Magneto-Viscous Effects on Unsteady Nano-Ferrofluid Flow Influenced by Low Oscillating Magnetic Field in the Presence of Rotating Disk

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1 Magneto-Viscous Effects on Unsteady Nano-Ferrofluid Flow Influenced by Low Oscillating Magnetic Field in the Presence of Rotating Disk PARAS RAM 1, VIMAL KUMAR JOSHI 2 AND SHASHI SHARMA 3 1,2 National Institute of Technology, Kurukshetra (INDIA 3 Indian Institute of Technology, Roorkee (INDIA 1 parasram_nit@yahoo.co.in, 2 joshiambition2008@gmail.com Abstract:- The purpose of the present problem is to theoretically examine the magneto-viscous effects on axisymmetric unsteady flow of an incompressible non-conducting Nano-Ferrofluid on a rotating disk under the influence of low oscillating magnetic field. The problem has been formulated by employing the basic idea of the Shliomis theory to the equation of motion and magnetization equation for Nano-Ferrofluid flow due to a rotating disk. The system of nonlinear partial differential equations governing the unsteady flow is expressed in cylindrical coordinates taking z-axis as axis of rotation are further solved by Newton method in MATLAB by taking the initial guess with the help of Flex PDE. The nature of radial, tangential and axial velocities in the presence of low oscillating magnetic field with different values of effective magnetization parameter is discussed quantitatively and qualitatively. It has been found that the change in the axial velocity due to magneto-viscosity is dominant in comparison to the radial and tangential velocity. Further, the axial velocity component reaches the steady state region slower than the redial velocity component and much slower than the tangential velocity component because of the centrifugal effect. The effective magnetization parameter creates an additional resistance on the axial component of velocity provided the applied magnetic field and vorticity of the flow are not collinear, and reaches to a maximum when the applied magnetic field is perpendicular to the vorticity of the flow. Further, the axial velocity does noepend on the distance from the axis but on the distance from the plate. In net shell, we may conclude that the present problem is a theoretical motivation for the study of strong changes in the viscous properties of fluid in electronically controllable signals and damping systems. Keywords: Nano-Ferrofluid, Rotating Disk, Magnetization, Oscillating Magnetic Field, Magneto-Viscous Effects, Vorticity. 1 Introduction Magnetic Nano-Ferrofluids are a special category of magnetically controllable smart nano-materials. These fluids are colloidal suspension of nano-sized particles of, or C in carrier liquids such as kerosene, heptanes or water coated with surfactants (antimony to prevent from agglomeration. The nano particle feels force exerted by external magnetic field and thus the Magnetic fluid can be made to move even in zero gravity. The stable suspensions of these artificial magnetic liquids were first synthesized in 1960 at NASA, USA. The strongest magnetic properties are inherent in saturated molecular solutions of transition and rare-earth metals whose magnetic susceptibility does not exceed the order of [1, 2, 3]. During the last 40 years, strong efforts have been made by the researchers to produce liquid materials thao not undergo strong changes in viscous properties under the influence of an electronically controllable signal. The viscosity controllable magnetic fluid can be used in magnetically controlled damping systems. Magneto viscous effects also have an important role in other applications of magnetic fluids. Sealing of rotary shafts can be used after a substantial enhancement of viscosity of the used fluid in the presence of strong magnetic field in the sealing gap for improving the overall performance of the device [4]. Thus, knowledge about the magneto viscous behavior of the fluid is an important part of the information needed for the proper design of magnetic fluid applications [5, 6, 7]. There have been physical reasons for the appearance of magnetic field-induced changes in the viscosity of a magnetic fluid [6, 7, 8]. Under the influence of shear flow, the suspended magnetic nano particles rotate in the flow with their axis of rotation parallel to the vorticity of the flow. When a magnetic field is applied to the suspension under shear, two extreme situations arise. In the first case, the magnetic field is applied collinearly with the vorticity, then the magnetic moment will be aligned in the direction of the applied magnetic field which ISBN:

2 is identical with the axis of the rotation of the particle. In this case, there is no change in the fluid viscosity due to magnetic field. However in the second case, when the applied magnetic field is perpendicular to the vorticity of the flow, the magnetic field will try to align the magnetic moment with the field direction while the viscous torque exerted by the flow tries to rotate the particle causing a misalignment of magnetic moment and magnetic field. Thus, the appearing magnetic torque is equilibrated by the viscous torque and the rotational viscosity depends not only on the magnetic field strength but also on the direction of the magnetic field relative to the flow [9, 10, 11]. The present investigation deals with the ferrofluid flow over a flaisk rotating about an axis perpendicular to its plane with uniform angular velocity. The boundary layer, influenced by friction, is thrown outwards owing to the action of centrifugal force which is neutralized by the Magnetic fluid particles towards the axis of the disk. Here, we have studied the field induced changes in the viscosity of a magnetic fluid on the velocity profiles. The Shliomis magnetization equation for irreversible thermodynamics is used and solved for quiescent Magnetic fluid in the absence of true magnetic field and the reduced magnetization curve for different values of particles spinning is graphically expressed.. This problem about the magneto viscous effects on the Magnetic fluid flow due to a rotating disk, to the best of our knowledge, has not been investigated yet. 2 Formulation of the Problem The system of Ferro-hydrodynamic governing equations consisting of equation of continuity, equation of motion, magnetization equation, equation of rotational motion and Maxwell relations in vector form are: (Maxwell Relations (1 (2 (3 (4, (5 Equation (1 & (5 indicates that the considered fluid is incompressible and non-conducting. Langevin has described the relationship between the magnetization of the fluid on the strength of a magnetic field as: where and is the saturation magnetization of the liquid, being determined by the volume concentration of the magnetic component and its instantaneous magnetization [3, 4, 5]. Then for constant field, the instantaneous equilibrium magnetization is given as:,, (6 where is the number of particles, is the magnetic moment of the particle, is the Langevin function, is the Langevin parameter (ratio of magnetization energy to the thermal energy, is the Boltzmann constant and is the temperature. Since, is small, the inertial term is negligible in comparison with relaxation term i.e. therefore, the equation (4 can be written as: Now, modified equations (1 and (3 due to (7 are as: Here, is the mathematical expression for Brownian relaxation time. Magnetic torque and viscous torque are acting on the particles denoted by and respectively. The equilibrium of both torques, which leads to the hindrance of the particle rotation, can thus be written from the equation (7 as and at equilibrium: (7 (8 (9 (10 (11 where is the volume fraction and denotes the effective magnetic field parameter (the ratio of effective magnetic force to the thermal force i.e.. In the presence of slow and oscillating ISBN:

3 magnetic field, the equation (9 can be written as [10, 14]: (12 The effective field is expressed by the equation of zero approximation from the equation (12 as: (13 Here, the parameter is the amplitude of the real magnetic field, is the effective Langevin function, is the frequency of the field. From Bacari [14], using linear approximations in and the expression where and is a unit vector along the applied field, equation (12 reduces to (14 Since magnetic torque is not equal to zero, so from equations (10 and (14, the expression for mean magnetic torque becomes as:, (15 Here, is the effective magnetization parameter, if (in the low oscillating field the Magnetic fluid viscosity under an alternating magnetic field can become larger than the Magnetic fluid viscosity in zero field. Now, as i.e. reduced pressure. In the investigation, unsteady, axially-symmetric, incompressible flow of an electrically nonconducting fluid has been considered, assuming that the viscosity is dependent on the Magnetic field and thermal effects are not taken into consideration. The disk rotates about axis with a constant angular velocity ω, where is the vertical axis in the cylindrical coordinate system with and the radial and tangential axis respectively. The components of the flow velocity are ( in the direction of increasing ( respectively and the flow is considered axi-symmetric and incompressible, so the equation of motion and continuity equations can be written in the cylindrical form as [18, 19, 20]: (18 (19 (20 (21 For the flow due to an infinite rotating disk about axis with a constant angular velocity, the approximate initial and boundary conditions used by [21, 22, 23, 24] are given as follows: (16 Thus the rotational viscosity is and from equations (10 and (15, the angular velocity of the particle becomes;. Now the equation (8 with the help of equation (16 can be written as: (22 To non dimensionalize (18 - (21, we make use of the following similarity transformations as used by [25, 26, 27, 28] in a similar type of equations: (23 (17 With the help of (23 from (18-(22, we get the system of nonlinear coupled differential equations in dimensionless form as: ISBN:

4 (24 Integrating (32, we have (25 (33 (26 (27 Here (28 3 Solution of the problem If we solve (24 (27 equations directly, the solution oscillates due to discontinuity which occurs between initial and boundary conditions in (28. So, the equations are transformed by taking a suitable coordinate as suggested by Ames and Attia in a similar kind of problems [28, 29]. Now, the transformed equations are (29 (30 (31 Equations (29-(31 are solved with the help of (28 in a PDE solver for different values of aifferent time. In a quiescent Magnetic fluid, magnetization and effective magnetic field intensity are always parallel to applied magnetic field intensity, and if the angular velocity of the particle is not zero, then from (3, (32 4 Results and Discussion: The set of transformed coupled equations (30 - (32 with the boundary condition (29 are solved with the help of PDE solver. The computation is carried out with for different values of. The problem considered here involves a number of parameters, on the basis of which a wide range of numerical results has been derived. Of these results, a small section is presented here for brevity. In this work, we have applied the basic idea of the Shliomis theory to the equation of motion and magnetization equation for Magnetic fluid flow due to a rotating disk. Fig. 1{(e 1, (e 2, (e 3 and (e 4 } shows the radial velocity profile with the variation of dimensionless parameter. From here we observe, in the ordinary case, i.e.,, after getting a peak value, the radial velocity suddenly decays to the steady state, however, in the graph (e 2, its decay is slower than (e 1 and, in graph the (e 3, it is much slower than (e 1, (e 2. Finally, in the graph (e 4, its decay is slowest among all. Moreover, (e 4 is the case where the radial velocity is affected maximum by the magnetoviscosity. It is also observed from the figure that, the peak values of the radial velocity aifferent time values are not globally affected by the field-induced viscosity and instead, have local effects. Further, these results are indicating, how the behavior of the radial velocity can be changed without affecting its peak values by changing the strength of externally applied alternating magnetic field? Here, for, the problem reduces to an ordinary case where there is no external magnetic field resulting in no magneto-viscous effects. In this case, at,, the radial velocity reaches the steady state region ISBN:

5 6 2 Φ =0 g=0 (e g=0.3 (e 2 E 6 2 g= (e g=0.9 (e 4 β Fig.1 [(e 1, (e 2, (e 3, (e 4 ]: The radial velocity profile with the variation of dimensionless parameter β. around, dimensionless axial distances from the ground, however, for, and, it reaches the steady state around, ;, ;,, respectively. For increasing values of time, the peak values of the radial velocity increases, and the flow remains radially outwards throughout the motion. The range of the parameters and are taken from [4, 5, 14]. Fig. 2{(f 1, (f 2, (f 3 and (f 4 } shows the growth of circumferential velocity profile with the variation of dimensionless axial distance for various values of time. Since, magnetoviscous effects increases on the tangential velocity with increasing time values, so the effects of the field induced viscosity on the tangential velocity at, in the graph are more prominent than at,. In the ordinary case, at,, the tangential velocity reaches steady state region around, 1.7 unit axial distance from the ground, however, for, and, it reaches the steady state around, 2 and, 2.1 and, 2.2 unit axial distances from the ground, respectively. ISBN:

6 1.0 Φ =0 g=0 (f (f 2 g=0.3 F 1.0 g= (f 3 β 1.0 g=0.9 (f 4 Fig. 2[(f 1, (f 2, (f 3, (f 4 ]: The tangential velocity profile with the variation of dimensionless parameter β. From the trends, we infer that the tangential velocity can be controlled by changing applied magnetic field and with increasing values of the parameter, decays of the tangential velocity to the steady state region is slower than the ordinary case. Further, the centrifugal force generated here is due to rotation of the disk being carried by fluid particles and at large distance its effeciminishes resulting in a decrease of the tangential velocity. Fig. 3{(g 1, (g 2, (g 3 and (g 4 } represent the axial velocity profile for different values of dimensionless time parameter at,, and, respectively. Comparing these figures, the axial velocity attains finite value around unit axial distance from the plate at, however, for, and, the finite values attained by the axial velocity are around, and unit axial distances from the plate, respectively. In ordinary case, at, the axial velocity attains a finite value between and as far from the plate, however, at the same time, it attains the finite values between and, and, and for,,, respectively. ISBN:

7 Φ =0 g=0 (g 1 g=0.3 (g G - g= (g 3 β - g=0.9 (g Fig. 3[(g 1, (g 2, (g 3 and (g 4 ]: The axial velocity with the variation of dimensionless parameter β. From all above, we conclude that the change in the axial velocity due to magneto-viscosity is dominant in comparison to the radial and tangential velocity. Figures 1, 2 and 3 indicate that the axial velocity component reaches the steady state region slower than the radial velocity component and much slower than the tangential velocity component. It happens because the centrifugal effect is a source of radial motion which is the source of the axial motion. And, due to the effect of effective magnetization parameter, all the three velocity components reach steady state slower than ordinary case with the variation of dimensionless axial distance. Here in the fig. 4(a, it is clear that there is a significant change in the axial velocity for increasing time values at far from the ground and also increasing the values of the parameter, the axial velocity gets larger values in comparison to the ordinary case. Further, negative values of the axial velocity indicate that magnetic fluid is flowing towards the disk. ISBN:

8 Η g=0, Φ =0 g=0.3, g=, g=0.9, (a M/M (b ω p t=0 ω p t=0.1 ω p t= ω p t= t/τ Β Fig. 4: Fig. 4(a is the axial velocity profile at infinite axial distance from the plate with the variation of time aifferent values of magnetization parameter. Fig. 4(b is the curve for reduced magnetization for different values of rotation of the Magnetic -particle when the true field is switched off at the moment t=0. Fig. 4(b shows the reduced magnetization curve when the applied magnetic field is suddenly switched off. In this case fluid remains at rest, so magnetization and effective magnetic field are always parallel to the applied magnetic field intensity and here, the rotation of the fluid particle is not zero. The result indicates that increasing the rotation of the colloidal particle decreases the magnetization in Magnetic fluid. Thus, the decay of magnetization is directly proportional to the rotation of the nano-magnetic particles. 5 Conclusions It is interesting to notice that when provided the applied magnetic field and vorticity of the flow are not collinear, the effective magnetization parameter creates an additional resistance on the components of velocity. This additional resistance of the Magnetic fluid flow due to a rotating disk reaches to a maximum when the applied magnetic field is perpendicular to the vorticity of the flow and rotation of the particles in the flow is completely inhibited. Further, the axial velocity does noepend on the distance from the axis buepends on the distance from the plate. In nut shell, we conclude that the strong changes in the viscous properties produce significant effects on electronically controllable signals and damping systems. References: [1] R.P. Feynman, R.B. Leighton, M. Sands, The Feyman Lecturers on Physics, Addison- Wesley Reading, Vol.1, [2] M.I. Shliomis, Ferrofluids as thermal ratchets, Physical Review Letters, Vol.92, 2004, pp [3] E. Blums, A. Cebers and M. M. Maiorov, Magnetic Fluids, Walter de Gruyter, Berlin and New York, [4] S. Odenbach, Magneto viscous Effects in Ferrofluids, Springer-Verlag, Berlin, [5] R. E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, ISBN:

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