is studied with the Nusselt number. Local Nusselt numbers for the inner and outer wall are given by the relationships:

$${\mathrm{Nu}}_{\mathrm{i}}={\left.\frac{\partial \tilde{T}}{\partial r}\right|}_{\tilde{r}=\tilde{R}}\mathrm{and}{\mathrm{Nu}}_{\mathrm{o}}={\left.\frac{\partial \tilde{T}}{\partial r}\right|}_{\tilde{r}=1},$$

(24)

while the average Nusselt numbers for the inner and outer wall are given by:

$${\overline{\mathrm{Nu}}}_{\mathrm{i}}=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\mathsf{\int }}}{\mathrm{Nu}}_{\mathrm{i}}\left(\theta \right)d\theta }\mathrm{and}{\overline{\mathrm{Nu}}}_{\mathrm{o}}=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\mathsf{\int }}}{\mathrm{Nu}}_{\mathrm{o}}\left(\theta \right)d\theta }$$

(25)

Equations (14)–(18) that govern the ferrofluid and thermal flow were solved using a methodology involving the enhanced C.V.P. numerical variational method for the coupling of the continuity, the energy, the Navier–Stokes equation and the magnetization equation. In this method, the partial differential equations are discretized and solved numerically using a pseudo-transient marching algorithm. Non-uniform stretched meshes are used to accurately compute the thin boundary layers near the walls. The C.V.P. method’s characteristics (extreme accuracy, robustness, easy convergence and easy implementation to complex geometries) make it unique for magnetic fluid applications. The numerical method is analytically presented and validated elsewhere [16]. The analytical solution of the magnetization equation for a ferrofluid flow inside a straight circular duct has also been validated previously [14].

3. Results and Discussion

Computations were carried out for a wide range of physical parameters of the ferrofluid flow for $\tilde{R}=0.5$. The relative flow, heat transfer and magnetization parameters varied in ranges, whereas $0\le \varphi \le 0.1$, $0<\xi \le 1$, $0\le \mathrm{Gr}\le {10}^3$, $\mathrm{Pr}=6.8$, $\mathrm{Re}=100$, ${\mathrm{R}}_{\mathrm{M}}=1.8\times {10}^{-4}$ and ${\mathrm{R}}_{\mathrm{H}}=2.77\times {10}^{-6}$. The numerical results are presented to illustrate the effect of the above parameters on the fluid flow and heat transfer to demonstrate the effect of the magnetic field on the velocity and temperature distribution.

The distributions of the axial velocity ${\tilde{v}}_{\zeta }$ in the cross-sectional area of the annular duct are presented in the contour plots of Figure 2 for various values of $\xi$ and $\mathrm{Gr}$ for concentration $\varphi =0.1$. For field strength $\xi =0$, the distribution of the axial velocity is uniform, while as the Grashof number increases, the axial velocity peak is shifted to the inner wall, due to the effect of the buoyancy forces. For $\mathrm{Gr}=0$, as the field strength $\xi$ increases, the axial velocity is suppressed near the specific angles $\theta =0°,90°,180°,270°$ and its maximum value is redistributed in four symmetric poles near angles $\theta =45°,135°,225°,315°$. For $\mathrm{Gr}={10}^3$, due to the effect of the buoyancy forces, we have the formulation of another two poles at angles $\theta =0°,180°$, of which the values are decreased as the field strength increases.

The contour plots of the streamlines of the transverse components of the velocity for various values of $\xi$ and Gr and for concentration $\varphi =0.1$ are shown in Figure 3 Due to the effect of the field strength, the transverse components of the velocity form two pairs of vortices (one clockwise and one anticlockwise) on each region of the duct near angles $\theta =45°,135°,225°,315°$. As the field strength increases, the value of the stream function $\psi$ also increases. Grashof number increase has a minor effect on the streamline values. Only for $\xi =0.1$ is the absolute value of the stream function $\psi$ slightly increased. For $\xi \ge 0.5$, the Grashof number has zero effect on the streamlines. This is due to the vertical alignment of the duct, because buoyancy forces are formulated in the axial direction.

The contour plots of the temperature for various values of $\xi$ and $\mathrm{Gr}$ for concentration $\varphi =0.1$ are shown in Figure 4. For field strength $\xi =0$, the distribution of the temperature is uniform. As the value of the stream function $\psi$ increases due to the increase in the field strength, the temperature is redistributed. Temperature profiles are the same for both values of the Grashof number, because the secondary flow is not affected by the buoyancy forces.

Increasing the field strength $\xi$ and concentration $\varphi$ significantly improves the heat transfer at the walls, as expressed by the Nusselt number, due to the induction of the secondary flow. On the contrary, the increase in the Grashof number has a negligible effect on the heat transfer mechanism. As is presented in Figure 5, for $\varphi =0.01$, the heat transfer increases around 168% on the inner wall and 145% on the outer wall, for field strength $\xi =2.0$ in comparison to $\xi =0$. This effect is much higher for $\varphi =0.1$, where the heat transfer increases around 469% on the inner wall and 407% on the outer wall, for field strength $\xi =2.0$ in comparison to $\xi =0$.

The required pump force for the ferrofluid flow is increased as the field strength and/or concentration increases, as is presented in Figure 6. For $\varphi =0.01$, the variation of the axial pressure gradient, as the field strength increases, is negligible, while for $\varphi =0.1$, field strength affects the axial pressure gradient. For $\xi =2.0$, required pump force increases around 17% in comparison to $\xi =0$. The effect of the Grashof number on the axial pressure gradient is much more significant. For $\mathrm{Gr}={10}^3$, the axial pressure gradient is 107–120% higher in comparison to $\mathrm{Gr}=0$.

4. Conclusions

The effect of the magnetic field on the ferrofluid flow and the heat transfer in a vertical annular cylindrical duct has been studied in the present work. The numerical solution of the constitutive equations is based on the C.V.P. method, which is applied to a conveniently fixed cylindrical coordinate system. From the results, it is observed that the velocity distribution, heat transfer and pressure drop of the ferrofluid flow are highly affected by the external magnetic field and the concentration of magnetic particles. The axial flow is redistributed in four symmetric poles, where its maximum value is observed. A secondary flow is generated, due to the effect of the magnetic field, which significantly improves the heat transfer between the walls and the fluid. The axial pressure gradient, which is required to maintain the same mass flow, also increases as the field strength and concentration of magnetic particles increases, but with a lower rate in comparison to the increase in the heat transfer.