Material transport of a magnetizable fluid by surface perturbation

Journal of Magnetism and Magnetic Materials 395 (2015) 67–72

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Material transport of a magnetizable fluid by surface perturbation V. Böhm a, V.A. Naletova b, J. Popp a, I. Zeidis a,n, K. Zimmermann a a b

Faculty of Mechanical Engineering, Ilmenau University of Technology, Ilmenau D-98693, Germany Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Vorobyovy gory, 119899 Moscow, Russia

art ic l e i nf o

a b s t r a c t

Article history: Received 26 October 2014 Received in revised form 2 July 2015 Accepted 12 July 2015 Available online 14 July 2015

Within the research for apedal, contour variable locomotion systems, the influence of an alternating magnetic field on the shape of the free surface of a magnetizable fluid (magnetic fluid) is studied. In the framework of the Stokes approximation, for the case where the amplitude of the alternating component of the applied magnetic field is much less than the magnitude of the permanent component, it is shown analytically that a periodical traveling applied magnetic field can generate a transport of the fluid in a prescribed direction. Numerical computations are performed to calculate and analyze the flow rate of the fluid as a function of the parameters of the field and the fluid. This effect can be used in fluid transporting engineering mini- and microsystems. & 2015 Elsevier B.V. All rights reserved.

Keywords: Magnetic fluid Ferrofluid Traveling magnetic field Perturbation Surface deformation Locomotion Peristaltic

1. Introduction Deformation of the free surface of a magnetic fluid can be used for creating flows with nonzero flow rates both in the fluid itself and in the ambient medium. Various peristaltic pumps based on magnetic fluids have been already used in medicine for pumping biological fluids, in particular, blood, since such pumps preserve the structure of the fluids to be pumped. A number of new designs for the peristaltic pumps based on magnetic fluids are proposed by [1–3]. In addition, the deformation of the free surface of a magnetic fluid subjected to a time variable inhomogeneous magnetic field can be used for creating propulsion devices, stepper motors [4–6] and various kinds of valves, flow-rate meters and breakers. The peristaltic flow of a viscous, incompressible fluid layer on a horizontal substrate induced by a sinusoidal deformation wave traveling along the fluid surface was studied by [7]. The perturbation of the surface was defined kinematically and, therefore, the nature of forces causing perturbations like these was not discussed. For the case of a magnetic fluid, these perturbations can be realized by magnetic forces in an inhomogeneous magnetic field. These forces can create a traveling surface deformation wave in the fluid as was described previously. In a homogeneous vertical magnetic field, the horizontal surface of a magnetostatic fluid layer on a horizontal substrate can n

Corresponding author. E-mail address: [email protected] (I. Zeidis).

http://dx.doi.org/10.1016/j.jmmm.2015.07.036 0304-8853/& 2015 Elsevier B.V. All rights reserved.

become unstable at a certain critical value of the field [8]. Steadystate spikes can appear on the surface without the simultaneous occurrence of a flow inside the fluid. An inhomogeneous magnetostatic field also cannot create a flow of a magnetic fluid with a flow rate. However, a traveling inhomogeneous magnetic field induces a traveling surface wave, which affects a flow with a nonzero flow rate inside the magnetic fluid. This phenomenon was observed in experiments [9–11], which created traveling waves on the surface of a magnetic fluid with a temporally and spatially varying magnetic field. Either a flow of a nonzero flow rate [9] was observed in these experiments or a sloping surface of the magnetic fluid appeared [10,11]. Analytical studies of the flow in a magnetic fluid layer subjected to a traveling magnetic field were performed by [12,9] using the perfect fluid model. Viscosity was taken into account by [12], who investigated analytically the flow of an infinitely deep layer of a magnetic fluid in such a field. Thin layered flows of heavy viscous magnetic fluids on horizontal or cylindrical substrates in traveling magnetic fields were investigated by [13,14]. The surface tension was taken into account. In these studies, closed-form expressions were obtained for the magnetic field that created a prescribed sinusoidal traveling wave on the surface of a thin layered magnetic fluid. The thickness of the layer was assumed small in comparison with the wavelength. The problem of determining the magnetic field for a prescribed flow can be regarded as the inverse problem. The direct problem, in which the flow of a thin layered heavy viscous

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magnetizable fluid on a horizontal substrate is determined for a given traveling periodic magnetic field, was solved analytically by [15]. In this study, analytical expressions for the average flow rate in the finite-thickness layer were obtained. In the present paper, the problem of determining the 2D flow of a finite-thickness layer of a heavy viscous magnetic fluid in a traveling periodic magnetic field is solved analytically for the case where the amplitude of the magnetic field oscillations is small. The surface tension is taken into account. The problem is solved in the Stokes approximation under the assumption that the magnetic permeability of the fluid is constant. Analytical expressions are obtained for the velocity, pressure, and average flow rate of the fluid.

2. Statement of the problem

=

H0*2 (z*)

+

A2 (z*)

sin (ζ *),

(1)

i, j = x, z,

components of the stress tensor, F x* = 0, F z* = − g is the mass density of the gravity force acting on the fluid, ∇j stands for differentiation with respect to the corresponding coordinate, the summation with respect to doubly repeated indices is assumed. The stress tensor pij* can be represented by

pij* = − p*δij + τij* + pijH ,

(2)

Here, H0*2 is the static component of Ha*2, A2 is the amplitude of the superimposed periodic perturbation, kn is the wavenumber, ω* is the angular frequency, tn is time, xn and zn are the horizontal and vertical coordinates, respectively. The asterisks denote the dimensional variables and parameters. When the magnetizable fluid is subjected to the magnetic field, its free surface is deformed and acquires the shape described by the equation z* = h* (x*, t*). Let ν,

ϱ, and μ denote the kinematic viscosity, the density, and the magnetic permeability of the fluid, respectively, g the acceleration due to gravity, and pa the atmospheric pressure. All these parameters are assumed to be constant and (μ − 1)⪡1 (noninduction approximation). In what follows, all physical quantities are measured in CGS units. 2.1. Equations of motion and boundary conditions In the dimensional variables, the equations of motion of a magnetizable incompressible fluid in a specified coordinate system can be represented as follows:

ρ

dvi* = ∇j pij* + ρFi*, dt *

(3)

(5)

where p is the fluid pressure, is the viscous stress tensor, δij is the Kronecker delta. The magnetic stress tensor pH ij has the form [8] n

τij*

Hi* B *j 4π

H *B* δij, 8π

−

(6)

where Hi are the components of the magnetic field H * when the magnetizable fluid distorts the applied magnetic field Ha*; B *j are the components of the vector B* defined by B* = μH * inside the magnetizable fluid and B* = H * outside the magnetizable fluid. Using Maxwell's equations div B* = 0 and rot H * = 0 we obtain 2

∇j pijH = −

where

ζ * = k*x* − ω*t *.

(4)

where vx* = u* and vz* = w* are the horizontal and vertical components of the velocity vector v , respectively, tn is time, pij* are the

pijH =

Consider a planar flow (2D flow) of an incompressible magnetizable fluid bounded from below by a horizontal impermeable plane, as depicted in Fig. 1. A periodic traveling magnetic field Ha* is applied. Here Ha* is a applied magnetic field when a magnetizable fluid is absent. The square of the applied magnetic field strength is defined by

Ha*2

∇i vi* = 0,

H* ∇i μ . 8π

(7)

For fluids with constant μ this term disappears from (3), however, a surface force density fm appears due to the step change in μ at the interface. The expression for fm is [8]

⎡ B *2 ⎛ ⎞ H *2 ⎛ ⎞⎤ 1 fm = ⎢− n ⎜⎜ − 1⎟⎟ + τ ⎜⎜μ − 1⎟⎟ ⎥ n for z* = h* (x*, t *). ⎢⎣ 8π ⎝ μ 8π ⎝ ⎠ ⎠ ⎥⎦

(8)

Here Bn* is the component of the vector B* normal to the fluid surface, Hτ* is the component of the magnetic field H * tangential to the fluid surface (Bn* and Hτ* are continuous on the surface), n is the unit vector of the outer normal to the fluid surface. In the noninduction approximation, when we have (μ − 1)⪡1, * (1 + O (μ − 1)2) and H * = H * (1 + O (μ − 1)2). Therefore, exBn* = Han τ aτ pression (8) for fm can be written as

fm =

Ha*2 (μ − 1 + O (μ − 1)2) n for z* = h* (x*, t *). 8π

(9)

The dynamic condition at the fluid surface in the dimensional variables has the form

−p*n + τij* n j ei =

R=

γn + fm R

for z* = h* (x*, t *),

(1 + h *2* )3/2 ′x , h* * * ′x x

(10)

(11)

where R is the radius of curvature of the surface at the respective point, γ is the coefficient of surface tension, and e i is the unit vector of the ith coordinate axis (e1 = e x , e2 = e z ), the lower prime stands for differentiation with respect to the corresponding variable. The dynamic condition must be augmented by the kinematic condition at the fluid surface and the no-slip condition at the rigid base

dh* = w *, dt * Fig. 1. Schematic of the system under consideration.

u* = w * = 0,

z* = h* (x*, t *),

z* = 0.

(12) (13)

V. Böhm et al. / Journal of Magnetism and Magnetic Materials 395 (2015) 67–72

69

follows: 2.2. Dimensionless variables

τxx = 2 Let h0 be the height of the unperturbed fluid layer. Using the parameters ω*, ν, and ϱ, we define the characteristic time Tc, the characteristic velocity Uc, the characteristic pressure Pc, and the characteristic magnetic field magnitude Hc by

Tc = 1/ω*,

Uc = h0 ω*,

Pc = ν ϱUc /h0 = ω*ν ϱ ,

Hc = H0* (h0 )

h* = hh0 ,

(u*, w *) = (u, w ) Uc ,

k* = k/ h0 ,

t * = tTc ,

Uc h0 , ν

Sr =

h0 , Tc Uc

Eu =

Pc

, 2

ϱUc

Fr =

Uc2 . gh0

(16)

(18)

where δ = Re/Fr = Eq. (17) is the Stokes approximation for the Navier–Stokes equation, and (18) is the two-dimensional continuity equation (4). The last equation allows introducing the stream function Ψ = Ψ (x, z, t ) defined by

w=

∂Ψ . ∂x

(19)

ε=

A2 (h0 ) Hc2

(20)

The boundary conditions (12) and (13) in the dimensionless form become

u (x, z, t ) = 0,

w (x, z, t ) = 0

for z = 0,

h t + u (x, z, t ) h x − w (x, z, t ) = 0 ′ ′

for z = h (x, t ).

(21) (22)

Use relations (19) to express the boundary conditions (21) and (22) in terms of the stream function as follows:

−Ψ z (x, z, t ) = 0, ′

Ψ x (x, z, t ) = 0 ′

for z = 0,

h t − Ψ z (x, z, t ) h x − Ψ x (x, z, t ) = 0 ′ ′ ′ ′

,

(28)

∞

Ψ (x, z, t ) = Ψ0 +

∑ εnΨn (z, ζ ),

(29)

n= 1 ∞

h (x, t ) = 1 +

∑ εnhn (ζ ).

(30)

n= 1

We assume that the function is such that has an order of ε. Therefore the expression for Ha2 at the surface z = h (x, t ) can be represented in the form ∞

Ha2 (x, t ) = 1 + ε sin (ζ ) +

(31)

n= 2

For ε = 0, which corresponds to the static magnetic field, the pressure gradient in the x-direction is zero and, hence, Ψ0 = const . Since the magnetic excitation is periodic with respect to ζ, we will seek the functions Ψn (z, ζ ) periodic with respect to ζ and represent them by the Fourier series

⎛ ⎞ ⎜⎜ηnm (z ) cos (mζ ) + ξnm (z ) sin (mζ ) ⎟⎟, ⎠ m=1 ⎝ ∞

∑

Ψn (z, ζ ) =

n ∈ .

(32)

By substituting the expression (32) for Ψn (z, ζ ) into (29) and the resulting expression for Ψ (x, z, t ) into Eq. (20) we arrive at the ordinary differential equations for the coefficients ηnm (z ) and ξnm (z ):

φ zzzz (z ) − 2m2k 2φ zz (z ) + m4 k 4φ (z ) = 0, ′ ′ where φ (z ) = ηnm (z ) or φ (z ) = ξnm (z ). Note that

(33) 2π

∫0 Ψn dζ = 0.

(24)

)

z = h (x, t ),

∑ εnGn (ζ ).

3.1. The average flow rate

for z = h (x, t ).

⎡−p − We h 1 + h′2x − 3/2 − WeH Ha2 ⎤⎦ ⎣ ′ xx n + τij n j ei = 0,

d (H02 ) /dz|z = 1

(23)

In the dimensionless form, the dynamic boundary condition (10) in the noninduction approximation becomes

(

(27)

which is the ratio of the squared perturbation amplitude at the undeformed fluid surface z* = h0 to the static component of the squared magnetic field strength. We assume that the parameter ε is small (ε⪡1). Then we will seek the stream function Ψ and the functions h and Ha2 at the surface z = h (x, t ) in the form of the power series of ε

Substitute (19) into (17) and eliminate the pressure p to obtain

ΔΔΨ (x, z, t ) = 0.

(26)

n z = (h 2x + 1)−1/2 . ′

H02 (z )

gh02 /(νUc ).

∂Ψ , ∂z

n x = − h x (h 2x + 1)−1/2 , ′ ′

(17)

∂u ∂w + = 0, ∂x ∂z

u=−

∂w . ∂z

Introduce the parameter

The Reynolds number and the product of the Strouhal and of the Reynolds numbers are assumed to be small (Re⪡1, SrRe⪡1). The velocity scale factor Uc is chosen so that EuRe = 1 and Sr = 1. In this case, the inertial forces are substantially less than the forces due to viscosity, pressure, and gravity. The magnetic field is supposed to be weak, which allows the magnetic permeability of the fluid to be considered constant. The equations of motion of the fluid can be approximated as follows:

∇p = Δv − δe z ,

τzz = 2

The x and z components of the normal vector n are given by

(15)

Such a nondimensionalization leads to the following definitions for Reynolds, Strouhal, Euler and Froude numbers:

Re =

∂u ∂w , + ∂z ∂x

3. Solution

(p* , τij* ) = (p, τij ) Pc ,

(Ha*, H0* ) = (Ha, H0 ) Hc .

τxz = τzx =

(14)

and then introduce the dimensionless variables as follows:

(x*, z*) = (x, z ) h0 ,

∂u , ∂x

The flow rate for the two-dimensional flow is defined by

Q (ζ ) =

∫0

h (x , t )

u (z, ζ ) dz.

(34)

This expression can be rewritten as

(25)

where We = γ /(h0 Pc ) = γ /(ω*h0 ϱν ) is the Weber number, and WeH = ϰHc2/(ω*ϱν ) is the magnetic Weber number, ϰ = (μ − 1) /(8π ). The components of the viscous stress tensor are expressed as

Q (ζ ) = −

∫0

h (ζ )

∂Ψ (z, ζ ) dz = − Ψ (h, ζ ) + Ψ (0, ζ ). ∂z

(35)

With reference to (29) and (30), the flow rate Q (ζ ) can be represented as a power series

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V. Böhm et al. / Journal of Magnetism and Magnetic Materials 395 (2015) 67–72

Q (ζ ) = Q 0 + εQ 1 + ε2Q 2 + ⋯,

(36)

where

(37)

Q 1 = Ψ1 (0, ζ ) − Ψ1 (1, ζ ),

(38)

Q 2 = Ψ2 (0, ζ ) − Ψ2 (1, ζ ) − Ψ 1 z (1, ζ ) h1.

We are interested in the time-average of the flow rate defined by T

Q (x, t ) dt

(51)

The formula (51) implies that Q¯ > 0, since sinh (2k ) > 2k for k > 0. Therefore, the direction of the material transport of the fluid coincides with the propagation direction of the excitation wave.

(39)

′

∫0

(δ + We k 2)2 (sinh (2k ) − 2k )2 + 4k 2 (1 + cosh (2k ) − 2k 2)2 + O (ε3).

Q 0 = Ψ0 (0, ζ ) − Ψ0 (1, ζ ) = 0,

1 Q¯ = T

ε2WeH2 k 2 (sinh (2k ) − 2k )

Q¯ =

where T = 2π .

(40)

3.2. Analysis of the equation for the average flow rate In the dimensional variables, the average flow rate Q¯ * defined in accordance with (51) is given by the expression

If Q¯ ≠ 0, the permanent transport of the fluid occurs. Substitute (36) into (40) to obtain

Q¯ * =

Q¯ = εQ¯ 1 + ε2Q¯ 2 + O (ε3),

where

(41)

B0 , B1/ω* + B2 ω*

(52)

2

where

B0 = A4 ϰ 2k* h02 (sinh (2k*h0 ) − 2k*h0 ),

Q¯ 1 = 0,

(42)

2

B1 = (ϱg + γk* )2 (sinh (2k*h0 ) − 2k*h0 )2 , 2

1 Q¯ 2 = − T

∫kx − 2π h1(ζ ) Ψ 1′z (1, ζ ) dζ .

(43)

By substituting the expansions (29) and (30) into the kinematic Eq. (24) we find that the functions Ψ1 and h1 are related by

h1 (ζ ) = − kΨ1 (1, ζ ).

(44)

Since the inequality B1/ω* + B2 ω* ≥ 2 B1B2 holds for nonnegative B1 and B2 and positive ω*, the maximum value of the flow rate, defined by * Q¯ max =

Taking into account (44) we find

k Q¯ 2 = 2π

(45)

From (41) it is apparent that to calculate the first non-zero term ε2Q¯ 2 in the expression for the average flow rate, it is necessary to have determined the first order term Ψ1. Substitute the expansions (29)–(31) into the boundary conditions (23) and (25) and match the coefficients of similar powers of ε. Use the equation of motion (17) to exclude the pressure p. Then we find the following boundary conditions for the function Ψ1:

Ψ 1 x (0, ζ ) = Ψ 1 z (0, ζ ) = 0,

(46)

Ψ 1 xx (1, ζ ) − Ψ 1 zz (1, ζ ) = 0,

(47)

′

′

3Ψ 1 xxz (1, ζ ) + Ψ 1 zzz (1, ζ ) + δh1 x − We h1 xxx − WeH k cos (ζ ) ′

′

′

(48)

= 0.

The solutions of (33) for the functions η1m (z ) and ξ1m (z ) are defined as follows:

η1m (z ) =

a1(1m) +

ξ1m (z ) =

cosh (mkz ) +

4) a1(m z

b1(1m)

a1(2m) z

cosh (mkz ) +

3) a1(m

sinh (mkz )

sinh (mkz ),

cosh (mkz ) +

2) b1(m z

4) + b1(m z sinh (mkz ).

=

A4 ϰ 2k*h02 , 2 2 * 4ϱν (ϱg + γk )(1 + cosh (2k*h0 ) − 2k* h02 )

(53)

* ωmax =

B1 B2

2

=

(ϱg + γk* )(sinh (2k*h0 ) − 2k*h0 ) . 2 2k*ϱν (1 + cosh (2k*h0 ) − 2k* h 2 )

(54)

0

* If k = k*h0 ⪡1, then the expressions (52), (53) and (54) for Q¯ *, Q¯ max * and ωmax can be represented as follows:

5 3A4 ϰ 2k* h05 ω*

Q¯ * =

2

6

,

2

2

4 ((ϱg + γk* )2k* h06 + 9k* ϱ2ν 2ω* )

(55)

2

′

′

B0 2 B1B2

* , where occurs for ω* = ωmax

kx

∫kx − 2π Ψ1(1, ζ ) Ψ 1′z (1, ζ ) dζ .

′

2

B2 = 4k* ϱ2ν 2 (1 + cosh (2k*h0 ) − 2k* h02 )2 .

kx

cosh (mkz ) +

1.6 1.4 1.2 1

(49) 3) b1(m

1.8

sinh (mkz )

0.8 0.6

(50)

(i ) In view of the expressions (49) and (50), only the coefficients a11 (i ) and b11 for i = 1, … , 4 do not vanish. Substitute now the expansion (32) for n ¼1 and m¼ 1, where η11 (z ) and ξ11 (z ) are given by expressions (49) and (50), respectively, into the boundary conditions (46)–(48) and match the coefficients of similar harmonics. These coefficients are defined as a solution of a system of linear algebraic equations. Hence, the average flow rate is defined as follows:

0.4 0.2 0

10

20

30

40

Fig. 2. Average flow rate Q¯ ⁎ versus angular frequency ω⁎ (k⁎ = 5 cm−1, h0 = 0.1 cm , γ = 30 g s−2 ).

V. Böhm et al. / Journal of Magnetism and Magnetic Materials 395 (2015) 67–72

71

4

2 1.8 1.6

3 1.4 1.2 1

2

0.8 0.6

1

0.4 0.2 0

2

4

6

8

10

12

0

14

Fig. 3. Average flow rate Q¯ ⁎ versus wave number kn (ω⁎ = 5.86 s−1, h0⁎ = 0.1 cm , γ = 30 g s−2 ).

10

20

30

40

50

60

Fig. 5. Average flow rate Q¯ ⁎ versus coefficient of surface tension γ (k⁎ = 5.15 cm−1, ω⁎ = 5.86 s−1, h0 = 0.12 cm ).

2.5

for the fixed values of the remaining parameters. Fig. 2 shows Q¯ * versus ωn for h0 ¼0.1 cm and k* = 5 cm−1. The * maximum value of the flow rate Q¯ max = 1.769·10−5 cm2 s−1 occurs

2

* for ωmax = 5.86 s−1. The inequality Re ⪡1 in the dimensional variables has the form ω*⪡ν/h 2. For the layer thickness h0 = 0.1 cm and 0

the kinematic ω*⪡189 s−1.

1.5

viscosity

ν = 1.89 cm2 s−1,

this

implies

that

1

* Fig. 3 represents Q¯ * versus kn for ω* = ωmax = 5.86 s−1 and h0 = 0.1 cm . The maximum value of the flow rate * * = 5.15 cm−1. = 1.772·10−5 cm2 s−1 occurs for k max Q¯ max

0.5

* Fig. 4 depicts Q¯ * as a function of h0 for ω* = ωmax = 5.86 s−1 and 1 − * * k = k max = 5.15 cm . The maximum value of the flow rate * = 2.074·10−5 cm−2 s−1 occurs for h0 max = 0.12 cm . Q¯ max

versus γ for h0 = h0 max = 0.12 cm , * * ω* = ωmax = 5.86 s−1, and k* = k max = 5.15 cm−1. The average flow * rate Q¯ over the period decreases as the coefficient of surface tension γ increases. Figs. 2–4 show that the optimal values of the parameters ω*, kn, and h0, for which the average flow rate attains a Fig.

0

0.1

0.2

0.3

0.4

0.5

Fig. 4. Average flow rate Q¯ ⁎ versus unperturbed fluid height h0 (k⁎ = 5.15 cm−1, ω⁎ = 5.86 s−1, γ = 30 g s−2 ).

2

* ωmax =

2

(ϱg + γk* ) k* h03 , 3ϱ ν

* Q¯ max =

A4 ϰ 2k*h02 2 . 8ϱν (ϱg + γk* )

(56)

The calculations were performed for magnetizable fluid APG S12n (Ferrotec GmbH, Germany) characterized by density ϱ = 1.32 g cm−3, kinematic viscosity ν = 1.89 cm2 s−1, coefficient of the surface tension γ = 30 gs−2, and magnetic permeability μ ¼ 1.118. The parameters of the applied magnetic field at the * surface were specified as Hc = H0 (h0 ) = 100 Oe and A (h0 ) = 30 Oe; 2 2 therefore ε = A (h0 ) /Hc = 0.09. Figs. 2–5 show the computational results for the dimensional average flow rate Q¯ * (Eq. 52) as a function of the dimensional frequency ω* (s−1), the wave number k* (cm−1), the layer thickness h0 (cm), and the coefficient of surface tension γ (g s−2), respectively,

5

shows

Q¯ *

maximal value, exist.

4. Conclusion A two-dimensional problem of magnetizable fluid transportation by a traveling periodic magnetic field causing a traveling periodic deformation of the free fluid surface was considered. The analysis of the problem was performed in Stokes approximation by using the perturbation method. An analytical expression for the average flow rate is presented. The optimal values of the parameters, for which the average flow rate is a maximum, are found. The occurrence of a maximum is due to the fact that the flow rate depends on the propagation velocity and the amplitude of the surface wave of the magnetizable fluid, the amplitude decreases as the velocity of the wave increases.

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V. Böhm et al. / Journal of Magnetism and Magnetic Materials 395 (2015) 67–72

Acknowledgment This work is supported by Deutsche Forschungsgemeinschaft (DFG project Zi 540-17/1), the Russian Foundation for Basic Research (project 14-01-91330) and the free state of Thuringia, Germany (postgraduation scholarship). We appreciate Prof. Bolotnik and Prof. Schumacher for their kind assistance.

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