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Drift velocity and deformation of polarized drops in magnetic fields
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 289 (2005) 318–320 www.elsevier.com/locate/jmmm
Drift velocity and deformation of polarized drops in magnetic fields Y.D. Sobral, F.R. Cunha Departamento de Engenharia Mecaˆnica, Faculdade de Tecnologia, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, 70910-900 Brası´lia – DF, Brazil Available online 30 November 2004
Abstract The motion and deformation of a drop composed by oil and magnetic microparticles is characterized by scaling arguments based on the governing equations for the flow of magnetic fluids. The scaling predicts that the drift velocity of a magnetic drop is proportional to the square of the applied magnetic field since viscous forces dominate inertia forces on the motion. Drop deformation, induced by applied field gradients in the scale of the drop radius, is found to vary linearly with the applied field. Experiments have been carried out and the proposed scaling verified. r 2004 Elsevier B.V. All rights reserved. PACS: 83.80.G; 03.40.D Keywords: Magnetic fluids; Drift velocity; Drop deformation; Magnetic recovery
1. Introduction The goal of this work is to give further understanding to the problem of magnetic separation, in particular to the problem of separating oil from water. One is aware of the major importance this subject has in petroleum industry. If an accident occurs either during production, transportation or storage of oil, immediate procedures should be put into practice to avoid major damages to the local environment. Under this perspective, the separation of oil from water using magnetic particles seems to be a promising method [1]. The response of magnetized oil drops to an external applied field was used to provide some insight in the evaluation of the efficiency of magnetic separators. To this end, we give an estimation of the drift velocity and deformation for a drop of magnetic particles in a magnetic field by scaling arguments based on the Corresponding author. Tel.:/fax: +55 61 3072314x229.
E-mail address: [email protected] (F.R. Cunha).
governing equations of the motion of magnetic fluids. Experimental observations have been carried out with a magnetic drop freely suspended in pure water, composed of oil and magnetic micro-particles, were used to verify the scalings predictions.
2. Governing equations The governing equations of flows of magnetic fluids are the magnetostatic set of equations for applied field and magnetic induction, as well as continuity, linear and angular momentum equations [1,2]. For the present study, one should only consider the linear momentum equation for a magnetic fluid: @u r (1) þ u ru ¼ rp þ Zr2 u þ m0 M rH; @t where r is the specific mass, u is the velocity, p is the modified pressure and Z the shear viscosity of the magnetic fluid. In addition, M represents the
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.090
ARTICLE IN PRESS Y.D. Sobral, F.R. Cunha / Journal of Magnetism and Magnetic Materials 289 (2005) 318–320
magnetization of the fluid when the magnetic field H is applied. The last term in Eq. (1) denotes the magnetic effects that arise in the flow of a magnetic fluid in the presence of a non-uniform applied field. Further details concerning the mathematical modelling of magnetic fluids flows are presented in Ref. [1,2].
319
It is thus expected that the deformation of the magnetic drop would change linearly with the applied field at OðRm Þ: Further details concerning the scaling analysis presented in this section can be found in Ref. [1].
4. Results and discussion 3. Scalings Consider a circular drop of oil of radius rd immersed in water, with density r; shear viscosity Z and suppose that magnetic spherical micro-particles are also present in the drop in a volume concentration f: We shall focus on the limit where viscous forces balance magnetic forces. From Eq. (1), a typical scale for viscous force per unit of volume is given by jZr2 uj ZU d =r2d ; where Ud states for the drift velocity of the drop. In analogy, the scale for the magnetic force per unit of volume is obtained from the Kelvin force term on Eq. (1), that is jm0 M rHj m0 MH=rd m0 w2 fH 2 =rd ; where a characteristic scale for the magnetization, based on the saturation susceptibility of the magnetic particles ws ; is used, that is M ws fH: When viscous force balance magnetic forces, it follows that the drift velocity Ud scales non-linearly with the applied field as 2
U d =U Rm ws fH^ ;
(2) 1=2
is a typical scale for the drift where U H 0 ðm0 =rÞ velocity obtained from the Bernoulli equation for magnetic fluids [2], H^ ¼ H=H 0 is the dimensionless absolute value of the applied magnetic field and the parameter Rm denotes the magnetic Reynolds number, that compares viscous effects with magnetic effects, is defined as Rm ¼ ðrm0 Þ1=2 rd H 0 =Z:
(3)
In order to validate the scalings in Eqs. (2) and (5), an experimental procedure was carried out. The motion of the magnetic drops was recorded along the direction of the applied field with a digital camera and the applied field, created by a permanent magnet, was measured with a digital gaussmeter. Thus, the instantaneous velocity and drop shape were determined as time evolved and were associated to local measurements of applied field. The experiments were carried out at conditions that Rm ¼ 760. The Reynolds number based on the particle radius and average velocity of the particle on the early stages of its motion was Re ¼ rUrd =Z ¼ 6: A detailed description of such experiments can be found in Ref. [1]. Fig. 1 illustrates the time evolution of the drop motion and deformation for a typical run of our experiments. Fig. 2 shows the dimensionless velocity of the magnetic drop as a function of the magnetic parameter 2 2 Rm H^ : For Rm H^ p70; the results led to propose the following law for the drift velocity: 2 U d =U ¼ 9:0 105 Rm H^ :
(6)
It should be noted that the value of the constant may change according to the magnetic fluid that composed the drop on the experiments, since it should take into account physical parameters of the system such as f and ws : As the applied field intensity increases, however, the velocity no longer follows the predictions of Eq. (2). It is
The evaluation of the drop deformation follows closely the analysis developed in Ref. [3] for a nonmagnetic emulsion under shear flows. The drop is approximated instantaneously by an ellipsis with major and minor axes A and B; respectively. Thus, the dimensionless deformation of the drop may be calculated as D ¼ ðA BÞ=ðA þ BÞ:
(4)
Considering the scale of the total magnetic force on the drop to be f m m0 fws H 2 r2d ; a typical scale for the shear stress caused by magnetic forces should be given by Sm m0 fws H 2 =p: Balancing the magnetic stress Sm by the viscous shear stress S v Z_g; with g_ ¼ D=ðrd =UÞ denoting the rate of strain of the flow, where now U is a scale defined as U Hðm0 =rÞ1=2 ; the first-order approximation for D is then expressed by ^ þ OðR2 Þ: D fws Rm H=p m
(5)
Fig. 1. A typical sequence of the motion of the drop under the action of an applied magnetic field. In (a) undeformed drop at t ¼ 0 s. In (b) at t ¼ 12 s, elliptical shape after motion and deformation.
ARTICLE IN PRESS 320
Y.D. Sobral, F.R. Cunha / Journal of Magnetism and Magnetic Materials 289 (2005) 318–320
Fig. 2. Dimensionless drift velocity. The line represents the prediction given by Eq. (6).
verified that the actual drift velocity of the magnetic drop is higher than that predicted by Eq. (2), as it can be seen in Fig. 2. Fig. 1 reveals that the drop deforms during its motion and it can no longer be modeled as being closely a circle and Eq. (2) does not apply. Local magnetic field gradients produce an internal motion of the magnetic particles inside the drop towards the higher field intensity regions and the drop assumes a shape of a slender body. A direct consequence is that viscous resistance to its motion is reduced, allowing the magnetic forces to be more effective and the drop to move faster. From the results for the drop deformation, in Fig. 3, we conclude that the scaling for the drift velocity of the magnetic drop in Eq. (2) fits well the observations in the limit of a slightly deformed drop. In addition, a very good agreement of the experimental data with the first-order deformation theory is observed, even for moderate deformation regimes of the drop, as it can be seen in Fig. 3. We propose the deformation of the drop as being described by the following equation: D ¼ 5:0 104 Rm H 0 (7) ^ For values of Rm H41200; the deformation of the drop is no longer linear with respect to H^ and the OðRm Þ theory fails at large stretchings (close to drop breakup). The same argument concerning the constant in Eq. (6) applies to Eq. (7). 4.1. An application The flow rate for recovering a typical oil-slick is predicted as a direct application of the scalings developed in this work. Consider a circular magnetic oil-slick of radius rd and thickness d: The volume rate of magnetic fluid is expressed as the product of a velocity
Fig. 3. Deformation of a magnetic drop as a function of the local applied field. The line represents the prediction given by Eq. (7).
and the lateral area S of the slick, say dV (8) ¼ U d S: dt Now, using the scaling from Eq. (2) for the capture velocity Ud, it follows that the slick radius is predicted according to rd ¼
2dr0 Z ; 2dZ þ m0 ws fr0 H 2 t
(9)
being r0 is the initial radius of the slick. This indicates that the radius, and consequently, the area occupied by the drop, both decay algebraically in time. A direct calculation with Eq. (9) predicts that a typical slick of r0 ¼ 100 m is reduced in about 90% of its initial area in two days for Rm ¼ 700. Despite there is no experimental data available to validate our theory, the results are quite promising. For large oil-slicks, however, inertial effects become important and the theory should be changed to incorporate these effects. In the present context, the magnetic Reynolds number Rm is defined as the product of a magnetic parameter and a hydrodynamic parameter, namely the magnetic pressure coefficient C pm ¼ m0 H 20 =rU 2 and Re. Thus, one may expect that in arbitrary flow regimes, the drift velocity and deformation of the drop should be ^ where described by a general law such as C apm Reb f ðHÞ; ^ ^ f ðHÞ is a function of H; since velocity and deformation scalings depend on the local value of applied field. This generalization remains to be investigated from observations and numerical simulations in future works. References [1] F.R. Cunha, Y.D. Sobral, Physica A 343C (2004) 36. [2] R.E. Rosensweig, J. Appl. Phys. 57 (1985) 4259. [3] G.I. Taylor, Proc. Roy. Soc. London A 146 (1934) 501.
Journal of Magnetism and Magnetic Materials 289 (2005) 318–320 www.elsevier.com/locate/jmmm
Drift velocity and deformation of polarized drops in magnetic fields Y.D. Sobral, F.R. Cunha Departamento de Engenharia Mecaˆnica, Faculdade de Tecnologia, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, 70910-900 Brası´lia – DF, Brazil Available online 30 November 2004
Abstract The motion and deformation of a drop composed by oil and magnetic microparticles is characterized by scaling arguments based on the governing equations for the flow of magnetic fluids. The scaling predicts that the drift velocity of a magnetic drop is proportional to the square of the applied magnetic field since viscous forces dominate inertia forces on the motion. Drop deformation, induced by applied field gradients in the scale of the drop radius, is found to vary linearly with the applied field. Experiments have been carried out and the proposed scaling verified. r 2004 Elsevier B.V. All rights reserved. PACS: 83.80.G; 03.40.D Keywords: Magnetic fluids; Drift velocity; Drop deformation; Magnetic recovery
1. Introduction The goal of this work is to give further understanding to the problem of magnetic separation, in particular to the problem of separating oil from water. One is aware of the major importance this subject has in petroleum industry. If an accident occurs either during production, transportation or storage of oil, immediate procedures should be put into practice to avoid major damages to the local environment. Under this perspective, the separation of oil from water using magnetic particles seems to be a promising method [1]. The response of magnetized oil drops to an external applied field was used to provide some insight in the evaluation of the efficiency of magnetic separators. To this end, we give an estimation of the drift velocity and deformation for a drop of magnetic particles in a magnetic field by scaling arguments based on the Corresponding author. Tel.:/fax: +55 61 3072314x229.
E-mail address: [email protected] (F.R. Cunha).
governing equations of the motion of magnetic fluids. Experimental observations have been carried out with a magnetic drop freely suspended in pure water, composed of oil and magnetic micro-particles, were used to verify the scalings predictions.
2. Governing equations The governing equations of flows of magnetic fluids are the magnetostatic set of equations for applied field and magnetic induction, as well as continuity, linear and angular momentum equations [1,2]. For the present study, one should only consider the linear momentum equation for a magnetic fluid: @u r (1) þ u ru ¼ rp þ Zr2 u þ m0 M rH; @t where r is the specific mass, u is the velocity, p is the modified pressure and Z the shear viscosity of the magnetic fluid. In addition, M represents the
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.090
ARTICLE IN PRESS Y.D. Sobral, F.R. Cunha / Journal of Magnetism and Magnetic Materials 289 (2005) 318–320
magnetization of the fluid when the magnetic field H is applied. The last term in Eq. (1) denotes the magnetic effects that arise in the flow of a magnetic fluid in the presence of a non-uniform applied field. Further details concerning the mathematical modelling of magnetic fluids flows are presented in Ref. [1,2].
319
It is thus expected that the deformation of the magnetic drop would change linearly with the applied field at OðRm Þ: Further details concerning the scaling analysis presented in this section can be found in Ref. [1].
4. Results and discussion 3. Scalings Consider a circular drop of oil of radius rd immersed in water, with density r; shear viscosity Z and suppose that magnetic spherical micro-particles are also present in the drop in a volume concentration f: We shall focus on the limit where viscous forces balance magnetic forces. From Eq. (1), a typical scale for viscous force per unit of volume is given by jZr2 uj ZU d =r2d ; where Ud states for the drift velocity of the drop. In analogy, the scale for the magnetic force per unit of volume is obtained from the Kelvin force term on Eq. (1), that is jm0 M rHj m0 MH=rd m0 w2 fH 2 =rd ; where a characteristic scale for the magnetization, based on the saturation susceptibility of the magnetic particles ws ; is used, that is M ws fH: When viscous force balance magnetic forces, it follows that the drift velocity Ud scales non-linearly with the applied field as 2
U d =U Rm ws fH^ ;
(2) 1=2
is a typical scale for the drift where U H 0 ðm0 =rÞ velocity obtained from the Bernoulli equation for magnetic fluids [2], H^ ¼ H=H 0 is the dimensionless absolute value of the applied magnetic field and the parameter Rm denotes the magnetic Reynolds number, that compares viscous effects with magnetic effects, is defined as Rm ¼ ðrm0 Þ1=2 rd H 0 =Z:
(3)
In order to validate the scalings in Eqs. (2) and (5), an experimental procedure was carried out. The motion of the magnetic drops was recorded along the direction of the applied field with a digital camera and the applied field, created by a permanent magnet, was measured with a digital gaussmeter. Thus, the instantaneous velocity and drop shape were determined as time evolved and were associated to local measurements of applied field. The experiments were carried out at conditions that Rm ¼ 760. The Reynolds number based on the particle radius and average velocity of the particle on the early stages of its motion was Re ¼ rUrd =Z ¼ 6: A detailed description of such experiments can be found in Ref. [1]. Fig. 1 illustrates the time evolution of the drop motion and deformation for a typical run of our experiments. Fig. 2 shows the dimensionless velocity of the magnetic drop as a function of the magnetic parameter 2 2 Rm H^ : For Rm H^ p70; the results led to propose the following law for the drift velocity: 2 U d =U ¼ 9:0 105 Rm H^ :
(6)
It should be noted that the value of the constant may change according to the magnetic fluid that composed the drop on the experiments, since it should take into account physical parameters of the system such as f and ws : As the applied field intensity increases, however, the velocity no longer follows the predictions of Eq. (2). It is
The evaluation of the drop deformation follows closely the analysis developed in Ref. [3] for a nonmagnetic emulsion under shear flows. The drop is approximated instantaneously by an ellipsis with major and minor axes A and B; respectively. Thus, the dimensionless deformation of the drop may be calculated as D ¼ ðA BÞ=ðA þ BÞ:
(4)
Considering the scale of the total magnetic force on the drop to be f m m0 fws H 2 r2d ; a typical scale for the shear stress caused by magnetic forces should be given by Sm m0 fws H 2 =p: Balancing the magnetic stress Sm by the viscous shear stress S v Z_g; with g_ ¼ D=ðrd =UÞ denoting the rate of strain of the flow, where now U is a scale defined as U Hðm0 =rÞ1=2 ; the first-order approximation for D is then expressed by ^ þ OðR2 Þ: D fws Rm H=p m
(5)
Fig. 1. A typical sequence of the motion of the drop under the action of an applied magnetic field. In (a) undeformed drop at t ¼ 0 s. In (b) at t ¼ 12 s, elliptical shape after motion and deformation.
ARTICLE IN PRESS 320
Y.D. Sobral, F.R. Cunha / Journal of Magnetism and Magnetic Materials 289 (2005) 318–320
Fig. 2. Dimensionless drift velocity. The line represents the prediction given by Eq. (6).
verified that the actual drift velocity of the magnetic drop is higher than that predicted by Eq. (2), as it can be seen in Fig. 2. Fig. 1 reveals that the drop deforms during its motion and it can no longer be modeled as being closely a circle and Eq. (2) does not apply. Local magnetic field gradients produce an internal motion of the magnetic particles inside the drop towards the higher field intensity regions and the drop assumes a shape of a slender body. A direct consequence is that viscous resistance to its motion is reduced, allowing the magnetic forces to be more effective and the drop to move faster. From the results for the drop deformation, in Fig. 3, we conclude that the scaling for the drift velocity of the magnetic drop in Eq. (2) fits well the observations in the limit of a slightly deformed drop. In addition, a very good agreement of the experimental data with the first-order deformation theory is observed, even for moderate deformation regimes of the drop, as it can be seen in Fig. 3. We propose the deformation of the drop as being described by the following equation: D ¼ 5:0 104 Rm H 0 (7) ^ For values of Rm H41200; the deformation of the drop is no longer linear with respect to H^ and the OðRm Þ theory fails at large stretchings (close to drop breakup). The same argument concerning the constant in Eq. (6) applies to Eq. (7). 4.1. An application The flow rate for recovering a typical oil-slick is predicted as a direct application of the scalings developed in this work. Consider a circular magnetic oil-slick of radius rd and thickness d: The volume rate of magnetic fluid is expressed as the product of a velocity
Fig. 3. Deformation of a magnetic drop as a function of the local applied field. The line represents the prediction given by Eq. (7).
and the lateral area S of the slick, say dV (8) ¼ U d S: dt Now, using the scaling from Eq. (2) for the capture velocity Ud, it follows that the slick radius is predicted according to rd ¼
2dr0 Z ; 2dZ þ m0 ws fr0 H 2 t
(9)
being r0 is the initial radius of the slick. This indicates that the radius, and consequently, the area occupied by the drop, both decay algebraically in time. A direct calculation with Eq. (9) predicts that a typical slick of r0 ¼ 100 m is reduced in about 90% of its initial area in two days for Rm ¼ 700. Despite there is no experimental data available to validate our theory, the results are quite promising. For large oil-slicks, however, inertial effects become important and the theory should be changed to incorporate these effects. In the present context, the magnetic Reynolds number Rm is defined as the product of a magnetic parameter and a hydrodynamic parameter, namely the magnetic pressure coefficient C pm ¼ m0 H 20 =rU 2 and Re. Thus, one may expect that in arbitrary flow regimes, the drift velocity and deformation of the drop should be ^ where described by a general law such as C apm Reb f ðHÞ; ^ ^ f ðHÞ is a function of H; since velocity and deformation scalings depend on the local value of applied field. This generalization remains to be investigated from observations and numerical simulations in future works. References [1] F.R. Cunha, Y.D. Sobral, Physica A 343C (2004) 36. [2] R.E. Rosensweig, J. Appl. Phys. 57 (1985) 4259. [3] G.I. Taylor, Proc. Roy. Soc. London A 146 (1934) 501.