Thermomagnetic effects on the stability of Taylor-Couette flow of a ferrofluid in the presence of azimuthal magnetic field

Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Research articles

Thermomagnetic effects on the stability of Taylor-Couette flow of a ferrofluid in the presence of azimuthal magnetic field Yavuz Emre Kamısß, Kunt Atalık ⇑ _ Department of Mechanical Engineering, Bog˘aziçi University, 34342 Bebek, Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 14 September 2017 Received in revised form 19 December 2017 Accepted 24 January 2018

2010 MSC: 00-01 99-00 Keywords: Ferrofluids Stability Taylor-Couette Ferrohydrodynamics

a b s t r a c t A stability analysis is conducted for the non-isothermal Taylor-Couette flow of a non-conductive ferrofluid under the action of an azimuthal magnetic field. The transition from steady flow to Taylor vortex flow is investigated in terms of the critical Taylor number and with respect to the changes in the magnetic field strength, radial temperature gradient, gap ratio, ferroparticle concentration and size. For the linear stability analysis, infinitesimal perturbations to the velocity, temperature and magnetization fields are considered and the resulting linear system of stability equations is solved using Chebyshev collocation method. Also, the original nonlinear system of equations is solved numerically using a finite element analysis software, and the results are compared with the linear stability analysis results. A significant stabilization is observed under strong magnetic fields for all cases. It is also observed that radial temperature difference has a destabilizing effect on the flow and this effect is amplified when the magnetic field strength is increased. Higher ferroparticle volume fraction and size lead to a strong degree of magnetization of the fluid and amplifies the stabilizing effect of magnetic fields. The stabilization under azimuthal magnetic forces is observed to be smaller for the narrow gap case compared to the wide gap case. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction Ferrofluids are colloidal suspensions of nano-sized ferroparticles (e.g.magnetite Fe3O4 particles) in a carrier liquid. These fluids, having strong susceptibility to external magnetic field, are used in many applications in heat and flow control. A common usage is in the high-power loudspeakers where the ferrofluid can effectively remove heat from the speaker coils and improve the sound quality by damping the resonances [1,2]. Ferrofluids can also be used in damping applications in civil engineering where damping of seismic activity is essential or in automotive industry where damping of rough road conditions is important [2]. Recent studies suggest the magnetic forces can be used as a body force to control flow fields [3], where non-contact mixing can be achieved, as in boundary layer control for high Reynolds number flow [4]. Ferrofluids are also popular in biomedical applications, especially in magnetic drug targeting [5–7]. The rotational flow between two concentric cylinders known as Taylor-Couette flow is a classical flow problem in fluid mechanics. Taylor-Couette devices are used as mixing devices in applications ⇑ Corresponding author. E-mail addresses: [email protected] (Y.E. Kamısß), [email protected] (K. Atalık). https://doi.org/10.1016/j.jmmm.2018.01.077 0304-8853/Ó 2018 Elsevier B.V. All rights reserved.

such as catalytic chemical reactors, rotating filtration devices and bioreactors [8–10]. Centrifugal effects generated by these devices are also needed to obtain stable suspensions of nano-sized particles in a solvent [11]. Taylor-Couette is known to have multiple stages of transition from laminar to turbulent regimes and have been subject to many theoretical and experimental investigations in the literature [12]. Andereck et al. [13] conducted TaylorCouette flow experiment to reveal the different transition modes from laminar to turbulent stages. For the gap ratios used in this study, Table 1 summarizes the measured Taylor number ranges for intermediate transition stages when the outer cylinder is stationary. The primary transition from Couette Flow (CF) to Taylor Vortex Flow (TVF) was studied both experimentally and numerically by Taylor in his benchmark study [12]. Fasel et al. [15] conducted a flow analysis using finite difference technique for wide gap at supercritical Taylor numbers to investigate in more detail the axisymmetric Taylor vortex flow regime. Jones conducted stability analyses for the transition from Taylor vortex flow to wavy vortex flow where stationary axisymmetric Taylor vortices were used as the basic state and non-axisymmetric perturbations were imposed in the small-gap limit [16] and for a wider range of gap ratios [17] to investigate the onset of wavy vortex flow. Modulated Taylor-Couette flow was investigated by Jones and Barenghi [18], where they consider sinusoidally varying rotational frequencies

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

197

Table 1 Taylor number (Ta) ranges for intermediate transition stages [13,14]. (The values are given for gap ratios (r 1 =r 2 ) of 0:3 and 0:9.) Flow regime

r 1 =r 2 ¼ 0:3

r 1 =r2 ¼ 0:9

Couette flow Taylor vortex flow Wavy vortex flow Modulated wavy vortices Turbulent Taylor vortices

Ta < 140:5 140:5 < Ta < 210:8 210:8 < Ta < 1558:1 1558:1 < Ta < 2108 2108 < Ta

Ta < 30:67 30:67 < Ta < 46 46 < Ta < 340 340 < Ta < 460 460 < Ta

of the cylinders. Modulated Taylor vortices have also been studied by Coughlin both analytically [19] and numerically [20]. In most of the publications in the literature, the transition from CF to TVF and to further flow regimes has been studied extensively for isothermal cases. There are also studies investigating this primary instability under the effect of radial temperature [21–23] and concentration gradients [11]. The first major study on Taylor-Couette stability under magnetic effects was done by Niklas [24] where an equation of motion was derived for a ferrofluid in Taylor-Couette flow with magnetic field dependent rotational viscosity and the effects of magnetic fields applied in different directions were discussed. The work was later extended to account for non-axisymmetric perturbations [25]. Stiles [26] considered an isothermal Taylor-Couette problem with axial magnetic field and developed analytical expressions for the narrow gap case that relate the criticial Taylor number to axial magnetic field strength. Odenbach [27] made an experimental study of magnetized Taylor-Couette flow with azimuthal magnetic field resulting from an alternating surface current flowing along the inner cylinder in the axial direction. Most of the studies on Taylor-Couette stability of ferrofluids under magnetic forces consider isothermal flow cases. Therefore, thermomagnetic effects on Taylor-Couette stability are needed to be investigated in more detail. Stiles et al.[28] investigates the non-isothermal case of a ferrofluid Taylor-Couette flow with a radial temperature difference and radial magnetization, to report significant drop in critical Taylor number wtih small temperature changes. In this study, we examine the instability mechanisms of non-isothermal Taylor-Couette flow of a ferrofluid under the action of an azimuthal external magnetic field with radial temperature gradients. The main purpose of this study is to underline the effects of azimuthal magnetic field strength, radial temperature difference, ferroparticle concentration and size on the stability of Couette flow. The gap effects on the stability are also demonstrated. The mathematical formulation is given in Section 2, the governing system of equations is linearized and the resulting system is solved using Chebyshev collocation method as explained in Section 3. Linear stability results are shown for isothermal and non-isothermal cases in terms of neutral stability curves and streamline plots, and the results are compared with the results in the literature. The original system of equations is also solved using the finite element analysis software package COMSOL and the results are compared with the linear stability results. The stability results for the thermomagnetic case are shown and discussed in Section 4 in terms of the variation of critical Taylor number with magnetic field strength, radial temperature gradient, particle concentration and size for different gap ratios. 2. Mathematical formulation The flow geometry is given in Fig. 1, and represents TaylorCouette ferrofluid flow under radial temperature gradient and with a line current I flowing along the z-axis. The unit normal vectors in the radial and axial directions are given by r and z, respectively. The inner cylinder of radius r 1 is rotating with a velocity U ¼ U 1 while the outer cylinder of radius r2 is stationary. The ratio of

Fig. 1. Flow geometry.

the radii is denoted by f ¼ r 1 =r 2 , which is also called the gap ratio. The inner and outer walls are kept at constant temperatures T ¼ T 1 and T ¼ T 2 , respectively. The line current results in an azimuthal magnetic field decaying radially, making the problem similar to that studied experimentally by Odenbach[27]. The governing mass, momentum, energy, magnetic relaxation equations and Gauss’ Law for magnetic field are given respectively as follows,

ru¼0

qf

ð1Þ

Du l u2 ¼ rp þ gr2 u þ l0 ðM  rÞH þ 0 r  ðM  HÞ þ q h r Dt 2 r ð2Þ




qc  l0 H 

 @M DT ¼ k f r2 T @T Dt

ð3Þ


 DM 1 l 1 H M  Meq ¼ ðr  uÞ  M þ 0 ðM  HÞ  M  Dt 2 H 6g/ sB

ð4Þ

rB¼0

ð5Þ

B ¼ l0 ðM þ HÞ

ð6Þ

where D=Dt ¼ @=@t þ ðu  rÞ is the material derivative. In this formulation t denotes the time, u ¼ ½ur ; uh ; uz  is the velocity vector with cylindrical components, p is the pressure field, T is the temperature field. H and M are the magnetic and magnetization field vectors respectively. The base fluid density is represented by qf ; g is the magnetic field dependent dynamic viscosity, l0 is the magnetic permeability of free space, qc is the heat capacity, kf is the thermal conductivity of the base fluid, / is the volume fraction of magnetic particles to base fluid, which is assumed to be independent of space. The characteristic time scale for mass diffusion is up to 5 degrees of magnitude longer than the characteristic time scale for momentum diffusion. Their ratio is the Schmidt number (Sc = g=qDB where DB is the Brownian diffusivity) and it is typically around 105 for a water

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

198

based ferrofluid and for the temperature and particle size values used in this study. One can therefore treat the fluid as pure since mass diffusion needs a much longer time to evolve [29–32]. Brownian relaxation time of the magnetic particles is denoted by sB . The third term on the right hand side of Eq. (2) represents the volumetric Kelvin force vector and the fourth term is the rotation vector of the magnetic torque. The last term represents the centrifugal force in the radial direction where the density, q, is given in accordance with the Boussinesq approximation [33], by the expression;

q ¼ /qp þ ð1  /Þqf ð1  bðT  T ref ÞÞ

ð7Þ

where b is the thermal coefficient of water and qp is the density of the nanoparticle. The problem studied in this paper considers a magnetic field resulting from external current. The general form of the Ampere’s Law is given as,

r  H0 ¼ Je þ Ji

ð8Þ

with the external current, Je and the induced current Ji ¼ rc ðE þ u  BÞ where rc is the conductivity of the medium and H0 is the external magnetic field. The ferrofluid is assumed to be non-conductive (rc ¼ 0), hence there are no induced currents (Ji ¼ 0). If we consider a magnetic field due to a permanent magnet, this leads to a curl-free magnetic field (r  H0 ¼ 0), as commonly used in the ferrofluid literature. In our problem, however, Eq. (8) reduces to r  H0 ¼ Je . The azimuthal magnetic field resulting from an external axial current Iz can be formulated by integrating r  H0 ¼ Je using Stokes’ Law, as

Z

I

H0  dl ¼ L

Je  dS ¼ I

ð9Þ

from which it can be deduced that Hh ¼ I=2pr with r 2 ð0; 1Þ. In the absence of induced or free surface currents, the tangential component of H0 is continuous across the radial boundaries of the fluid domain. For the linear stability problem, the perturbations in the local magnetic field H are neglected such that H ¼ H0 . With the parameter ranges covered in this study, the equilibrium susceptibility, which is the ratio of equilibrium magnetization to external magnetic field strength, is at most Oð102 Þ. Therefore the OððM0  H0 Þ=LÞ terms will be dominant over the OððM0  H0 Þ=LÞ terms, where the symbols with prime denote perturbations in the corresponding fields and L is a length scale. Then the perturbations in local magnetic field H0 may be neglected as also explained in [24,34]. Approximating the local field H with external field H0 allows to define a divergence free magnetization field in the perturbed state as the form of the magnetic field determined by (9) satisfies r  H0 ¼ 0 which leads to r  M ¼ 0 due to Eqs. (5) and (6). The equilibrium magnetization of the fluid aligned with the external magnetic field is denoted by M eq , and is governed by the Langevin relation given by,

M eq ¼/M d LðaL Þ
 1 ¼/M d cothðaL Þ 

aL

3 LðaL Þ is the Langevin function where aL ¼ p6 l0 dkBMT d H is the local Lan-

gevin parameter [1]. In this expression d represents the nanoparticle diameter and kB is the Boltzmann constant. The magnetic field dependent viscosity, g, is given in non-dimensional form as g which can be expressed as [35],

g 3 aL  tanhðaL Þ ¼ / g0 2 aL þ tanhðaL Þ

where g0 is the dynamic viscosity of the base fluid.

r xð1  fÞ þ ð1 þ fÞ 2r  ðr 1 þ r 2 Þ ¼ x¼ r2 2 Dr tg u pr 2 T  T1     t ¼ u ¼ p ¼ T ¼ U1 Dr qf U1 g T2  T1

r ¼

M ¼

M Md

H ¼

ð12Þ

Hh 1 ¼ H0 r 

where Dr ¼ r 2  r1 . The magnetic field strength scale, H0 , is the value of magnetic field strength at the outer cylinder (H0 ¼ I=2pr2 ). The basic flow state is the Couette flow which represents the 1-D steady state solution of the system of Eqs. (1)–(5) and is written in terms of the flow variables as,

u ¼ ð0; Vðr  Þ; 0Þ T  ¼ Hðr  Þ M ¼ ð0; M 0 ; 0Þ 







ð13Þ 



Using u ¼ 1; T ¼ 0 at r ¼ f and u ¼ 0; T ¼ 1 at r ¼ 1 as boundary conditions, the functions in the above basic state solution (13) can be expressed as,


 1 f with A ¼ 2 Vðr Þ ¼ A r   r f 1

Hðr  Þ ¼ 1 

ð14Þ

logðr  Þ logðfÞ

ð15Þ

M0 ¼ /LðaL Þ with aL ¼

c M H cT H þ cT2

ð16Þ

where cM ¼ p6 l0 d M d H0 ; cT ¼ kB ðT 2  T 1 Þ and cT2 ¼ kB T 1 . To investigate the critical Taylor number for the transition from CF to TVF, infinitesimal perturbations are imposed on the basic flow in the following form, 3

^r ; u ^h ; u ^ z ÞðxÞert u ¼ð0; Vðr  Þ; 0Þ þ ðu ^ðxÞert p ¼p

 þikz

 þikz



  T  ¼Hðr  Þ þ Tb ðxÞert þikz b r; M b h; M b z ÞðxÞert þikz M ¼ð0; M 0 ; 0Þ þ ð M

ð17Þ

where r is the dimensionless amplification factor and k is the dimensionless wavenumber. The variables with hat (^) symbol denote the related perturbation amplitudes. The temperature dependent density given in (7) is also scaled with qf to give the non-dimensional density q as,

!

q ¼

qp q þ ð1  /Þð1  cq;T T  Þ ¼/ qf qf

ð18Þ

 ) and a perturbed state (q0) resulting and decomposed into a basic (q from the perturbations in temperature (T0) with T ref taken as T 1 , which is finally expressed as,

!

ð10Þ

where M d is the bulk magnetization of the magnetic particle and

g ¼

All of the variables in the above system of Eqs. (1)–(5) are nondimensionalized by introducing the following non-dimensional variables,

ð11Þ

q q ¼ / p þ ð1  /Þð1  cq;T HÞ  ð1  /Þcq;T T 0 qf |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð19Þ

q0

q

 in (19) reprewhere cq;T ¼ bðT 2  T 1 Þ. The term underbraced as q sents the variation of the density in the basic state as a function of temperature which is mostly neglected in the literature. The linear analysis shows that accounting for this variation results in a rise in the critical Taylor number which is also confirmed by the solution of the full nonlinear system of equations, which will be presented in Section 3. Therefore we choose to include this term for the rest of the calculations presented in this paper. When the variable expressions in Eq. (17) and Eq. (19) are substituted into Eqs.

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

(2)–(4), the nonlinear terms are neglected, and the perturbation b z are eliminated using the divergence-free ^; u ^ z and M amplitudes p properties of velocity and magnetization fields. Finally we obtain the following set of linear ordinary differential equations for the perturbation amplitudes, 2

rðDD  k2 Þu^Re ¼ g ðDD  k2 Þ u^Re


  RM 1  f  b f ^h  q0 VÞ u H M h   Ta2 Vð2q þk  r r Re 

2

ð20Þ

ru^h ¼ 2ð1  fÞAu^Re þ g ðDD  k2 Þu^h

ð21Þ

b hÞ b r  RH M rPr Tb ¼ PrDHu^Re þ ðD D  k2 Þ Tb þ jPrðReð1  fÞA M ð22Þ
 1f b Re 2 ^h r  M ðM0 ðDD  k Þ þ DM0 D Þu r ¼ r 2 


 1f 1f ^h þ Re  þ M D u DM M 0 0 0 r r 


  1f 1f bh þ Re ð1  fÞAD þ DV  V þ VD M r r

  1f br þ DF MH  ð23Þ F MH M r




b h ¼ DM0 u ^ Re þ Reð1  fÞ A  rM

 V b bh M r  RH M r

ð24Þ

The operator symbols and non-dimensional parameters appearing in Eqs. (20)–(24) are given as follows,

d d 1f D ¼ 2 þ  dx dx r qf U 1 Dr 2 Re ¼ Re ð1  fÞ ¼ Ta2 f D¼2

g0

Pr ¼ RH ¼

g0 c kf

RM ¼

qf Dr2 g0 sB

^ Re ¼ Reu ^r u

l0 qf Dr2 Md H0 lMH j¼ 0 d 0 qcðT 2  T 1 Þ g20

F MH ¼

RM M 0 H0 þ RH 6g/

ð25Þ

where Re is the Reynolds number, Pr is the Prandtl number and the definition of the Taylor number (Ta) follows the definition in Walowit [21]. RM is the ratio of magnetic forces to viscous forces and RH is the ratio of viscous time scale to magnetic relaxation time. The boundary conditions for the perturbation amplitudes at x ¼ 1 are Dirichlet and clamped conditions given by,

^ Re ¼ Du ^ Re ¼ u ^ h ¼ Tb ¼ 0 u

ð26Þ

In alignment with r  M ¼ 0, the normal component of B must be continuous at the boundaries therefore the perturbation amplitudes of the magnetization at the boundary must also be zero and they are given as,

br ¼ M bh ¼0 M

ð27Þ

3. Numerical method and validation The system of linear stability Eqs. (20)–(24), constitutes an eigenvalue problem of the form Ax ¼ rBx which is solved by Cheb; M b r; M b h , are ^h ; T ^ Re ; u byshev collocation method. The variables u assigned trial functions in the form of Chebyshev polynomials. Functions chebdif.m and cheb4c.m [36] were utilized to solve the system using MATLAB, in which the differentiation matrices Dm;n are constructed as follows,

Dð1Þ m;n ¼

8 cm ð1Þnþm > ; > cn ðxm xn Þ > > > x > 1 m <2 ; ð1x2 Þ m

199

n–m n¼m–1

2ðN1Þ þ1 > > ; n¼m¼1 > 6 > > > 2ðN1Þ2 þ1 : ; n¼m¼N  6 2

 l where c1 ¼ cN ¼ 2; c2 ¼ . . . ¼ cN1 ¼ 1 and DðlÞ ¼ Dð1Þ . The interpolation points are the Chebyshev nodes of the second kind given as


 pðN þ 1  2mÞ xm ¼ sin 2ðN  1Þ

ð28Þ

where m ¼ 1 . . . N and N is the number of collocation points. After constructing the matrices based on Eqs. (20)–(24), the generalized eigenvalues are calculated using QZ-algorithm and the neutral stability curves are obtained in terms of the Taylor (Ta) and wavenumbers (k) where the real part of the leading eigenvalue is zero. In our calculations we observed that a value of N ¼ 100 is sufficient to obtain numerical convergence with an accuracy of the order of 104 . For all the cases we studied, rr ¼ ri ¼ 0 was observed where r ¼ rr þ iri ; which means the instability sets in cellular convection also known as Taylor vortices. To consider the effects of nonlinear terms on the stability in terms of the value of the critical Taylor number for the primary transition from CF to TVF, the full nonlinear system of Eqs. (1)– (5) was also solved using COMSOL software package with finite element method. Since ri ¼ 0, the instability is stationary rather than oscillatory in time and therefore stationary solvers have been used to solve the magnetic relaxation equation in 2D axisymmetric geometry. The axial length of the annulus is set using an aspect ratio of C ¼ L=h ¼ 10. The conditions in axial boundaries are set to be periodic in all variables. To observe the onset of cellular convection (i.e. Taylor vortices), we conducted parametric sweep starting from %50 of the critical Taylor number obtained from the linear analysis and increased until the vortices are clearly visible. The grid independence is reached after around 3  105 degrees of freedom and our simulation used a mesh with 341296 degrees of freedom (corresponding to 48552 quadratic elements). Fig. 2 shows the variation of the nondimensional kinetic energy (EK ) integrated over R  Cð1fÞ R 1 u2   the r  z plane with respect to degrees of dr dz 0 f 2 freedom of the finite element simulation and the resulting gridindependent mesh. The linear stability solutions are tested with the stability results of Walowit [21] for isothermal and non-isothermal cases under radial temperature gradient in the absence of the magnetic field. Fig. 3a shows the neutral curves for the onset of transition from CF to TVF, where the minimum points reveal the pairs of critical wavenumber and Taylor number, (kcr ; Tacr ), for each gap ratio, f. The results are in good agreement with the isothermal results of Walowit [21]. Taylor [12] had observed that the critical values obtained from linear stability analysis agree well with experimental results, which points that a nonlinear analysis should give the same critical Taylor number for the transition from CF to TVF. This hypothesis is also validated with the numerical solution of the full system of equations using COMSOL. Fig. 3b shows the streamline plots obtained using COMSOL which verify the existence of Taylor vortices at Ta = 111.85. One can observe from Fig. 3b that instabilities and flow bifurcation begins at earlier stages due to nonlinear effects however the settling of the Taylor Vortex Flow occurs at the critical Taylor number which is obtained from the linear analysis. The critical point in k-Ta space for f ¼ 0:3 is (3.217,111.85) in the corresponding neutral stability curve. Fig. 3a illustrates also the effect of gap width on the stability of Couette flow. It can be

200

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

Fig. 2. (a) The convergence of the nondimensional kinetic energy integrated over the r-z plane with respect to degrees of freedom of the finite element simulation. (b) A screenshot of the mesh from the lower portion of the axisymmetric geometry.

observed that narrowing the gap width destabilizes the flow significantly, shifting the neutral stability curve downwards, which is also in agreement with the results in the literature [12]. The isothermal results and their validations are also gathered in Table 2. To test the non-isothermal results, we use the values in Avramenko [11] for the parameter cq;T ð¼ bðT 2  T 1 Þ), which is a nondimensional coefficient representing the density change due to temperature difference, and we demonstrate the effects of the  in Eq. (19). Fig. 4a inclusion of the basic state density variation q shows the neutral stability curves and Fig. 4b shows the resultant Taylor vortices for the non-isothermal case for the gap ratio f ¼ 0:3. We use a sample case from Avramenko’s study (cq;T ¼ 1) to illustrate the effects of the variation of the density in the basic state. The critical Taylor number for cq;T ¼ 1 is Ta = 91.9 without including the variation of the density in the basic state, while Ta = 94.6 is the critical value taking this variation into account, shown in the neutral curves in Fig. 4a and the corresponding vortices in Fig. 4b. Table 3 compares the results at a gap ratio of f ¼ 0:3. The inclusion of the base state density variation reduces the destabilizing effect of the positive temperature gradient and reduces the stabilizing effect of the negative temperature gradient (cq;T < 0).

Fig. 3. (a) Neutral stability curves for isothermal flow with different gap ratios. (b) Streamline plots from nonlinear study at f ¼ 0:3.

Table 2 Comparison of the isothermal results with those of Walowit [21]. f

0.1 0.3 0.5 0.7 0.9

This paper

Data of Walowit [21]

kcr

Tacr

kcr

Tacr

3.35 3.21 3.15 3.13 3.15

421.39 111.85 68.15 52.04 43.87

3.3 3.2 3.16 3.14 3.13

422.79 111.89 68.18 52.04 43.88

The values of the parameter cq;T used in Ref. [11] correspond to very large values of DT if the value taken for b is that of water’s (given in Table 4). Therefore the effect of cq;T , which scales the density change in the fluid due to temperature difference, is magnified both in Ref. [11] and in Fig. 4a. Using the values in Table 4 and the

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

201

Table 4 Thermophysical and magnetic properties used in the calculations. Density of the base fluid

qf

997 kg/m3

Specific heat of the base fluid Thermal conductivity Viscosity of the base fluid

cf k

4180 J/kg  K 0.58 W/m  K

Thermal coefficient

b

g0

9  104 Pa  s

Density of the nanoparticle

qp

2:57  104 1/K 5200 kg/m3

Bulk magnetization of the magnetic particle

Md

sB

4:5  105 A=m

Brownian relaxation time of the magnetic particles

106 s

[11]. The non-dimensional critical wavenumber can also be deduced from the streamline plots by using the relation Ckcr ¼ pnv ortex where C is the aspect ratio and nv ortex is the number of vortices in the plots [37]. The streamline plots given in Figs. 3 and 4 reveal that kcr  p, which confirms the critical wavenumber in the neutral curves. 4. Results and discussion To investigate the effects of azimuthal magnetic field strength, radial temperature difference and particle concentration and size on the stability of Couette flow in the presence of magnetic field, we choose a typical water-magnetite suspension with the thermophysical and magnetic properties given in Table 4. We conduct a parametrical study by changing the azimuthal magnetic field strength at the outer wall, H0 , the temperature difference, DT ¼ T 2  T 1 , the ferroparticle concentration / and size d. Each of these dimensional parameters appear in different nondimensional numbers defined in Eq. (25). The parametrical study is conducted with respect to these dimensional parameters with values chosen from practical ranges to demonstrate the realistic effects which could be verified with experiments. The gap effects are considered for a wide gap case (f ¼ 0:3) and a narrow gap case (f ¼ 0:9). The magnetic field strength values are chosen from the interval of 0 6 H0 ½A=m 6 105 , the temperature difference values are chosen from the interval of 10 6 DT½K 6 50, the volume fraction values of magnetic particles (magnetite) are chosen from the interval 0 < / < 0:1 and finally the size of the ferroparticles is varied in the interval of 5 6 d½nm 6 25. These value ranges correspond to the ranges used in the experiments in the literature [27]. With the values given in the table, Pr = 6.48 and Fig. 4. (a) Neutral stability curves for nonisothermal flow with different values of the parameter cq;T at f ¼ 0:3. (b) Streamline plots for cq;T ¼ 1 at f ¼ 0:3.

RH ¼ 1:77  107 are fixed. The values of the rest of the nondimensional parameter are provided in Appendix A. 4.1. Thermomagnetic effects

Table 3 Comparison of the non-isothermal critical values at gap ratio of f ¼ 0:3 with the results of Avramenko [11]. cq;T

1 1 2

This paper

Data of Avramenko [11]

kcr

Tacr

kcr

Tacr

3.14 3.23 3.25

123.98 94.63 83.31

3.02 3.231 3.245

152.79 91.9 79.76

values of DT studied later in this paper, the parameter cq;T ranges from 2:6  103 to 1:3  102 , which is relatively low compared to the values investigated by Avramenko et al. [11]. It can be concluded that an increase in the parameter cq;T destabilizes the flow. Higher values of cq;T point to a density decrease towards the outer wall which in turn favors flow bifurcation and vortical structures

The variation of the critical Taylor number with respect to magnetic field strength is shown in Figs. 5 and 6, in the wide gap case (f ¼ 0:3) and the narrow gap case (f ¼ 0:9), respectively, for different radial temperature differences and ferroparticle volume concentrations, and a particle diameter of d ¼ 10 nm. It is seen that strong magnetic fields are required to observe a significant increase in the critical Taylor number, i.e. a significant stabilization for both gap ratios. The magnetic field strength appears in the nondimensional numbers RM ; j and also the dimensional parameter cM which appears in the local Langevin parameter, aL (defined with Eq. (10)), and represents the magnetic energy of a single ferroparticle. The magnetic force is scaled with RM =Re, which corresponds to the ratio of magnetic forces and inertial forces, therefore an increase in RM would stabilize the flow as it increases the effect of radial Kelvin force over centrifugal force in the momentum equation, Eq. (2). An upward shift in the local Langevin parameter, aL , increases the viscosity through the expression (11), which also

202

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

Fig. 5. Variation of the critical Taylor number with respect to magnetic field strength at different radial temperature differences and different concentrations for f ¼ 0:3, the wide-gap case. (a) / ¼ 0:04 (b) / ¼ 0:06 (c) / ¼ 0:08.

Fig. 6. Variation of the critical Taylor number with respect to magnetic field strength at different radial temperature differences and different concentrations for f ¼ 0:9, the narrow-gap case. (a) / ¼ 0:04 (b) / ¼ 0:06 (c)/ ¼ 0:08.

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

203

has a stabilizing effect. The parameter j, which appears in Eq. (22) and represents the ratio of magnetocaloric energy effects over thermal energy effects as defined in Eq. (25), is the macroscopic equivalent of the local Langevin parameter, aL . In the literature, that appears in Eq. (3) is evaluated using a linear magthe term @M @T netic equation of state and named as pyromagnetic coefficient [1,38]. However when magnetic relaxation is taken into account, as in this study, M becomes implicitly dependent on the temperature T therefore the linear magnetic state equation assumption is not suitable. This term appears in Eq. (3) as a source term that is scaled with j. It is observed that increasing j has a destabilizing effect on the flow. The effect of the radial temperature difference can also be seen in Figs. 5 and 6; an increased temperature nonuniformity leads to flow destabilization. To illustrate the combined effects of the magnetic field strength and the temperature difference, we plotted a typical case for a concentration of

/ ¼ 0:06 in Fig. 7a and b, for the wide and narrow gap ratios, respectively. We can see that destabilizing effect of increasing positive temperature gradient is more pronounced at strong magnetic fields for both gap ratios. Although the temperature effects on the critical Taylor number ratios using these DT values seem small, the destabilization trend is clearly detectable. The effect of DT appears strongest due to the density change parameter, cq;T , whose significance on the stabilization is underlined in Section 3. The critical wavenumbers, kcr , corresponding to the nonmagnetic flow cases have been validated in Section 3. In the literature, no significant change in the dimensionless critical wavenumber is observed for isothermal flows under azimuthal magnetic fields [24,25]. For the highest magnetic field strength used in this study (H0 ¼ 100 kA/m), our results indicate a slight decrease of 10% of the critical wavenumber with respect to nonmagnetic case where the critical wavenumber kcr  p.

Fig. 7. Effect of the radial temperature difference on the stability at / ¼ 0:06. Tacr0 corresponds to the isothermal critical Taylor number at / ¼ 0:06. (a) wide gap (f ¼ 0:3) (b) for narrow gap (f ¼ 0:9).

Fig. 8. Effect of the volume fraction of ferroparticles on the stability. Tacr0 corresponds to the isothermal critical Taylor number in the absence of magnetic field effects. (a) wide gap (f ¼ 0:3) (b) for narrow gap (f ¼ 0:9).

204

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

The numerical solution of the full system of Eqs. (2)–(5) given by finite element analysis software COMSOL, show that nonlinear terms have small contribution to the instability mechanisms of the ferrofluid flow. Although the instability and the formation of vortical structures are observed at slightly lower Taylor numbers, the results agree with linear stability analysis results showing that the critical Taylor number does not change significantly up to a magnetic field strength of 1 kA/m and having the onset of cellular convection at the Taylor number values given by the linear analysis.

4.2. Gap effects It is well known from the literature that in the absence of magnetic effects, narrowing the gap width has a profound destabilizing effect and decreases the critical Taylor number [12], for both isothermal and non-isothermal flows. This is also validated in Section 3 (Fig. 3a). It can be observed from Fig. 6 that the range of critical Taylor number has decreased to less than 50 percent in the narrow gap case compared to its value in the wide gap case shown in Fig. 5. As also shown in Fig. 7b, although the thermomagnetic effects on the stability are similar, these effects are smaller compared to the wide gap case (Fig. 7a). To illustrate the gap effects more clearly, we plot the critical Taylor number ratio for the isothermal case at a fixed concentration for the wide and narrow gap ratios in Fig. 9. Though the trend of stabilization with increasing magnetic field is similar, the stabilizing effects are weaker in the narrow gap case compared to the wide gap case. This is primarily attributed to the fact that for narrow gap ratios (f ! 1), the centrifugal effects become much more dominant than magnetic effects, leading to flow instabilities at much lower Taylor numbers. Also, the wide gap allows a larger change in the magnetic field strength which is essential for the Kelvin force to compete with the centrifugal force. Using the non-dimensional definition of the magnetic field due to an axial line current in Eq. (12), the range of values for the non-dimensional magnetic field strength is between [1,3.33] in wide gap (f ¼ 0:3) whereas in narrow gap (f ¼ 0:9) it is between [1,1.11].

Fig. 9. Gap effects on the stabilization at / ¼ 0:06, Tacr0 corresponds to the critical Taylor number for the isothermal case at / ¼ 0:06 in the absence of the magnetic field.

4.3. Particle concentration and size effects The concentration effects are shown in Fig. 8a and b for the isothermal case. Higher volume fraction of ferroparticles lead to a stronger stabilization under strong magnetic fields. The volume fraction / is a multiplication factor of the magnetic effects, therefore for the limiting case of / ! 0 the magnetic effects would vanish, hence higher concentrations magnify the stabilizing effects of the magnetic field strength. The effect of ferroparticle size is demonstrated for an isothermal flow in Fig. 10. An increase in the particle size has a considerable effect on the stabilization of the flow under strong magnetic field, especially in the wide gap case, as shown in Fig. 10a. The size of the ferroparticle has an influence on the degree of magnetization of the ferrofluid such that larger particles would result in higher local Langevin parameters and higher degree of magnetization, leading to an increase in both the Kelvin force and the viscosity. One obvi-

Fig. 10. The effect of ferroparticle size on the stability at different values of magnetic field strength. (a) wide gap (f ¼ 0:3) (b) narrow gap (f ¼ 0:9).

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

ous drawback of having a suspension with relatively larger particle size is the magnetic agglomeration and the loss of colloidal stability of the suspension against gravity. Particle sizes of colloids that are stable to above mentioned mechanisms range up to about 10 nm [1]. We can verify again from Fig. 10b that for narrow gap ratios, the stabilizing effects are weaker.

205

Radial temperature difference has a destabilizing effect on the flow and this effect is magnified with increasing magnetic field strength. Ferroparticle volume fraction increases the stabilizing effect of the applied magnetic field by strongly coupling the magnetic effects to the flow. The size of the ferroparticle has also a stabilizing effect which is amplified under stronger magnetic field. These effects are smaller for the narrow gap case where centrifugal effects dominate over magnetic effects.

5. Conclusion In this paper we investigated the thermomagnetic effects on the stability of the Taylor-Couette flow of a ferrofluid in the presence of an azimuthal magnetic field generated by an axial DC current. A linear stability analysis using Chebyshev collocation method has been applied, also the full nonlinear system of equations has been solved using a finite element software package to reveal the critical Taylor numbers and wavenumbers for the transition from Couette flow to Taylor vortex flow. It is found that significant stabilization may be achieved under the application of strong magnetic fields.

Appendix A. Values of the non-dimensional parameters used in the studies The parametric study of linear stability was conducted with dimensional parameters like the magnetic field strength H0 ½A=m, the temperature difference between two concentric cylinders, DT½K. Tables A.5,A.6,A.7,A.8 show the values of the nondimensional an scaling parameters introduced in Section 2 at the parameter ranges studied in this paper.

Table A.5 Values of cq;T and cMT1 at the parameter range studied in the paper.

DT½K

cq;T cMT1 ½J

10

0

10

30

50

0.00257 1.38  1022

0 0

0.00257 1.38  1022

0.00771 4.14  1022

0.01285 6.90  1022

Table A.6 Values of RM at the parameter range studied in the paper. H0 ½A=m 0 RM

0

1

10 )

10

1.11  10

5

2

10

1.11  10

6

3

104

1.11  10

7

4  104

1.11  10

8

7  104

4.45  10

8

105

7.78  10

8

1.11  108

Table A.7 Values of cM ½J at the parameter range studied in this paper. H0 ½A=m d½nm 5 10 15 20 25

0 0 0 0 0 0

101

102 25

3.70  10 2.96  1024 9.99  1024 2.37  1023 4.63  1023

103 24

3.70  10 2.96  1023 9.99  1023 2.37  1022 4.63  1022

104 23

3.70  10 2.96  1022 9.99  1022 2.37  1021 4.63  1021

4  104 22

3.70  10 2.96  1021 9.99  1021 2.37  1020 4.63  1020

7  104 21

1.48  10 1.18  1020 4.00  1020 9.47  1020 1.85  1019

105 21

2.59  10 2.07  1020 7.00  1020 1.66  1019 3.24  1019

3.70  1021 2.96  1020 1019 2.37  1019 4.63  1019

Table A.8 Values of j at the parameter range studied in this paper. H0 ½A=m

DT½K

0

101

102

103

104

4104

7104

105

10

0

1:36  107

1:36  106

1:36  105

1:36  104

5:43  104

9:50  104

1:36  104

10

0

1:36  107

1:36  106

1:36  105

1:36  104

5:43  104

9:50  104

1:36  104

30

0

4:52  108

4:52  107

4:52  106

4:52  105

1:81  104

3:17  104

4:52  104

50

0

2:71  108

2:71  107

2:71  106

2:71  105

1:09  104

1:90  104

2:71  104

206

Y.E. Kamısß, K. Atalık / Journal of Magnetism and Magnetic Materials 454 (2018) 196–206

References [1] R.E. Rosensweig, Ferrohydrodynamics, Courier Corporation, 2013. [2] A. Afifah, S. Syahrullail, N. Sidik, Magnetoviscous effect and thermomagnetic convection of magnetic fluid: a review, Renew. Sustain. Energy Rev. 55 (2016) 1030–1040. [3] K.J. Solis, J.E. Martin, Torque density measurements on vortex fluids produced by symmetry-breaking rational magnetic fields, Soft Matter 10 (33) (2014) 6139–6146. [4] F. Cornat, D. Beck, I. Jacobi, H. Stone, Flow control using ferrofluids, in: APS Meeting Abstracts, 2013. [5] R. Ganguly, A.P. Gaind, S. Sen, I.K. Puri, Analyzing ferrofluid transport for magnetic drug targeting, J. Magn. Magn. Mater. 289 (2005) 331–334. [6] B. Shapiro, Towards dynamic control of magnetic fields to focus magnetic carriers to targets deep inside the body, J. Magn. Magn. Mater. 321 (10) (2009) 1594–1599. [7] A. Nacev, C. Beni, O. Bruno, B. Shapiro, The behaviors of ferromagnetic nanoparticles in and around blood vessels under applied magnetic fields, J. Magn. Magn. Mater. 323 (6) (2011) 651–668. [8] M.E. Ali, D. Mitra, J.A. Schwille, R.M. Lueptow, Hydrodynamic stability of a suspension in cylindrical couette flow, Phys. Fluids 14 (3) (2002) 1236–1243. [9] D. Dherbécourt, S. Charton, F. Lamadie, S. Cazin, E. Climent, Experimental study of enhanced mixing induced by particles in taylor–couette flows, Chem. Eng. Res. Des. 108 (2016) 109–117. [10] A. Akonur, R.M. Lueptow, Chaotic mixing and transport in wavy taylor–couette flow, Physica D 167 (3) (2002) 183–196. [11] A. Avramenko, A. Tyrinov, I. Shevchuk, N. Dmitrenko, Centrifugal instability of nanofluids with radial temperature and concentration non-uniformity between co-axial rotating cylinders, Eur. J. Mech.-B/Fluids 60 (2016) 90–98. [12] G.I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Philos. Trans. R. Soc. London Ser. A, Containing Papers of a Mathematical or Physical Character 223 (1923) 289–343. [13] C.D. Andereck, S. Liu, H.L. Swinney, Flow regimes in a circular couette system with independently rotating cylinders, J. Fluid Mech. 164 (1986) 155–183. [14] M. Azaditalab, A. Houshmand, A. Sedaghat, Numerical study on skin friction reduction of nanofluid flows in a taylor–couette system, Tribol. Int. 94 (2016) 329–335. [15] H. Fasel, O. Booz, Numerical investigation of supercritical taylor-vortex flow for a wide gap, J. Fluid Mech. 138 (1984) 21–52. [16] C. Jones, Nonlinear taylor vortices and their stability, J. Fluid Mech. 102 (1981) 249–261. [17] C. Jones, The transition to wavy taylor vortices, J. Fluid Mech. 157 (1985) 135– 162. [18] C. Barenghi, C. Jones, Modulated taylor–couette flow, J. Fluid Mech. 208 (1989) 127–160.

[19] K. Coughlin, P. Marcus, Modulated waves in taylor-couette flow part 1. Analysis, J. Fluid Mech. 234 (1992) 1–18. [20] K. Coughlin, P. Marcus, Modulated waves in taylor-couette flow part 2. Numerical simulation, J. Fluid Mech. 234 (1992) 19–46. [21] J. Walowit, S. Tsao, R. DiPrima, Stability of flow between arbitrarily spaced concentric cylindrical surfaces including the effect of a radial temperature gradient, J. Appl. Mech. 31 (4) (1964) 585–593. [22] M. Ali, P. Weidman, On the stability of circular couette flow with radial heating, J. Fluid Mech. 220 (1990) 53–84. [23] J.M. Lopez, F. Marques, M. Avila, Conductive and convective heat transfer in fluid flows between differentially heated and rotating cylinders, Int. J. Heat Mass Transfer 90 (2015) 959–967. [24] M. Niklas, Influence of magnetic fields on taylor vortex formation in magnetic fluids, Zeitschrift für Physik B Condensed Matter 68 (4) (1987) 493–501. [25] M. Niklas, H. Müller-Krumbhaar, M. Lücke, Taylor-vortex flow of ferrofluids in the presence of general magnetic fields, J. Magn. Magn. Mater. 81 (1–2) (1989) 29–38. [26] P.J. Stiles, M. Kagan, J.B. Hubbard, On the couette-taylor instability in ferrohydrodynamics, J. Colloid Interface Sci. 120 (2) (1987) 430–438. [27] S. Odenbach, H. Gilly, Taylor vortex flow of magnetic fluids under the influence of an azimuthal magnetic field, J. Magn. Magn. Mater. 152 (1–2) (1996) 123– 128. [28] P. Stiles, P. Blennerhassett, Stability of cylindrical couette flow of a radially magnetized ferrofluid in a radial temperature gradient, J. Magn. Magn. Mater. 122 (1–3) (1993) 207–209. [29] M. Shliomis, M. Souhar, Self-oscillatory convection caused by the soret effect, EPL 49 (1) (2000) 55. [30] W. Köhler, S. Wiegand, Thermal Nonequilibrium Phenomena in Fluid Mixtures, vol. 584, Springer, 2008. [31] M.I. Shliomis, B.L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. Magn. Mater. 252 (2002) 197–202. [32] M.I. Shliomis, B.L. Smorodin, Onset of convection in colloids stratified by gravity, Phys. Rev. E 71 (3) (2005), 036312. [33] J.M. Lopez, F. Marques, M. Avila, The boussinesq approximation in rapidly rotating flows, J. Fluid Mech. 737 (2013) 56–77. [34] J. Hart, Ferromagnetic rotating couette flow: the role of magnetic viscosity, J. Fluid Mech. 453 (2002) 21–38. [35] M. Shliomis, Effective viscosity of magnetic suspensions, Zh. Eksp. Teor. Fiz 61 (2411) (1971) s1971d. [36] J.A. Weideman, S.C. Reddy, A matlab differentiation matrix suite, ACM Trans. Math. Soft. (TOMS) 26 (4) (2000) 465–519. [37] P.K. Kundu, I.M. Cohen, Fluid Mechanics, Elsevier, 2008. [38] B. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40 (04) (1970) 753–767.