The influence of a magnetic field on the mechanical behavior of a fluid interface
R. G. Gontijo, S. Malvar, Y. D. Sobral & F. R. Cunha
Meccanica An International Journal of Theoretical and Applied Mechanics AIMETA ISSN 0025-6455 Meccanica DOI 10.1007/s11012-016-0488-x
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Author's personal copy Meccanica DOI 10.1007/s11012-016-0488-x
The influence of a magnetic field on the mechanical behavior of a fluid interface R. G. Gontijo . S. Malvar . Y. D. Sobral . F. R. Cunha
Received: 17 March 2015 / Accepted: 7 July 2016 Ó Springer Science+Business Media Dordrecht 2016
Abstract This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is
integrated numerically using a fourth order RungeKutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field. Keywords Meniscus shape Free surface Magnetic fluid Non-linear response Asymptotic solution
1 Introduction R. G. Gontijo S. Malvar F. R. Cunha (&) Departamento de Engenharia Mecaˆnica - VORTEX, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, Brası´lia, DF 70910-900, Brazil e-mail:
[email protected] Y. D. Sobral Departamento de Matema´tica, Universidade de Brası´lia, Campus Universita´rio Darcy Ribeiro, Brası´lia, DF 70910-900, Brazil
Recently the study of magnetic fluids has gained increasing importance in the global research landscape due to its many possible application fields. These fluids can be used in a great number of industrial processes, biomedical applications, optimization of heat transfer rates [1–4] and so on. Recent advances in magnetic fluid flows are described in the current literature [5, 6]. Theoretical works and optical
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experimental measurements of ferrofluid free surfaces for tangential and perpendicular magnetic fields developed in recent years have been considered of special importance [7–11]. Some studies have shown the possibility of displacing magnetic fluids inside porous media by the combination of capillary and magnetic pressures [8, 12]. The possibility of using a magnetic fluid for promoting a more effective capillary displacement is promising for applications of oil industries. Oil reservoirs constitute a porous medium where capillary pressure plays an important role on the dynamics of fluid displacement inside small porous containing immiscible fluids. Bragard and Lebon [12] investigated the capillary rise of a non-magnetic fluid in a porous media and developed a scaling law that establishes a relationship between the microstructure of the porous medium and the height that a fluid rises by capillary pressure, here called the free surface equilibrium height. The first experimental evidence on the possibility of pumping a magnetic fluid due to an external applied magnetic field was shown by [13]. In their pioneer work the authors studied the effect of magnetic pumping at high frequency oscillating magnetic fields finding a negative viscosity, which depends on the orientation of the applied field. A first attempt of developing a study for capillary rise of magnetic fluids by the effect of magnetic pressure in a porous medium was done by [14]. These authors examined the equilibrium height of a magnetic fluid column inside a cylindrical capillary in the presence of an external uniform magnetic field. They found that due to the fluid meniscus deformation, the surface pressure drop in the fluid decreases in the longitudinal and transverse direction of the field with respect to the capillary axis. Some works [11, 15–17] have done numerical simulations and measurements of ferrofluid meniscus shape around a vertical cylindrical wire carrying electric current and observed the influence of the viscosity, the contact angle and the surface tension on the shape of the meniscus. Possible instabilities formed on a ferrofluid free surface due to the presence of external applied magnetic fields have been also investigated [7, 18, 19]. Boudouvis et al. [20] were the first to examine the deformation of a free surface of a ferrofluid by numerical simulation using a Galerkin method and carrying out some experimental observations. Their main focus was to investigate the effect of the contact
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angle on the deformation of the ferrofluid’s free surface pool and to observe the deformation of a captive ferrofluid drop at different magnetic field strengths. A related problem using the Young–Laplace equation coupled to Maxwell equations in order to investigate stability of polarized droplets between two faces of charged parallel plates in the presence of a magnetic field or electric field has been examined by Wohlhuter and Barasan [21]. More recently the evolution and interfacial instability of a thin ferrofluid film on a substrate in the presence of a magnetic field has been investigated [22]. The present theoretical work aims to study the behavior of the shape and the equilibrium height of a ferrofluid meniscus in the presence of a vertical magnetic field against the direction of gravity. The meniscus shape is investigated for different combinations of the magnetic parameters: magnetic Bond number, susceptibility and its derivative with respect to the field. We consider both uniform and nonuniform fields as the magnetic field depends on the wavenumbers in the longitudinal direction. In addition, we present an asymptotic expression for the ferrofluid meniscus shape under conditions of small values of the magnetic Bond number and for constant magnetic susceptibility. Magnetic fluid rise with contact angles higher than p=2 by combining the values of the magnetic Bond number and the magnetic susceptibility is also examined. One of the motivations for this work is to investigate possibilities of capillary rise of a magnetic fluid by controlling the identified magnetic physical parameters of the system. The highlights of this article are: (1) to present a mechanical model for describing the shape and predict the equilibrium height of a magnetic fluid, (2) to examine the influence of the physical parameters on the meniscus shape, (3) to show several analytical solutions in different asymptotic regimes, (4) to use these analytical solutions in order to validate a numerical code and (5) to discuss the results from a physical point of view.
2 Formulation of the problem The problem to be examined here considers a magnetic fluid between two parallel flat plates as illustrated in Fig. 1. In the lower side of the interface we have a magnetic fluid, called in this formulation Fluid 1 and
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in the upper side of the interface we have a non magnetic fluid, called Fluid 2. The spacing between the plates is 2b. The magnetic fluid has density q1 , dynamic viscosity g1 and magnetic susceptibility v1 . The non-magnetic fluid has properties q2 , g2 and v2 ¼ 0. A meniscus between fluid 1 and fluid 2 is formed. The surface tension is denoted by c. The shape of the meniscus is described by the curve y f ðxÞ ¼ 0. Here d denotes the vertical distance from the origin of the coordinate system xy to the bottom of the container. In in this problem d is also called the meniscus equilibrium rise. The mean curvature of the function y ¼ f ðxÞ is denoted by Ck and a represents the contact angle between the magnetic liquid and the plate solid walls.
r B ¼ 0;
ð3Þ
r r þ qg ¼ 0:
ð4Þ
Here r denotes the gradient vector operator, H is the applied magnetic field, B represents the magnetic induction vector field, M is the medium magnetization, l0 is the magnetic permeability of the free space, r is the fluid stress tensor, q is the density of the fluid and g denotes the gravity acceleration vector field. 2.2 Constitutive equation for a magnetic fluid
Maxwell’s equations are considered in the magnetostatic limit and Cauchy’s equations in the equilibrium (i.e. no flow). Therefore, we have the following governing equations of the problem:
On the discussions on the constitutive equation for the magnetic stresses there are some controversies in the current literature about what would be the more appropriate constitutive equation to describe the behavior of a magnetic fluid [7, 23–27]. In this work the constitutive equation for describing the stress tensor of a magnetic fluid in the absence of shear stresses is written in a slightly different form in terms of the dyadic HB with B ¼ l0 ðH þ MÞ. We use
r H ¼ 0;
ð1Þ
r ¼ rh þ rm ¼ ph I pm I þ l0 ðHM þ HHÞ:
B ¼ l0 ðM þ HÞ;
ð2Þ
2.1 General governing equations
ð5Þ
y
Fluid 2 α x
Fluid 1
d
b
Fig. 1 A sketch of the problem used in the mathematical formulation
The tensor rh denotes the hydrodynamic contribution represented by ph I for the particular case of an inviscid or static fluid. The second rank tensor rm is the magnetic contribution of the total stresses. Here, ph denotes the hydrodynamic pressure and pm ¼ l0 ðH HÞ=2 is called the magnetic pressure. Now substituting (5) into (4) we obtain the modified Euler equation for a ferrofluid rph þ l0 M rH þ l0 r ðM HÞ þ qg ¼ 0: ð6Þ So, the divergent of Eq. (5) results separately in two different magnetic contributions relevant to the problem of describing an arbitrary ferrofluid. The first one, l0 M rH, represents a magnetic surface force per unit of volume due to field gradients, whereas the second one l0 r ðM HÞ denotes a magnetic force per unit of volume due to the magnetic torque. In a general case both terms may be important for describing the symmetric and non-symmetric contributions of magnetic stresses, respectively. In the present work, however, we use the standard superparamagnetic assumption, i.e. M ¼ vðHÞH, where
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vðHÞ is the fluid susceptibility and H ¼j H j. Under this symmetric condition of the magnetic fluid, Eq. (6) can be simplified since the contribution l0 r ðM HÞ is always null, resulting in 2 H ð7Þ rph ¼ l0 vðHÞr þ qg; 2 In this work we just consider a first order correction of the v field dependence so that a Taylor series around H0 truncated at OðDHÞ can be applied: vðHÞ ¼ vðH0 Þ þ b0 DH þ OðDH 2 Þ;
ð9Þ where A is an integration constant. 2.3 Two immiscible fluids Let’s consider the system shown in Fig. 1 filled with two immiscible fluids where the liquid 1 is a magnetic and fluid 2 is not magnetic. Now, applying Eq. (9) for each side of the interface we obtain 3 l v H2 H H0 H 2 p1h ¼ q1 g x þ 0 0 1 þ l0 b0 þ A1 ; 2 3 2
Hence Eq. (12) becomes p2 p1 ¼ ðq1 q2 Þgð y þ dÞ 3 v H2 H H0 H 2 l0 0 1 þ b 0 : 2 3 2
ð13Þ
2.4 Jump of traction on the interface Now let’s examine the jump of normal traction on the interface between those two fluids given by the so called Young–Laplace equation: h i ðnÞ ðnÞ ^ f ðnÞ ¼ f 1 f 2 ¼ 2Ck cn; ð14Þ where the symbol [ ] denotes the jump of a certain property on the interface, f is the traction, given by f ¼ n^ r, n denotes the normal component of the traction f , Ck represents the mean curvature of the surface and c is the surface tension of the meniscus. The fluids 1 and 2 are considered two immiscible incompressible ferrofluids. The stress tensor in both sides of the interface are written respectively as r1 ¼ p1 I
l0 H12 I þ B1 H 1 ; 2
ð15Þ
ð10Þ
r2 ¼ p2 I
l0 H22 I þ B2 H 2 : 2
ð16Þ
ð11Þ
Now calculating the traction due to fluid 1 and fluid 2 and using the jump of normal stresses on the interface, Eq. (14) results in
for convenience the terms p1h and p2h will be simply called p1 and p2 , respectively. Thus the pressure jump at the interface between the fluids is given by p2 p1 ¼ðq2 q1 Þgx 3 v0 H12 H H0 H 2 l0 þ b0 þ ðA2 A1 Þ: 2 3 2 ð12Þ In order to calculate the term ðA2 A1 Þ the boundary condition at y ¼ d where pressures p2 ¼ p1 ¼ p0 and H1 ¼ 0 is applied. In this case it is considered that the external magnetic field is applied in the meniscus
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ðA2 A1 Þ ¼ ðq1 q2 Þgd:
ð8Þ
where DH ¼ H H0 and b0 ¼ ðdv=dHÞH0 . Here v0 represents vðH0 Þ. Taking a scalar product of Eq. (7) by dx and integrating, we obtain 3 l0 v0 H 2 H H0 H 2 ph ¼ qg x þ þ l0 b0 þ A; 2 3 2
p2h ¼ q2 g x þ A2 ;
region and that it decays far away from the surface, where it is null. Note that g x ¼ gy, where g is the magnitude of the vector g. We find
l0 2 H2 H12 n^ 2 ðB2n H2 B1n H1 Þ;
2Ck cn^ ¼ ðp2 p1 Þn^ þ
ð17Þ
where B1n and B2n denote the normal components of vector B on fluids 1 and 2, respectively. The magnetic boundary conditions given in [1] are B1n ¼ B2n and H1t ¼ H2t , which means that the normal components of B and the tangential component of H are continuous across the interface. Applying the magnetic boundary conditions and after some algebraic manipulations we obtain
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2Ck c ¼ ðp2 p1 Þ þ
l0 2 2 H2n H1n Bn ðH2n H1n Þ: 2 ð18Þ
Since B ¼ l0 ðM þ HÞ and also considering the fluid 1 a magnetic liquid and fluid 2 a non-magnetic fluid (so the magnetization in 2 is null), we obtain H1n ¼
Bn M1n l0
ð19Þ
characteristic length scale. For making magnetic quantities non-dimensional the reference H0 is used. Here H0 represents a typical applied field. Therefore Eq. (24) written in terms of non-dimensional quantities is given by
2 3=2 h ~ 1 þ v0 H ~ 21n Y 00 ¼ 1 þ Y 02 BoðY þ DÞ Bom v0 H !# ~3 H ~ 21 H1 2Bom b0 ; 3 2
and
ð25Þ
H2n ¼
Bn : l0
ð20Þ
Replacing (19) and (20) in (18) we obtain p2 p1 ¼ 2Ck c þ
2 2 l0 2 2 v H þ b20 H1n 2H1n H0 H1n : 2 0 1n ð21Þ
Now substituting Eq. (21) into (13) we have h lv 2 2Ck c ¼ Dqgð y þ dÞ 0 0 H12 þ v0 H1n 2 3 H1 H0 H12 l0 b0 ; 3 2
y00 2ð1 þ y02 Þ3=2
;
Bo ¼
Dqgb c=b
ð26Þ
and
ð22Þ
The average curvature of an arbitrary curve y ¼ f ðxÞ is given by Ck ¼
~ ¼ H=H0 , Bo and where Y ¼ y=b and D ¼ d=b. Also, H Bom are two important physical parameters of the problem called the Bond number and the magnetic Bond number defined respectively as
ð23Þ
where y0 ¼ dy=dx. Using Eq. (23) into (22) finally results l v 00 02 3=2 Dqgð y þ d Þ 2 0 0 H12 þ v0 H1n y ¼ 1þy c 2c l0 b0 H13 H0 H12 : c 3 2 ð24Þ In the present work, Eq. (24) represents a modified Young–Laplace equation with non-constant curvature in the presence of magnetic effects.
Bom ¼
l0 H02 : 2c=b
ð27Þ
The Bond number can be interpreted as the relation between the hydrostatic pressure and the interfacial tension, whereas the magnetic Bond number represents the ratio between magnetic pressure and interfacial tension. The following boundary conditions of the problem are considered Yð1Þ ¼ cot a;
and
Y 0 ð1Þ ¼ cot a
ð28Þ
The solution of the ordinary nonlinear differential ~ 1 ; a; Bo; Bom ; v; equation of second order Y 00 ¼ FðX; H b0 Þ (i.e. Eq. 24) with the boundary conditions (28), gives the shape of the meniscus formed on the interface between two immiscible fluids under action of a perpendicular magnetic field. Here fluid 1 is taken as the magnetic liquid.
3 Theoretical solutions 2.5 Non-dimensional governing equation It is important to write Eq. (24) in terms of nondimensional quantities. For this purpose, we consider a typical length scale of the problem as being the gap between the plates. According to Fig. 1 the size b is the
Analytic formulas are useful for validating numerical results. In this section, we develop and review several possible analytic solutions for a ferrofluid meniscus rising under actions of capillary and magnetic pressure. For all analytic solutions we use the assumption
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Y 02 1 so that the meniscus mean curvature is reduced to Ck ¼
Y 00 : 2
ð29Þ
3.1 Geometric solution for the meniscus shape under constant curvature A specific geometric solution of the meniscus shape is obtained for a constant curvature Ck ¼ cos a=2, according to the sketch illustrated in Fig. 2. When Eq. (29) is integrated with boundary conditions Yð0Þ ¼ 0 and Y 0 ð1Þ ¼ cot a, yields YðXÞ ¼ X ðcot a cos aÞ þ X 2
cos a : 2
ð30Þ
Note that in this case the solution is symmetric with respect to the Y axis and that the boundary condition Y 0 ð1Þ ¼ cot a is automatically satisfied due to the symmetry of the problem. Moreover, the condition Yð0Þ ¼ 0 is only a geometrical restriction associated with the definition of the origin of the problem coordinate system and it does not represent an extra boundary condition. The model is well posed. The first derivative of Eq. (30) can be used to find a geometric constraint in terms of contact angle for the application of analytic solutions based on Eq. (29). LðaÞ ¼ ðX 1Þ cos a þ cot a 1
ð31Þ
Equation (31) and its derivative points out that the minimum value of LðaÞ occurs for a ¼ p=2 in all X. However, at X ¼ 1 that implies LðaÞ ¼ cot a the value
R sin( α)
R
Another analytic solution is obtained by direct integration of (25), in the limit of Y 02 ! 0 and v0 ¼ 0 under the same boundary conditions Yð1Þ ¼ cot a and Y 0 ð1Þ ¼ cot a. We have pffiffiffiffiffiffi pffiffiffiffiffiffi cotðaÞ cosh BoX csch Bo pffiffiffiffiffiffi YðXÞ ¼ D þ Bo ð32Þ Equation (32) gives the meniscus shape under condition of non-magnetic fluid (i.e, v0 ¼ 0) with contact angles satisfying the constraint cotðaÞ 1, that is a p=2. 3.3 Prediction of D for a non-magnetic fluid meniscus with constant curvature Again, consider the sketch of the meniscus shown in Fig. 2. We have that f ¼ Rð1 sin aÞ and cos a ¼ b=R. For a meniscus with a constant mean curvature we assume that Y 00 is constant so that the non-dimensional mean curvature Ck is given simply by Ck ¼
Y 00 1 cos a ¼ : ¼ 2R 2 2
1 sin a D cos a;
α ζ
3.2 Meniscus shape for a non magnetic fluid for cotðaÞ 1 and non-constant curvature
ð33Þ
The necessary and sufficient condition for assuming a constant meniscus curvature occurs when f d, where d ¼ bD is the dimensional equilibrium rise of the meniscus. In non-dimensional terms we can write that
y
b α
x
Fig. 2 A sketch of the free surface’s geometrical parameters
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of this functions is maximum. So, if cot a 1 the constraint (29) is satisfied at any X of the meniscus shape.
ð34Þ
Now, the non-dimensional governing Eq. (25) under condition of constant curvature condition is given by
2 ~ 1 þ v0 H ~ 21n 2Ck ¼ BoðY þ DÞ Bom v0 H ! ~3 H ~ 21 H1 2Bom b0 ; ð35Þ 3 2 Evaluating Eq. (35) at the origin for a non-magnetic fluid (v0 ¼ 0) we find that the capillary rise Dc for the condition of constant curvature is given by
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cos a : Bo
ð36Þ
The sign of the capillary rise in Eq. (36) is determined by the value of the contact angle a When a\p=2 the meniscus rises, when a [ p=2 the meniscus submerges, and when a ¼ p=2 the meniscus remains flat at the level of the free surface outside the plates. Substituting Eq. (36) into (34) yields Bo 1 þ sin a;
ð37Þ
2 Zero level
Bom
Dc ¼
Zero level
1
that is the condition for the constant mean curvature approach to be applied. 0
3.4 Prediction of D for a non-magnetic fluid meniscus for cotðaÞ 1 and non-constant curvature Equation (32) provides the shape of a non-magnetic fluid meniscus in the limiting case of cotðaÞ 1. As seen by Eq. (31), this is the necessary condition for the assumption that Y 02 1. Using the geometrical condition that Yð0Þ ¼ 0, we obtain an analytic expression for the equilibrium displacement D0 , given by " pffiffiffiffiffiffi# pffiffiffiffiffiffi csch Bo D0 ¼ BoDc ð38Þ sinðaÞ 3.4.1 Prediction of D for a ferrofluid for cotðaÞ 1 and constant curvature Now, applying Eq. (35) evaluated at the origin for the case of a ferrofluid undergoing a vertical magnetic ey , yields field H~1 ðX; YÞ ¼ ð1 þ Y=DÞGðXÞ^ Bom 1 v0 ð1 þ v0 Þ b0 : D ¼ Dc þ ð39Þ 3 B0 Equation (39) states that a magnetic fluid in the small gap between two parallel plates can rise against gravity by magnetic pressure even if the effect produced by capillary pressure Dc is null, that occurs for a ¼ p=2 and the condition for Bo 1. In this case, there is a new contribution from the magnetic action corresponding to the second term on the right hand side of Eq. (39) that can produce the fluid meniscus elevation inside a small porous. This behavior can be better seen in Fig. 3.
0
χ0
1
2
Fig. 3 Combination of Bom and v0 in which a magnetic fluid in the small gap between two parallel plates can rise against gravity for contact angles higher than p=2. This curve was plotted using the constant curvature theory, Eq. (39). The inserts in this figure show two possible configuration of the meniscus. In this case we consider b ¼ 1=10 and Bo ¼ 1=10
It is possible to observe that even though different combinations of Bom and v0 may produce D [ 0 and D\0, the shape of the free surface practically does not change for different pairs of these parameters. The shapes illustrated in the details of Fig. 3 were obtained with numerical simulations using the methodology presented in the next section. In the next sections we will show that the application of a variable magnetic field like sin (kX), where k is a wavenumber, may drastically change the meniscus shape of a magnetic fluid. 3.5 An asymptotic solution for a ferrofluid meniscus shape: Bom 1, v0 1, cotðaÞ 1 and non-constant curvature Consider a ferrofluid meniscus where Bom 1, but not necessary null and cotðaÞ 1, so Y 02 is small as required by Eq. (31). In this case the governing equation of the problem is given by
~ 21 þ v0 H ~ 21n Y 00 ¼ BoðY þ DÞ Bom v0 H ! ~3 H ~ 21 H1 2Bom b0 ; ð40Þ 3 2 for a magnetic field like
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H1 ðX; YÞ ¼ ð1 þ Y=DÞe^y ;
ð41Þ
we have that 2Y Y 2 þ 2 Y 00 ¼ BoðY þ DÞ Bom v0 ð1 þ v0 Þ 1 þ D D 2 3 b 3Y 2Y þ 0 1 þ 2 þ 3 : 3 D D ð42Þ Equation (42) does not have an analytic solution, but it is possible to seek an asymptotic solution for small values of the magnetic Bond number. For this purpose consider a regular asymptotic expansion, [28–30], given by YðXÞ ¼ Y0 ðXÞ þ Bom Y1 ðXÞ þ Bo2m Y2 ðXÞ þ :::
ð43Þ
and D ¼ D0 þ Bom D1 þ Bo2m D2 þ :::;
ð44Þ
where Y0 ðXÞ is the leading order term for the nonperturbed equation, Y1 ðXÞ represents the first correction of the magnetic effect, Y2 ðXÞ is the second correction, and so on. Thus, we have D0 as the equilibrium rise for the case of a non magnetic fluid, D1 is the first correction of magnetic effects on the equilibrium rise D, and so on. Applying the regular asymptotic expansion, Eq. (43) at the governing Eq. (42), we find that Y000 ¼ BoðY0 þ D0 Þ; Y1
00
ð45Þ
3Y000 D1 3Y0 D1 þ ¼ Bo Y1 þ 4D1 þ F; D0 D0 ð46Þ
3 Y 00 D2 Y000 D2 þ 0 1 þ Y100 D1 þ Y2 00 D0 D0 D2 ¼ Bo Y2 þ 4D2 þ 6 1 D0 3 Y0 D2 þ Y0 D21 Y1 D1 þ D0 v ð1 þ v 0 Þ Y0 D1 Y 2 D1 2Y0 Y1 0 3D1 þ þ 2Y1 þ 0 2 þ D0 D0 D0 D0 b0 3Y02 D1 6Y0 Y1 Y02 Y1 3D1 þ þ þ6 2 ; 3D0 D0 D20 D0
ð47Þ
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where F is given by 2Y0 Y02 þ F ¼ v0 ð 1 þ v0 Þ 1 þ D0 D20 b0 3Y02 2Y03 þ 1 þ 2 þ 3 ; 3 D0 D0
ð48Þ
Equation (45) represents the leading order solution of the non magnetic problem and was already presented in Eq. (32). The equation that governs the behavior of the OðBom Þ correction is given by (46), while Eq. (47) is the governing equation for the OðBo2m Þ correction. The boundary conditions used for these asymptotic 0 0 solutions are Y0 ð1Þ ¼ cotðaÞ, Y0 ð1Þ ¼ cotðaÞ, 0 0 0 0 Y1 ð1Þ ¼ Y1 ð1Þ ¼ 0 and Y2 ð1Þ ¼ Y2 ð1Þ ¼ 0. The leading order solution is given by pffiffiffiffiffiffi pffiffiffiffiffiffi cotðaÞ cosh BoX csch Bo pffiffiffiffiffiffi Y0 ðXÞ ¼ D0 : Bo ð49Þ Details of the analytic solution of differential equation (46) are presented in the appendix of this article. In contrast the correction OðBo2 Þ, that was solved using the software Mathematica presents a too tedious calculation for the details to be given in this work. So, we have omitted this calculations by convenience. The comparison of the asymptotic solutions with numerical results will be presented in the next section. The values of D0 , D1 and D2 are obtained by applying the geometrical conditions given respectively by: Y0 ð0Þ ¼ 0, Y1 ð0Þ ¼ 0 and Y2 ð0Þ ¼ 0. As pointed out before, this procedure is tedious and generates huge expressions for D1 and D2 . All the manipulations done related to these asymptotic theories were made by using the Wolfram Mathematica 7 for Linux.
4 Numerical solution In order to validate our theories and to extend the solution of the problem to more complicated cases where we don’t have a theoretical solution, we developed a computational code written in FORTRAN called MENIS-2D. This code computes the equilibrium height D of a ferrofluid and the free surface shape ~ 1 ; a; Bo; Bom ; v; b0 Þ of a two-dimensional Y ¼ FðX; H ferrofluid meniscus between two vertical plates as the
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sketch shown in Fig. 1. The system of 2 ordinary differential equations dYk ~ 1 ; a; Bo; Bom ; v; b0 Þ; ¼ Gk ðX; H dt
where
k ¼ 1; 2;
ð50Þ resulting from the second order nonlinear differential equation (24) and boundary conditions (28) was integrated numerically using a fourth-order RungeKutta scheme. The method requires four Gk evaluations at each step of the numerical integration. The problem was transformed in an Initial Valor Problem (IVP) using an iterative scheme and a Newton– Rhapson method to accelerate the convergence of the solution [31]. The solution was advanced according to the following algorithm 2 1: Set Bo ; a; Bom ; v0 ; b0 ; and DX; 6 2: Set boundary conditions; 6 6 6 3: Set initial value for D; 6 6 4: Compute G ðX; H ~ 1 ; a; Bo; Bom ; v0 ; b0 Þ; 6 k 6 6 dU dY 6 5: ! G and ! U; 6 dX dX 6 0 6 6: Fc ðDÞ ! Y ð1Þ cotðaÞ; 6 6 6 7: Jc ½Yð0Þ ! Yð0Þ 0; 6 6 8: IfjFc ðDÞj\tol then ! END 6 6 6 9: IfjJc ½Yð0Þ j\tol then ! END 6 6 10: Fc ðD þ DDÞ ! Y 0 ð1Þ 1; 6 6 6 11: Jc ½Yð0Þ þ DYð0Þ ! Yð0Þ 0; 6 6 FcðDÞ 6 12: D ! D DD 6 FcðD þ DDÞ Fc ðDÞ 6 6 6 Jc½Yð0Þ 6 13: Yð0Þ ! Yð0Þ DYð0Þ 4 Jc½Yð0Þ þ DYð0Þ Jc ½Yð0Þ 14: Back to step 4: ð51Þ The algorithm (51) describes graphically the numerical integration scheme. The non-dimensional space step DX was sufficiently small to ensure an error in the numerical integration less than 103 . A typical value of the space step was b 104 . An initial guess for the equilibrium height D is required in order to initialize the iterative scheme of the transformed IVP. The maximum number of iterations needed for each space step integration is around 20. The integration starts at X ¼ 1 with boundary condition Y 0 ð1Þ ¼ cotðaÞ and with arbitrary
values of D and Yð1Þ. After each integration step the boundary condition Y 0 ð1Þ ¼ cotðaÞ and the geometrical restriction Yð0Þ ¼ 0 are verified. If these conditions are not satisfied, a Newton–Rhapson scheme is used in order to updated a new rise D that must satisfy Y 0 ð1Þ cotðaÞ ¼ 0
ð52Þ
and a new value of Yð1Þ that must satisfy Yð0Þ ¼ 0:
ð53Þ
The procedure is repeated until both conditions given by Eqs. (52) and (53) are satisfied within a tolerance of 106 . In this way the initial guess of the equilibrium rise D and the position Yð1Þ are carefully controlled in order to ensure the convergence of the method. The external magnetic field in the direction of gravity was considered as being a function of x and y; Hðx; yÞ ¼ h1 ðxÞh2 ðyÞ. For instance, an uniform field on the gravity direction is imposed with h1 ðxÞ ¼ 1 and h2 ðyÞ ¼ H0 e^y . Here H0 denotes a constant value of a reference applied field. In addition, an uniform field gradient is imposed with h1 ðxÞ ¼ 1 and h2 ðyÞ ¼ Hoð1 þ y=dÞe^y . In a more general form we h2 ðyÞ ¼ Hoð1þ propose h1 ðxÞ ¼ ½1 þ e sinðkxÞ y=dÞ^ ey , that depends on the wavenumbers k ¼ 2pn=b where n = 0, 1, 2,.... This proposition is done in order to test the response of this nonlinear system when we apply an harmonic periodic input. 4.1 Validation and preliminary results One test of the full numerical integration scheme was made in order to reproduce the exact solution obtained under conditions of non-constant curvature, small Bom and arbitrary values of the parameter b0 , as shown in Fig. 4. We can see that within the range of Bom in which the asymptotic solution is valid ðBom 1Þ, the numerical solution presents a very good agreement with the exact solution. The agreement between the OðBom Þ solution and the numerical values is very good with Bom ranging in the interval [0,0.15]. After this point, the OðBo2m Þ solution starts to agree much better than solution OðBom Þ correction. The agreement between the numerical and OðBo2m Þ solutions works very well for Bom up to 0.3. Another important test for the numerical solution and also to check the theoretical solution for constant
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Fig. 4 Equilibrium height versus the magnetic Bond number. The black circles represent numerical results, the solid line denotes the exact solution for small values of the magnetic Bond number given by the OðBom Þ asymptotic theory, while the dashed line denotes the OðBo2m Þ solution. The insert in this figure shows the detail of the asymptotic for smaller values of the magnetic Bond number. For this plot: Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10 and a ¼ p=2 1=10
curvature, Eq. (39), is a comparison of the equilibrium rise for several values of the magnetic Bond number under condition of constant curvatures. This result is shown in Fig. 5. It is possible to observe a great agreement between the numerical solution and the theory of constant curvature, specially for higher magnetic Bond number values. Under the conditions of the result shown in Fig. 5, for small values of Bom , capillary pressure is more important than magnetic pressure against hydrostatic pressure. So, as the capillary pressure is proportional to the mean curvature a model with a constant curvature will show some discrepancies at small Bom . On the other hand for moderate value of Bom 0:5 magnetic pressure dominates capillary pressure against gravity and variations of curvatures becomes unimportant. Figure 5 also suggests that for Bom 0:5, the magnetic effect on the shape makes curvature variations unimportant on the equilibrium rise. In this range of parameters the effect of a magnetic field tends to jump the meniscus shape from a non-constant curvature curve to a condition of constant curvature.
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Fig. 5 Equilibrium height as a function of the magnetic Bond number. The black circles represent numerical values, the solid line denotes the theoretical solution for the constant curvature condition. For this plot: Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10 and a ¼ p=2 1=10
5 Results and discussions In the previous section of this work, some of our theories were compared with the numerical results in order to test them. In this section we present numerical results that cannot be explored or examined using the analytic solutions developed here. Such as the influence of the wavenumber k and the parameter e used to impose an harmonic field, on the shape of the meniscus and on its equilibrium rise D. 5.1 Meniscus shape Figure 6 shows four configurations of the meniscus shape for different values of the harmonic field wavenumber. The parameter n is related to the wavenumber k ¼ 2pn=b. It is important to notice that the variations of n allow us to examine different length scales of the problem. The wavenumber may also be related to the spatial frequency, given by n ¼ k=2p. In other words, a characterization of a structure’s periodicity across any position in space. Figure 6 shows that increasing the wavenumber results in different meniscus shapes of the magnetic meniscus free surface, including asymmetrical ones. It
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Fig. 6 Meniscus shape for different values of n. a Stands for n ¼ 1, b for n ¼ 2, c for n ¼ 3 and d for n ¼ 4. For this plot: Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, a ¼ p=2 1=10, e ¼ 1 and magnetic Bond numbers varying from 0 to 1.5
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Fig. 7 Meniscus shape for n ¼ 100. The solid line represents Bom ¼ 0 while the dashed line considers Bom ¼ 3=2. For this plot: Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, a ¼ p=2 b0 , e ¼ 1
also indicates that the asymmetric degree between the left and right side of the curve, that could be defined by D ¼ Yð1Þ Yð1Þ, decreases with the increase of the wavenumber. We argue that in the limiting case of k ! 1 the shape tends to the case of a magnetic problem with a non-harmonic field, as indicated by Fig. 7. In the limiting case of k ! 1 the corresponding wavelength goes to zero, meaning that the harmonic contribution of the applied field tends to a constant behavior equivalent to the imposition of a non-harmonic magnetic field in the vertical direction. In the present work a space or wavelength scale is considered. As the wavenumber increases, the shape of the meniscus does not respond to the forcing for smaller length scales. So, we shall find more symmetrical and weaker nonlinear shapes. Fig. 8 presents the shape of the meniscus for three different magnetic Bond numbers. It is seen an increase of the
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Fig. 8 Meniscus shape for three different cases. The inserts in this figure show the Fourier transform of the meniscus shape represented by the amplitude as a function of the wavenumber. The solid line represents Bom ¼ 1=10, the dashed one features
Bom ¼ 1=4 and the dotted one represents Bom ¼ 1=2. a The FFT for Bom ¼ 1=10, b Bom ¼ 1=4 and c Bom ¼ 1=2. For this plot: e ¼ 1=2, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 þ 18, k=p ¼ 2
asymmetry degree of the meniscus shape as the magnetic Bond number is increased. In this plot we also show the shape Fourier transform corresponding to the three different magnetic Bond numbers examined. The results suggest that an increase of the magnetic Bond number also reflects in a more effective spectrum spreading. Actually, for the larger magnetic Bond number the total energy is a more distributed quantity at the different length scales of the spectrum. For instance, the result for Bo m = 0.5 indicates a small loss of energy in the peak represented by the dotted line in the insert (c). Another interesting finding in this problem are the bifurcations of the meniscus shape in the presence of magnetic field. Figure 9 shows the shape of the meniscus and the equivalent FFT plots in the inserts of the graphics for the case of a non-magnetic fluid ðBom ¼ 0Þ and for a small value of the magnetic bond number ðBom ¼ 0:2Þ. In this figure A represents the amplitude.
Figure 9 shows that for higher values of the harmonic magnetic field, the nonlinearity is strongly influenced by just small variations of the magnetic parameters of the system. This strong dependence on small perturbations of a physical parameter is a typical characteristic of problems governed by nonlinear differential equations. It is also interesting to notice that other shape configurations start to appear when magnetic pressure tends to dominate capillary pressure on the liquid-interface, indicating a more nonlinear and complex behavior of the meniscus shape. We note that the excitation wavenumber is present as the first harmonic. It means that even being very nonlinear, the meniscus still has a frequency signature. In both cases, most of the energy is concentrated in the first harmonic, in which k=p ¼ 2. We noted that when there is an applied magnetic field, the second, third and fourth harmonics are concentrated in the same wavenumber: k=p ¼ 4, k=p ¼ 6 and k=p ¼ 8, respectively. Nevertheless, when the magnetic field is
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Fig. 9 Meniscus shape for a non magnetic case Bom ¼ 0 (a) and Bom ¼ 1=5 (b). The inserts in (a) and (b) giving the amplitute as a function of the wavenumber represent the Fourier transform of
the meniscus shape for both cases studied. For this plot: e ¼ 3, 1 Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, a ¼ p=2 10 , k=p ¼ 2
applied, those harmonic steal energy from the first one, given a more non-linear response to the meniscus. In addition, many other harmonic can be seen at k=p ¼ 10 and k=p ¼ 12, for instance. The Fast Fourier Transform (FFT) is a very interesting tool but has the disadvantage that it has only wavenumber resolution and no space resolution. The wavelet transform or wavelet analysis is probably the most recent solution to overcome the shortcomings of the Fourier transform. Since the introduction of wavelets as a signal processing tool in the late 1980s, considerable attention has focused on this application [32, 33]. For instance, one can apply a rectangular (so-called Haar-basis) window function [34], to zoom-in on the singularity of a signal. On Fig. 10 it is possible to see two different combination of parameters that lead to nonlinear responses. In this figure, the wavelet transforms using Haar-basis are shown with the FFT transforms. The FFT does not show many relevant information in addition to the energy loss in the spectral signature harmonic. Note that this signature is in different positions, as shown by the dotted lines, due the excitation’s wavenumber. Haar basis best represents functions that consist of sharp peaks and discontinuities. In the upper case (a), the clearer parts of the wavelet transform correspond to bigger coefficients. In this case, The coefficients become larger in the vicinity of a singularity and in this case, a null derivative. The larger scales represent low
frequencies or large wavelengths. In this case, a low wavenumber. Once the system is excited with a low wavenumber, the coefficients are higher in larger scales and not present the the smaller ones. As the scale decreases, there is less energy on the harmonics, since the wavenumbers are bigger, which can be confirmed through the FFT. Furthermore, in this case, it is possible to determine that the right side of the meniscus must have a boundary condition of Y ¼ 0. This can be observed because the signals tend to have high coefficient values at the edges, due to the high derivatives and boundary conditions. In this case, this does not occur, demonstrating the proximity to the null value or to what could be considered a resting value for the in a situation without capillary forces. The comparison between the wavelet transform and the physical shape of the meniscus and wavenumber spectrum response can be seen with Fig. 11. In the case bellow (b), the excitation has a larger k, which can be observed by the coefficients on the wavelet transform. It is noted that the coefficients are large even in smaller scales when compared with the case (a). In addition, the singularities appear almost together and can be seen as a inflection. In this case, it is possible to determine that the shape of the meniscus approaches a paraboloid. Figure 12 shows the amplitude of the first harmonic in the wavenumber spectrum as a function of the Bom . it is possible to see that, the increase of the magnetic field, transfers energy into the system, which now has
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1 Fig. 10 Meniscus Haar wavelet transform. a k=p ¼ 2, BoM ¼ 7=5, v0 ¼ 1=10, b0 ¼ 0, e ¼ 1:0 and a ¼ p2 10 , b k=p ¼ 4, BoM ¼ 1, p 1 v0 ¼ 1=10, b0 ¼ 1=5, e ¼ 7=10 and a ¼ 2 10
harmonics with higher amplitudes. In this respect, the signal carries more power. The way this amplitude increases as a function of the Magnetic Bond can be described by a second-degree polynomial. To sum up, it is possible to see in Fig. 13 many different patterns formed by the meniscus shape. In general, the modification of the system’s control parameters, such as magnetic Bond number, oscillation amplitude and wavenumber, can modify the anisotropy in the meniscus’ shape. Furthermore, high magnetic fields at low values of oscillation can generate highly non-linear results. The ability to control the shape of the meniscus opens doors for the study of traction jump and even modifications on the capillary forces. The applications are diverse both in oil and bioengineering industry.
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5.2 Equilibrium rise of the meniscus In this section we examine the dependence of the equilibrium rise of the meniscus on the contact angle a. A comparison between Eq. 39 and numerical results is presented in the plot of Fig. 14. We can see that when the a ! p=2 both solutions provide the same result. As the values of a start to differ significantly from p=2, because the curvature can not be defined by the second derivative of the shape and it can not be assumed constant. For a ¼ 0, the error is about 15%. The plot in Fig. 14 is also pointing out the range of application of the theory given by Eq. (39). It should be said that that all analytical solutions consider the restriction that Y 02 1, that requires a p=2 1. In addition Fig. 14 shows the importance of the
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Fig. 11 Source of meniscus shape used to compute the wavelet transform. a Meniscus shape for k=p ¼ 2, BoM ¼ 7=5, 1 v0 ¼ 1=10, b0 ¼ 0, e ¼ 1:0 and a ¼ p2 10 . b Correspondent
FFT. c k=p ¼ 4, BoM ¼ 1, v0 ¼ 1=10, b0 ¼ 1=5, e ¼ 7=10 and 1 . d Correspondent FFT a ¼ p2 10
numerical solution for exploring other regimes within the full range of contact angles. An interesting result is the behavior of D when varying the amplitude of the harmonic contribution of the field. This result is particularly important because there is no theory to predict how the nonlinearities on the meniscus shape may change the equilibrium rise in which a magnetic interface can be displaced in a capillary under the influence of a magnetic pressure. Figure 15 shows the plot of D as a function of e. We can see that D is a rapid varying function of the amplitude e of the magnetic field. It is observed that the variation of D with e is Oðe3 Þ as showed by the best fit. This indicates that displacements of magnetic liquid interfaces against gravity may have much
higher equilibrium rise by monitoring the amplitude of an external magnetic field, in contrast with the displacement produced by capillary pressure. From this results we can see that while D is proportional to Bom and v20 it goes like Oðe3 Þ. In general, the results indicates that as the nonlinearities related to magnetic effect on the polar fluid free surface increase, the equilibrium rise of the magnetic meniscus may increase even in the absence of capillary pressure. 5.3 A brief discussion on the boundary conditions of the problem The formulation of the problem requires some assumptions related to the boundary condition used
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Fig. 12 Comparison between the amplitude of the first harmonic in the wavenumber spectrum and the Magnetic Bond number. The dotted curve can be approximated as second order polynomial given by A ¼ 408:38 þ 1288:59Bom þ 199:65Bo2m . For this plot: e ¼ 1, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 18, k=p ¼ 1
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Fig. 14 Equilibrium rise D as a function of the contact angle a. The solid line represents the theory with constant curvature and magnetic effects, Eq. (39), while the dashed line is the numerical solution. For this plot: e ¼ 0, Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, Bom ¼ 1=10
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Fig. 13 Comparison between the meniscus patterns formed by different controlling parameters. a e ¼ 1, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 18, k=p ¼ 2, Bom ¼ 3=2. b e ¼ 1, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 18, k=p ¼ 1, Bom ¼ 3=2. c e ¼ 3, v0 ¼ 1=10, b0 ¼ 0,
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a ¼ p=2 18, k=p ¼ 1, Bom ¼ 13=10. d e ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, a ¼ p=2 15, k=p ¼ 1, Bom ¼ 1=10. e e ¼ 1, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 þ 14, k=p ¼ 1, Bom ¼ 1=2. f e ¼ 3, v0 ¼ 1=10, b0 ¼ 0, a ¼ p=2 16, k=p ¼ 6, Bom ¼ 2
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in order to provide the shape of the meniscus. We must choose a physically consistent boundary condition in order to properly model the problem. In this section we show two options and discuss the differences between them. We define here the following options:
0
B.C.1: Y ð0Þ ¼ 0 and 0 X 1
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Y ð1Þ ¼ cotðaÞ; ð54Þ
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ε Fig. 15 Equilibrium rise D as a function of e. The insert in this figure shows two different meniscus shapes. In the insert the solid line represents e ¼ 0 while the dashed line denotes e ¼ 5=4. For this plot: n ¼ 1, Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, Bom ¼ 1:0. The solid line is well fitted by the relation: D ¼ c0 þ c1 e þ c2 e2 þ c3 e3 , where c0 ¼ 593=1000; c1 ¼ 3=1000; c2 ¼ 199=500 and c3 ¼ 3=1000
0
and Y ð1Þ ¼ cotðaÞ; ð55Þ
Boundary condition 1 requires the imposition of vertical symmetry in order to provide the full shape of the meniscus. Figure 16 shows the difference between the meniscus shape considering boundary conditions 1 and 2 for several magnetic Bond numbers. We can see from Fig. 16 that the assumption of vertical symmetry leads to an unrealistic behavior of the free surface. Indeed we do not know the derivatives values in the center of the meniscus. The only real boundary conditions that we know for sure from the physics of the problem are the values of the derivatives in the walls, since they depend exclusively on the contact angle, that is a property from the fluid and the wall material. It is interesting to notice in the insert of Fig. 16b that as we increase the magnetic Bond number the derivatives in the center of the meniscus increase in magnitude. For the non magnetic case, Bom ¼ 0 both boundary conditions provide the same result, since we obtain a symmetrical profile. We argue that the deviation of the symmetry condition in the magnetic free surface shape is a measurement of the intensity of the magnetic effects.
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Fig. 16 Meniscus shape considering symmetrical boundary conditions (a) and non-symmetrical boundary conditions (b). For this plot: n ¼ 1, Bo ¼ 3=10, v0 ¼ 1=10, b0 ¼ 1=10, e ¼ 1:0 and a ¼ p2 1=10
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6 Conclusions In this work we have presented a theoretical and numerical analysis on the behavior of two important variables regarding the behavior of a magnetic fluid free surface. The shape of the surface and the equilibrium rise were explored. The numerical code developed to solve the nonlinear modified Young– Laplace differential equation has been validated against several theories, including non-magnetic cases for constant and non-constant curvature, magnetic problem for constant curvature and an asymptotic theory valid of small magnetic effects and nonconstant curvature. We have shown that even for the condition of an interface with a small curvature it would be possible to make a displacement of a fluid interface by a combination of a low capillary pressure and a magnetic pressure. In this case the pressure jump on the interface is not null in the presence of a magnetic field gradient and, consequently the normal components of the magnetic field is not continuous on the boundaries and fluid interface. The idea behind it would be to promote the displacement of a fluid interface in capillaries of diameter not necessarily too small (e.g. porous media with higher permeability) by magnetic pressure since the effect of capillary pressure can be very small in this case. Actually, we have proposed as a real application the possibility of displacing magnetic fluids inside porous media by a combination of capillary pressure and magnetic pressure. The possibility of using a magnetic fluid for promoting a more effective displacement is promising for applications of oil reservoirs. Additionally, it was shown that variations of the magnetic liquid susceptibility with respect to the applied field decreases the effect of magnetic pressure on the magnetic interface for contact angle higher than p=2. Another interesting finding of this work regards the nonlinearities on the free surface shape with respect to the wavelength and the intensity of the applied field. It was also shown that for the case of e 6¼ 0, which cannot be solve by and analytic expression, a small variation of the magnetic Bond number may produce a significant change in the shape of the free surface, leading to additional shape configurations. Finally, we found out that the increase of the non-dimensional amplitude e leads to a
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nonlinear behavior of the equilibrium rise of the magnetic liquid. This magnetic effect has been described by a Oðe3 Þ shape dependence. Acknowledgments The work was supported in part by the Brazilian funding agencies CNPq- Ministry of Science, Technology and Innovation of Brazil, and by the CAPES Foundation—Ministry of Education of Brazil.
Appendix In this appendix we present more details on the full expression of the asymptotic solution OðBom Þ. pffiffiffiffiffiffi csch 4 Bo n pffiffiffiffiffiffi 12 BoD0 3Bo2 D20 D1 96Bo5=2 D30 pffiffiffiffiffiffi þ 2ðn b0 Þ cot2 ðaÞ 24 BoD0 2Bo2 D20 D1
pffiffiffiffiffiffi þ n b0 Þ cot2 ðaÞ cosh 2 Bo
pffiffiffiffiffiffi þ 12Bo5=2 D30 D1 cosh 4 Bo þ cot2 ðaÞ h pffiffiffiffiffiffi 8ðn þ b0 Þ BoD0 coshða4 Þ pffiffiffiffiffiffi þ 4 Bo 2nD0 b0 ½2D0 þ 3 cotðaÞð1 þ XÞ coshða3 Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi þ 4 BoD0 coshð2a3 Þ n b0 8n BoD0 cosh 2 BoX
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi þ 8b0 BoD0 cosh 2 BoX þ 8 BoD0 coshða1 Þ n b0 pffiffiffiffiffiffi pffiffiffiffiffiffi 12b0 Bo cotðaÞ coshða1 Þ½1 X þ 4n BoD0 coshð2a1 Þ pffiffiffiffiffiffi pffiffiffiffiffiffi 4b0 BoD0 coshð2a1 Þ 8 BoD0 coshða2 Þ n b0 hpffiffiffiffiffiffi i þ b0 cotðaÞ sinh Boð1 3XÞ þ 3b0 cotðaÞ½sinhða4 Þ
Y1 ðXÞ ¼
sinhða2 Þ 12b0 cotðaÞ½sinhða1 Þ sinhða3 Þ þ b0 cotðaÞ sinhða5 Þ
with pffiffiffiffiffiffi pffiffiffiffiffiffi Boð1 þ XÞ a2 ¼ Boð3 þ XÞ pffiffiffiffiffiffi a3 ¼ Boð1 þ XÞ pffiffiffiffiffiffi pffiffiffiffiffiffi a4 ¼ Boð3 þ XÞ a5 ¼ Boð1 þ 3XÞ
a1 ¼
n ¼ v0 ð1 þ v0 Þ
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