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Asymptotic solution for pressure-driven flows of magnetic fluids in pipes
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 289 (2005) 314–317 www.elsevier.com/locate/jmmm
Asymptotic solution for pressure-driven flows of magnetic fluids in pipes F.R. Cunha, Y.D. Sobral Departamento de Engenharia Mecaˆnica, Faculdade de Tecnologia, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, Brası´lia-DF, 70910-900, Brazil Available online 30 November 2004
Abstract This work focuses on laminar pipe-flows of a magnetic fluid composed of super-paramagnetic particles. The deviations from the equilibrium magnetization arise due to vorticity effects on the flow. In the regime of small deviations of magnetization, the problem is reduced to a weakly nonlinear ordinary differential equation for the velocity. The resulting equation is solved by a regular perturbation method, and explicit expressions for the velocity profile and the friction factor of the flow are presented as a function of magnetic and hydrodynamic physical parameters. r 2004 Elsevier B.V. All rights reserved. PACS: 03.40.G; 83.80.G; 02.30 Keywords: Magnetic fluids; Laminar flow; Asymptotic solution; Friction factor
1. Introduction In this work, the laminar flow of magnetic fluids in pipes in the presence of nonuniform axial magnetic fields is examined. Flows of magnetic fluids have been a subject of several studies [1–4]. Despite the simplicity of the geometry of laminar flow in pipes, the flow structure of magnetic fluids is complex due to the nonlinear coupling between hydrodynamics and magnetism. Theoretical works concerning the laminar flow of magnetic fluids in pipes are generally based on asymptotic solution and numerical simulations. An important step for solving such a problem is to identify the physical parameters of the coupled flow in order to make the suitable simplifications that lead to realistic solutions of the nonlinear governing equations. An explicit equation Corresponding author. Tel.:/fax: +55 61 3072314x229.
E-mail address: [email protected] (F.R. Cunha).
for the flow rate of magnetic fluid has been derived based on a magnetic potential theory for a magnetic fluid of homogeneous magnetization [2]. The aim of the present paper is to develop a slightly nonlinear theory in order to predict the velocity profile and the friction factor of a magnetic laminar pipe flow in terms of magnetic and hydrodynamic properties. This flow quantity is an important parameter in the design of oil–water magnetic-separator systems for oil-spills applications. We show an amount of drag reduction in the flow produced by magnetic field gradients and fluid magnetization in qualitative agreement with experimental observations.
2. Governing equations and solutions The governing equations of flows of magnetic fluids are the magnetostatic set of equations for applied field
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.089
ARTICLE IN PRESS F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
and magnetic induction, as well as continuity, linear and angular momentum equations [5,6]. For the present study, one should only consider the linear momentum equation for a magnetic fluid in terms of dimensionless variables for a unidirectional axisymmetric flow in a pipe of radius R and length L, with a constant pressure gradient G ¼ dp=dz; with G40: 1 d du dH r þ ReC pm M ¼ ReG; (1) r dr dr dz where Re ¼ rUR=Z is the Reynolds number based on the density of the magnetic fluid r; shear viscosity Z and on the average velocity U of the flow. Here, C pm ¼ m0 H 20 =rU 2 ; m0 being the free space permeability and H0 a typical applied field intensity. The parameter Cpm denotes the magnetic pressure coefficient measuring the relative importance between magnetic pressure and hydrodynamic pressure. In addition, M represents the total dimensionless magnetization of the fluid when the dimensionless magnetic field H is applied. The magnetization M is determined with a slightly different evolution equation based on the one proposed by Shliomis and Morozov in Ref. [3]. In this article, the characteristic magnetization relaxation time of the particles has been used as a more appropriate time scale for magnetization shifts from the equilibrium magnetization than the Brownian relaxation time [5]; as also suggested in Ref. [2]. Besides, the effects of the inertia of the particles on the evolution of the magnetization are negligible in the limit o ¼ ts U=R51; where ts is the magnetization relaxation time. This may be understood as a quasi-steady regime, where changes in the magnetization occur instantaneously during the motion of the particles. Thus, we are interested in exploring the limit in which significant changes on the magnetization of the magnetic parcels of fluid occur as a consequence of flow vorticity. Then, it follows that oO M ¼ M M 0 ;
(2)
where O ¼ ðdu=2drÞey is the vorticity of the flow and M0 is the static magnetization. Owing to the axisymmetry, the M y component of the magnetization vanishes and the equations for the r and z components are simply given by M z du=dr ¼ 2ðM r M 0r Þ=o
and
M r du=dr ¼ 2ðM z M 0z Þ=o:
ð3Þ M 0r
¼ 0: Now, since the applied field H0 ¼ H 0 ðzÞez ; then Solving Eq. (3) we found M z ¼ M 0z W 1 and M r ¼ ð1=2ÞoM z du=dr; with W ¼ 1 þ ð1=4Þo2 ðdu=drÞ2 : The calculation of the magnetic force depends on the absolute value of the magnetization. Under conditions in which o2 ðdu=drÞ2 51, an asymptotic solution for M is obtained after using a binomial
expansion, namely " # 1 2 du 2 0 M Mz 1 o : 8 dr
315
(4)
Substituting Eq. (4) into Eq. (1), the latter is rewritten as d2 u 1 du o2 du 2 F ðzÞ þ ¼ TðzÞ: (5) dt2 r dr 8 dr Here F ðzÞ ¼ ReC pm M 0z dH=dz; TðzÞ F ðzÞ ¼ GRe; since M0z is constant and dH/dz are functions of z0 only. Defining ¼ F ðzÞo2 =8 as being a small parameter of the coupled magnetic-hydrodynamic flow, Eq. (5) becomes a weakly nonlinear differential equation that can be solved by a regular perturbation expansion such as uðrÞ u0 þ u1 þ 2 u2 þ Oð3 Þ: After solving Eq. (5), the following asymptotic solution for the flow u(r) is determined: uðrÞ ¼
1 2 TðzÞð1 r2 Þ T ðzÞð1 r4 Þ 4 64 2 T 3 ðzÞð1 r6 Þ Oð3 Þ: þ 576
ð6Þ
The friction factor of flow is defined as the dimensionless shear stress at the wall of the tube [7]: 2 du f ¼ : (7) Re dr r¼1 Now, using the nondimensionalized velocity profile given in Eq. (6) for evaluating shear stress at the wall and the average velocity of the flow, Eq. (7) is expanded to OðÞ; giving the following expression for the friction factor fRe ¼ TðzÞ
2 2 3 T ðzÞ þ T ðzÞ Oð3 Þ: 8 48
(8)
For C pm ¼ 0; that means fluid free of magnetization, the friction factor obtained in Eq. (8) reduces to the wellknown Poiseuille law of a nonmagnetic fluid, namely f ¼ 8=Re: Under this condition, the pressure gradient is simply given by G ¼ 8=Re:
3. Results Fig. 1 presents a comparison between the different orders of the asymptotic solution for u(0) and the numerical integration of Eq. (5) based on a fourth-order Runge–Kutta algorithm. It is seen that the asymptotic solution up to Oð2 Þ provides very accurate results for values of the parameter ¼ Oð1Þ: Numerical integration and asymptotic produces results in excellent agreement. Fig. 2 shows the results obtained for the velocity profile in Eq. (5) for some values of the parameter Cpm, which indicates the intensity of the magnetic field. The sign of dH/dz states for the direction of the magnetic
ARTICLE IN PRESS 316
F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
Fig. 1. Comparison between the different order solutions for the velocity profile in Eq. (6). The comparison is made with the maximum velocity umax of the flow. The points represent the numerical solution of Eq. (5). The results are obtained for C pm ¼ 1; dH=dz ¼ 1; Re ¼ 1; G ¼ 1; and M 0z ¼ 0:3:
Fig. 2. Effect of the parameter Cpm on the velocity profiles obtained from the Oð2 Þ solution in Eq. (5). Results obtained for o2 ¼ 0:8; G ¼ 1; Re ¼ 1; M 0z ¼ 0:3 for C pm ¼ 5 and 10. When C pm ¼ 0; the magnetic fluid has no magnetization. Dotted lines were plotted for dH=dz ¼ 1 and heavy lines for dH=dz ¼ 1:
field gradient, either favorable, if dH=dz40; or unfavorable, dH=dzo0: It is observed that the flow rate increases for favorable applied fields. This is caused by an additional magnetic force acting on the fluid particles that can be understood as an additional pressure gradient on the flow. For adverse applied fields, the flow rate is reduced and, for C pm ¼ 5; the flow is reversed, as can be seen in Fig. 2. It is important to note that the proposed theory is valid for nonhysteretic dilute magnetic fluids, for which the o is indeed a very small parameter O(102–103). For typical values of F(z) varying in the range of O(1–100), would be typically 102. The theory sounds very promising for the characterization of superparamagnetic flows. We have
recently extended this theory for the situation of pipe core-flows composed of two immiscible fluids like a magnetic oil–water flow [5]. The effect of the magnetic parameter Cpm on the friction factor of the flow is reported in Fig. 3. A drastic reduction of f is seen when the magnetic effects become significant. The friction factor is reduced by 66% with Cpm increasing from 1 to 5 for Re ¼ 10: This prediction seems to agree qualitatively with experimental observations of magnetic flows presented in Ref. [7]. Actually, the magnetic force acting on the magnetic fluid parcels becomes more significant as Cpm increases. So, hydrodynamic pressure drop of the flow is reduced for a fixed Re number. In the present context, we construct an equivalent Moody diagram for laminar flow of a magnetic fluid shown in Fig. 4. This plot presents results for f as a
Fig. 3. Effect of Cpm on the friction factor obtained in Eq. (8), for different values of Re. The values of other parameters are: M 0z ¼ 1; dH=dz ¼ 1; o ¼ 102 :
Fig. 4. Effect of Re on the friction factor obtained in Eq. (8) for different values of Cpm. When C pm ¼ 0; f ¼ 8=Re: The values of other parameters are: M 0z ¼ 1; dH=dz ¼ 1; o ¼ 102 :
ARTICLE IN PRESS F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
function of the Reynolds number for different values of the magnetic pressure coefficient. For the condition C pm ¼ 0, the magnetic force is absent in the flow and the friction factor f is just that predicted by the Poiseuille law for a nonmagnetic laminar fluid, i.e. f ¼ 8=Re: On the other hand if C pm a0; the friction factor dependence on Reynolds number changes to a more rapid decay such as that predicted by Eq. (8). A reduction in the flow drag even for small values of Cpm is seen since Reynolds number is sufficiently large. In general, the results suggest that the dynamics of superparamagnetic flows satisfying the condition o51 may be fully described by two physical parameters: the Reynolds number Re and magnetic pressure coefficient Cpm. Finally it should be important to note that for 4Oð1Þ the nonlinear term in Eq. (5) becomes significant and the asymptotic solution proposed in this work fails. Beyond this limit, a numerical solution of the
317
full nonlinear coupled magnetic-hydrodynamic flow is required [8].
References [1] S. Kamiyama, K. Koike, Hydrodynamics of magnetic fluids, in: B.M. Berkovskii, V.G. Bashtovoi (Eds.), Magnetic Fluids and Applications Handbook, Begell House Publishers, 1996. [2] M. Zahn, J. Magn. Magn. Mater. 85 (1990) 181. [3] M.I. Shliomis, K.I. Morozov, Phys. Fluids 6 (1994) 2855. [4] B.U. Felderhof, Phys. Rev. E 64 (2001) 21508. [5] Y.D. Sobral, F.R. Cunha, Physica A 343 C (2004) 36. [6] R.E. Rosensweig, J. Appl. Phys. 57 (1985) 4259. [7] A. Gyr, H.W. Bewersdorff, Drag Reduction of Turbulent Flows by Additives, Kluwer Academic Publishers, Holland, 1995. [8] D.M. Ramos, M.Sc. Thesis, University of Brası´ lia, 2004.
Journal of Magnetism and Magnetic Materials 289 (2005) 314–317 www.elsevier.com/locate/jmmm
Asymptotic solution for pressure-driven flows of magnetic fluids in pipes F.R. Cunha, Y.D. Sobral Departamento de Engenharia Mecaˆnica, Faculdade de Tecnologia, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, Brası´lia-DF, 70910-900, Brazil Available online 30 November 2004
Abstract This work focuses on laminar pipe-flows of a magnetic fluid composed of super-paramagnetic particles. The deviations from the equilibrium magnetization arise due to vorticity effects on the flow. In the regime of small deviations of magnetization, the problem is reduced to a weakly nonlinear ordinary differential equation for the velocity. The resulting equation is solved by a regular perturbation method, and explicit expressions for the velocity profile and the friction factor of the flow are presented as a function of magnetic and hydrodynamic physical parameters. r 2004 Elsevier B.V. All rights reserved. PACS: 03.40.G; 83.80.G; 02.30 Keywords: Magnetic fluids; Laminar flow; Asymptotic solution; Friction factor
1. Introduction In this work, the laminar flow of magnetic fluids in pipes in the presence of nonuniform axial magnetic fields is examined. Flows of magnetic fluids have been a subject of several studies [1–4]. Despite the simplicity of the geometry of laminar flow in pipes, the flow structure of magnetic fluids is complex due to the nonlinear coupling between hydrodynamics and magnetism. Theoretical works concerning the laminar flow of magnetic fluids in pipes are generally based on asymptotic solution and numerical simulations. An important step for solving such a problem is to identify the physical parameters of the coupled flow in order to make the suitable simplifications that lead to realistic solutions of the nonlinear governing equations. An explicit equation Corresponding author. Tel.:/fax: +55 61 3072314x229.
E-mail address: [email protected] (F.R. Cunha).
for the flow rate of magnetic fluid has been derived based on a magnetic potential theory for a magnetic fluid of homogeneous magnetization [2]. The aim of the present paper is to develop a slightly nonlinear theory in order to predict the velocity profile and the friction factor of a magnetic laminar pipe flow in terms of magnetic and hydrodynamic properties. This flow quantity is an important parameter in the design of oil–water magnetic-separator systems for oil-spills applications. We show an amount of drag reduction in the flow produced by magnetic field gradients and fluid magnetization in qualitative agreement with experimental observations.
2. Governing equations and solutions The governing equations of flows of magnetic fluids are the magnetostatic set of equations for applied field
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.089
ARTICLE IN PRESS F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
and magnetic induction, as well as continuity, linear and angular momentum equations [5,6]. For the present study, one should only consider the linear momentum equation for a magnetic fluid in terms of dimensionless variables for a unidirectional axisymmetric flow in a pipe of radius R and length L, with a constant pressure gradient G ¼ dp=dz; with G40: 1 d du dH r þ ReC pm M ¼ ReG; (1) r dr dr dz where Re ¼ rUR=Z is the Reynolds number based on the density of the magnetic fluid r; shear viscosity Z and on the average velocity U of the flow. Here, C pm ¼ m0 H 20 =rU 2 ; m0 being the free space permeability and H0 a typical applied field intensity. The parameter Cpm denotes the magnetic pressure coefficient measuring the relative importance between magnetic pressure and hydrodynamic pressure. In addition, M represents the total dimensionless magnetization of the fluid when the dimensionless magnetic field H is applied. The magnetization M is determined with a slightly different evolution equation based on the one proposed by Shliomis and Morozov in Ref. [3]. In this article, the characteristic magnetization relaxation time of the particles has been used as a more appropriate time scale for magnetization shifts from the equilibrium magnetization than the Brownian relaxation time [5]; as also suggested in Ref. [2]. Besides, the effects of the inertia of the particles on the evolution of the magnetization are negligible in the limit o ¼ ts U=R51; where ts is the magnetization relaxation time. This may be understood as a quasi-steady regime, where changes in the magnetization occur instantaneously during the motion of the particles. Thus, we are interested in exploring the limit in which significant changes on the magnetization of the magnetic parcels of fluid occur as a consequence of flow vorticity. Then, it follows that oO M ¼ M M 0 ;
(2)
where O ¼ ðdu=2drÞey is the vorticity of the flow and M0 is the static magnetization. Owing to the axisymmetry, the M y component of the magnetization vanishes and the equations for the r and z components are simply given by M z du=dr ¼ 2ðM r M 0r Þ=o
and
M r du=dr ¼ 2ðM z M 0z Þ=o:
ð3Þ M 0r
¼ 0: Now, since the applied field H0 ¼ H 0 ðzÞez ; then Solving Eq. (3) we found M z ¼ M 0z W 1 and M r ¼ ð1=2ÞoM z du=dr; with W ¼ 1 þ ð1=4Þo2 ðdu=drÞ2 : The calculation of the magnetic force depends on the absolute value of the magnetization. Under conditions in which o2 ðdu=drÞ2 51, an asymptotic solution for M is obtained after using a binomial
expansion, namely " # 1 2 du 2 0 M Mz 1 o : 8 dr
315
(4)
Substituting Eq. (4) into Eq. (1), the latter is rewritten as d2 u 1 du o2 du 2 F ðzÞ þ ¼ TðzÞ: (5) dt2 r dr 8 dr Here F ðzÞ ¼ ReC pm M 0z dH=dz; TðzÞ F ðzÞ ¼ GRe; since M0z is constant and dH/dz are functions of z0 only. Defining ¼ F ðzÞo2 =8 as being a small parameter of the coupled magnetic-hydrodynamic flow, Eq. (5) becomes a weakly nonlinear differential equation that can be solved by a regular perturbation expansion such as uðrÞ u0 þ u1 þ 2 u2 þ Oð3 Þ: After solving Eq. (5), the following asymptotic solution for the flow u(r) is determined: uðrÞ ¼
1 2 TðzÞð1 r2 Þ T ðzÞð1 r4 Þ 4 64 2 T 3 ðzÞð1 r6 Þ Oð3 Þ: þ 576
ð6Þ
The friction factor of flow is defined as the dimensionless shear stress at the wall of the tube [7]: 2 du f ¼ : (7) Re dr r¼1 Now, using the nondimensionalized velocity profile given in Eq. (6) for evaluating shear stress at the wall and the average velocity of the flow, Eq. (7) is expanded to OðÞ; giving the following expression for the friction factor fRe ¼ TðzÞ
2 2 3 T ðzÞ þ T ðzÞ Oð3 Þ: 8 48
(8)
For C pm ¼ 0; that means fluid free of magnetization, the friction factor obtained in Eq. (8) reduces to the wellknown Poiseuille law of a nonmagnetic fluid, namely f ¼ 8=Re: Under this condition, the pressure gradient is simply given by G ¼ 8=Re:
3. Results Fig. 1 presents a comparison between the different orders of the asymptotic solution for u(0) and the numerical integration of Eq. (5) based on a fourth-order Runge–Kutta algorithm. It is seen that the asymptotic solution up to Oð2 Þ provides very accurate results for values of the parameter ¼ Oð1Þ: Numerical integration and asymptotic produces results in excellent agreement. Fig. 2 shows the results obtained for the velocity profile in Eq. (5) for some values of the parameter Cpm, which indicates the intensity of the magnetic field. The sign of dH/dz states for the direction of the magnetic
ARTICLE IN PRESS 316
F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
Fig. 1. Comparison between the different order solutions for the velocity profile in Eq. (6). The comparison is made with the maximum velocity umax of the flow. The points represent the numerical solution of Eq. (5). The results are obtained for C pm ¼ 1; dH=dz ¼ 1; Re ¼ 1; G ¼ 1; and M 0z ¼ 0:3:
Fig. 2. Effect of the parameter Cpm on the velocity profiles obtained from the Oð2 Þ solution in Eq. (5). Results obtained for o2 ¼ 0:8; G ¼ 1; Re ¼ 1; M 0z ¼ 0:3 for C pm ¼ 5 and 10. When C pm ¼ 0; the magnetic fluid has no magnetization. Dotted lines were plotted for dH=dz ¼ 1 and heavy lines for dH=dz ¼ 1:
field gradient, either favorable, if dH=dz40; or unfavorable, dH=dzo0: It is observed that the flow rate increases for favorable applied fields. This is caused by an additional magnetic force acting on the fluid particles that can be understood as an additional pressure gradient on the flow. For adverse applied fields, the flow rate is reduced and, for C pm ¼ 5; the flow is reversed, as can be seen in Fig. 2. It is important to note that the proposed theory is valid for nonhysteretic dilute magnetic fluids, for which the o is indeed a very small parameter O(102–103). For typical values of F(z) varying in the range of O(1–100), would be typically 102. The theory sounds very promising for the characterization of superparamagnetic flows. We have
recently extended this theory for the situation of pipe core-flows composed of two immiscible fluids like a magnetic oil–water flow [5]. The effect of the magnetic parameter Cpm on the friction factor of the flow is reported in Fig. 3. A drastic reduction of f is seen when the magnetic effects become significant. The friction factor is reduced by 66% with Cpm increasing from 1 to 5 for Re ¼ 10: This prediction seems to agree qualitatively with experimental observations of magnetic flows presented in Ref. [7]. Actually, the magnetic force acting on the magnetic fluid parcels becomes more significant as Cpm increases. So, hydrodynamic pressure drop of the flow is reduced for a fixed Re number. In the present context, we construct an equivalent Moody diagram for laminar flow of a magnetic fluid shown in Fig. 4. This plot presents results for f as a
Fig. 3. Effect of Cpm on the friction factor obtained in Eq. (8), for different values of Re. The values of other parameters are: M 0z ¼ 1; dH=dz ¼ 1; o ¼ 102 :
Fig. 4. Effect of Re on the friction factor obtained in Eq. (8) for different values of Cpm. When C pm ¼ 0; f ¼ 8=Re: The values of other parameters are: M 0z ¼ 1; dH=dz ¼ 1; o ¼ 102 :
ARTICLE IN PRESS F.R. Cunha, Y.D. Sobral / Journal of Magnetism and Magnetic Materials 289 (2005) 314–317
function of the Reynolds number for different values of the magnetic pressure coefficient. For the condition C pm ¼ 0, the magnetic force is absent in the flow and the friction factor f is just that predicted by the Poiseuille law for a nonmagnetic laminar fluid, i.e. f ¼ 8=Re: On the other hand if C pm a0; the friction factor dependence on Reynolds number changes to a more rapid decay such as that predicted by Eq. (8). A reduction in the flow drag even for small values of Cpm is seen since Reynolds number is sufficiently large. In general, the results suggest that the dynamics of superparamagnetic flows satisfying the condition o51 may be fully described by two physical parameters: the Reynolds number Re and magnetic pressure coefficient Cpm. Finally it should be important to note that for 4Oð1Þ the nonlinear term in Eq. (5) becomes significant and the asymptotic solution proposed in this work fails. Beyond this limit, a numerical solution of the
317
full nonlinear coupled magnetic-hydrodynamic flow is required [8].
References [1] S. Kamiyama, K. Koike, Hydrodynamics of magnetic fluids, in: B.M. Berkovskii, V.G. Bashtovoi (Eds.), Magnetic Fluids and Applications Handbook, Begell House Publishers, 1996. [2] M. Zahn, J. Magn. Magn. Mater. 85 (1990) 181. [3] M.I. Shliomis, K.I. Morozov, Phys. Fluids 6 (1994) 2855. [4] B.U. Felderhof, Phys. Rev. E 64 (2001) 21508. [5] Y.D. Sobral, F.R. Cunha, Physica A 343 C (2004) 36. [6] R.E. Rosensweig, J. Appl. Phys. 57 (1985) 4259. [7] A. Gyr, H.W. Bewersdorff, Drag Reduction of Turbulent Flows by Additives, Kluwer Academic Publishers, Holland, 1995. [8] D.M. Ramos, M.Sc. Thesis, University of Brası´ lia, 2004.