Proceedings of the 11th International Conference on Fluid Flow, Heat and Mass Transfer (FFHMT 2024)
Chestnut Conference Centre - University of Toronto, Toronto, Canada – June 16-18, 2024
Paper No. 078
DOI: 10.11159/ffhmt24.078

Bounded Pressure-Driven Flow of an Incompressible Electrically
Conducting Fluid: Porous Medium and Lubrication Applications
Érick Marcelino Miranda1, Marcos Fabrício de Souza Aleixo Filho1, Francisco Ricardo Cunha1
1

University of Brasilia, Faculty of Technology, Department of Mechanical Engineering, Fluid Mechanics of Complex
Flows Group - VORTEX
Brasilia, Brazil
erick99marcelino@outlook.com; mffilhoaleixo@gmail.com; frcunha@unb.br

Abstract - This work examines the effects of the interaction of a transversal and uniform magnetic field applied to a flow with anelectrically conducting and incompressible fluid. This fluid is bounded by two fixed parallel walls in a channel. The bottom wallconsists of a porous medium, where the jump of shear stress is given in terms of a suitable relative velocity by a semi-empiricalboundary condition proposed by Beavers and Joseph. The formulation of the flow problem is based on the incompressible
magnetohydrodynamic governing equations in terms of non-dimensional variables. The relevant physical parameter measuring therelative importance between magnetic and viscous forces is identified as the Hartmann number. The solution of the problem showsthe existence of a flow deceleration strongly dependent upon the Hartmann number. In addition, another interesting result is theobserved decrease in the magnitude of the longitudinal component of the magnetic flux density as Hartmann number increases. In
conclusion, the application of a transverse magnetic field in the flow of an electrically conducting fluid in tiny pores can produce
an effective effect like the flow deceleration produced as the porous medium permeability is decreased. Therefore, it seems to bepossible to produce such an effect by just monitoring the magnetic field instead of changing the complex microstructure of theporous medium. Finally, the problem of lubrication with an MHD fluid was addressed. Exact solutions were obtained for the velocity
and pressure field. Due to the braking effects caused by the Lorentz force, the pressure gradient is reduced and consequently the
support force on the bearing is also reduced. Therefore, the MHD effects for this model undermine the lubrication effects of thepurely hydrodynamic model.
Keywords: Magnetohydrodynamics, Porous Medium, Beavers and Joseph Boundary Condition, Flow Control byExternal Magnetic Field, Asymptotic solution, Lubrication Problem.

1. Introduction
Magnetohydrodynamics (MHD) studies the interaction between an electrically conducting fluid (non-polar andnon-magnetic) and a magnetic field. Flows with MHD effects are a part of fluid mechanics involving these fluids (suchas salted water, ionized gases, or liquid metal) and a magnetic field [1]. The industry utilizes this flow type for several
purposes, including convection inhibition, mixing, heating, deceleration, and fluid pumping.
The central governing equation in the MHD problem is the modified Navier-Stokes (with the Lorentz force term)and Maxwell equations [2]. Due to these complex equations, the asymptotic method becomes an important analysis tool.
We can consider every MHD problem as a combination of hydrodynamic and magnetic effects. Due to the asymptotic
method, we fully recover the hydrodynamic case when we set the magnetic effects to zero in an MHD problem.
Therefore, to control the intensity of the MHD effects in the problems, one can use parameters arising from the nondimensionalization process of the resulting equations. This way, critical non-dimensional parameters such as theHartmann number (𝐻𝑎 ) (ratio of Lorentz to viscous forces) and magnetic Reynolds number (ratio of advection to the
diffusion of 𝑩) are extracted. In this context, an example of MHD flow is an electrically conducting fluid flowing
bounded by two fixed flat parallel walls with a uniform transverse magnetic field applied. This problem has various
applications, such as in the petroleum industry, cooling systems, aerodynamic heating, and fluid droplets [3], [4].
The main scope of this paper is to solve the flow of an electrically conducting fluid in a channel bounded twowalls. Note that in one of the applications studied here, one of the walls is porous. The effect of the magnetic field on
the curve of pressure gradient versus flow rate is examined at different Hartmann numbers. The deviation from the linear
plot for pure hydrodynamic condition (no-MHD flow) is also explored. Additionally, we calculate the velocity andmagnetic field induction profiles, and we shall show how the maximum values of these quantities depend on the

078-1

Hartmann number. The effective viscosity (η𝑒𝑓𝑓 ) of the flow is also determined as function of 𝐻𝑎 . The studies here areimportant for engineering applications like drag reduction in channel flows with different boundary conditions bycontrolling the applied magnetic field.

2. Problem Formulation
We are going to study two problems. The first, called a parallel wall problem, is the MHD flow bounded by two
fixed parallel walls, and the second concerns a lubrication bearing problem. For the first, the three boundary conditionsof the problem were defined. They refer to the no-slip condition at the upper wall. Beaver and Joseph's condition, definedby a semi-empirical method, will be used at the bottom wall (porous medium), and the sliding velocity will be used at
the porous interface [5].
The fluid is bounded by two fixed walls with no-slip boundary conditions for the second problem, called the
lubrication problem. The bottom wall is parallel to the x-axis, and the upper wall varies linearly with this axis. Fromthen on, we studied these MHD problems, selected the governing equations, made them non-dimensional and solved
them. For the two problems, we employed the asymptotic method to split the MHD from the hydrodynamic effect in
the governing equations. In parallel to this, however, these problems were also studied in a purely hydrodynamic context.
This way, the governing equations were solved, and made non-dimensional. It is essential to point out that due to thehigh density of calculations, computational assistance was used in solving the equations for the MHD case.
The studied problems involve an electrically conducting Newtonian fluid interacting with a transverse andFig.1: Representation of the proposed problem for theflow between two fixed parallel walls spaced by h. The
uniform magnetic field transverse to the applied flow isrepresented by 𝑩.

Fig.2: Scheme representing a lubrication bearing
with an external magnetic field. The wall distanceis δ(𝑥) and varies from 𝛿0 until 𝛿𝑙 (𝑥).

uniform applied magnetic field. A schematic of the proposed problems can be seen in Fig. 1 and Fig. 2. The transverseand uniform magnetic field is denoted as B. Note that 𝐁 = Bx (y) 𝒆̂𝒙 + B0 𝒆̂𝒚 , where 𝐵𝑥 is the longitudinal componentof the induced magnetic field, and 𝐵0 is the transversal component of the applied field.
Non-dimensionalization simplifies equations by minimizing complexity and reducing the number of variables in
the model. Thus, the non-dimensional quantities for the parallel wall problem are given by:

𝑦∗ =

𝑦
;
ℎ

𝑣𝑥 ∗ =

𝑣𝑥
;
𝑢𝑚
𝑥

𝑝∗ =

𝑝ℎ
;
η𝑢𝑚

And, for the lubrication problem, by: 𝑥 ∗ = 𝑙 , 𝛿 ∗ (𝑥 ∗ ) =

𝐸0∗ =

𝐸0
;
𝑢𝑚 𝐵0

𝐵𝑥∗ =

𝐵𝑥
.
𝐵0

(1)

𝑝𝑙
𝛿(𝑥)
and 𝑝∗ = .
𝑙η𝑈Where the characteristic magnetic flux density, 𝐵0 , is given as a characteristic parameter of the problem. The pressure
field is denoted by 𝑝. The mean velocity is represented as 𝑈. The external electric field is 𝐸0 . The length of the walls is
given by 𝑙. The velocity in the x-direction is 𝑣𝑥 , and the film velocity in the porous medium is 𝑢𝑚 , which flows at a low
Reynolds number in a porous medium is calculated by Darcy’s law. First, however, we are going to use it to make the078-2

velocity non-dimensional. Furthermore, another essential non-dimensional parameter is the Hartmann number, definedas the ratio between the Lorentz force and the viscous force: 𝐻𝑎 = 𝐵0 ℎ√𝐾𝑒 /η, where 𝐾𝑒 and η are the electricalconductivity and viscosity of the fluid, respectively [2]. Another non-dimensional parameter is the Magnetic Reynolds
number (ratio between the diffusion time and the advection time of 𝐵): 𝑅𝑒𝑚 = ℎ𝑢𝑚 /ν𝑚 , where ν𝑚 is the magnetic
diffusion coefficient [2]. Note that we used a characteristic length for the definitions of the Hartmann number and themagnetic Reynolds number, which for the parallel wall problem was defined as ℎ. For the lubrication problem, we used
the mean velocity as the characteristic velocity in 𝑅𝑒𝑚 .

3. Parallel Wall Problem
The configuration shown in Fig.1 describes the problem with MHD effects. The current density is evaluated by
the classical Ohm's law as follows [2]:𝑱 = 𝐾𝑒 (𝑬 + 𝒗 × 𝑩).

(2)Using the magnetic field equation and Eq. (2), we obtain the Lorentz force:

𝒇𝐿 = 𝑱 × 𝑩 = 𝐾𝑒 (𝐸0 + 𝑣𝑥 𝐵0 )(−𝐵0 𝒆̂𝒙 + 𝐵𝑥 𝒆̂𝒚 ).

(3)

The governing equation for this problem is −∇𝑝 + 𝜂∇2 𝒗 + (𝑱 × 𝑩) = 0. This equation in 𝑥 and 𝑦 directions,presented in non-dimensional form, is given by Eq. (4) and Eq. (5), respectively.

σ2 +−𝑑2 𝑣𝑥∗𝑑𝑦 ∗ 2∗− 𝐻𝑎 2 𝐸0 ∗ − 𝐻𝑎 2 𝑣𝑥 ∗ = 0∂𝑝+ 𝐻𝑎 2 𝐵𝑥∗ (𝐸0∗ + 𝑣𝑥 ∗ ) = 0∂𝑦 ∗𝑑2 𝐵𝑥∗
𝑑𝑣𝑥 ∗
+ 𝑅𝑒𝑚
=0
𝑑𝑦 ∗
𝑑𝑦 ∗ 2

With the following boundary conditions: 𝑦 ∗ = 0 ⇒

(4)
(5)
(6)

𝑑𝑣𝑥 ∗
= ασ(𝑢𝑖 − 1), 𝑦 ∗ = 1 ⇒ 𝑣𝑥 ∗ (𝑦 ∗ ) = 0 and 𝐵𝑥 ∗ (𝑦 ∗ = 0) =𝑑𝑦 ∗𝐵𝑥 ∗ (𝑦 ∗ = 1) = 0, where σ = ℎ/√𝑘 and the non-dimensional quantity α is called the porosity of the porous medium.
Note that, in order to complete the MHD flow formulation, it is necessary to include in the model the equation ofmagnetic induction transport given by Eq. (6). The analytical solution of the Eq. (4), Eq. (5), and Eq. (6) were obtainedusing the MATLAB software. This way, the velocity field, magnetic flux density, pressure field, flow rate, and induced
electric field were determined.

4. Lubrication Problem
For the lubrication problem, we will not use the asterisk (*) to denote non-dimensional quantities to simplify thenotation. By scale analysis, the non-dimensional governing equations of the lubrication problem are:

𝜕 2 𝑣𝑥 (𝑦)
𝜕𝑝
𝜕𝑝
2
2
(𝑦)
−
𝐻
𝑣
=
+
𝐻
𝛽;
= 𝐻𝑎 2 𝛽𝐵𝑥 + 𝐻𝑎 2 𝑣𝑥 (𝑦)𝐵𝑥 ;
𝑎
𝑥
𝑎
2
𝜕𝑦
𝜕𝑥
𝜕𝑦

078-3

(7)

𝜕𝑣𝑥 (𝑦) 𝜕𝑣𝑦
𝜕𝑣𝑥 (𝑦)1 𝜕 2 𝐵𝑥
+
= 0;
+
=0
𝜕𝑥
𝜕𝑦
𝜕𝑦
𝑅𝑒𝑚 𝜕𝑦 2
conditions:𝑣𝑥 = 𝑣𝑦 = 0 ⇒ 𝑦 = 𝛿(𝑥),𝑣𝑥 = 1 ⇒ 𝑦 = 0
and𝐸𝑜
𝐵𝑥 = 0 ⇒ 𝑦 = 𝛿(𝑥). The parameter 𝛽 =represents the non-dimensional electric field. The solution for the velocity
𝐵0 𝑈
With

the

followingboundary

field is:e𝐻𝑎 𝛿𝑒 𝐻𝑎 𝑦𝑣𝑥 (𝑦) = e𝐻𝑎𝛿 e-𝐻𝑎y [𝜆 + 2𝐻 𝛿(𝜆 − e𝐻𝑎 𝛿 𝜆 + 1)] − 𝜆 − 2𝐻 𝛿(𝜆 − 𝑒 𝐻𝑎 𝛿 𝜆 + 1),
e 𝑎 −1𝑒 𝑎 −1
where 𝜆 =𝐺=

(8)

𝛽𝐻𝑎 2 −G𝜕𝑝
and 𝐺 = − . From this result we can obtain the expression for the pressure gradient:
𝐻𝑎 2
𝜕𝑥

2 𝐻𝑎 2 β + 𝐻𝑎 3 𝑄MHD − 𝐻𝑎 2 e𝐻𝑎 𝛿 + 𝐻𝑎 2 + 𝐻𝑎 3 β 𝛿 − 2 𝐻𝑎 2 β e𝐻𝑎 𝛿 + 𝐻𝑎 2 𝑄MHD e𝐻𝑎 𝛿 + 𝐻𝑎 3 β 𝛿 e𝐻𝑎 𝛿,
𝐻𝑎 𝛿 − 2 e𝐻𝑎 𝛿 + 𝐻𝑎 𝛿 e𝐻𝑎 𝛿 + 2(9)where 𝑄𝑀𝐻𝐷 is the MHD flow rate and δ is the wall distance which depends on the 𝑥.
To observe the MHD effects on the lubrication bearing, the pressure field was resolved considering four Hartmann
values, from a very small value (Ha = 0.01), which is close to the hydrodynamic case, up to a high Hartmann (Ha = 2).
From Fig. (6), a decay of the pressure distribution in the x direction is observed with the increase in the Hartmann
number. Until 𝐻𝑎 = 2, there is a negative increase in pressure leading to a case alike the positive profile ( 𝛿𝑜 < 𝛿𝑙 ), in
the pure hydrodynamic case. This leads to the assertion that an intense magnetic field is responsible for a pressure dropin x, weakening the lubrication effect observed in the purely hydrodynamic case. Therefore, the MHD effects for thisproblem are to inhibit or weaken the support force between bearing and shaft.

5. Asymptotic Solution
Although it is possible to find an exact solution for Eq. (4), an asymptotic analysis of the flow becomes useful for
splitting the solution of the problem into two contributions: purely hydrodynamic and correction by magnetic effect.
Such methodology has the advantage of splitting the solutions as the sum of the two contributions: hydrodynamic and
MHD. Using a regular perturbations method described in [5], [6], the velocity field can be expressed as:𝑣𝑥∗ (𝑦 ∗ ) = 𝑣0 ∗ (𝑦 ∗ ) + ε𝑣1∗ (𝑦 ∗ ) + ε2 𝑣2∗ (𝑦 ∗ ) + 𝒪(ε3 ),(10)

where, in this flow problem, the small parameter is ε = 𝐻𝑎 2. Replacing Eq. (10) in Eq. (4) and after a few algebraic
manipulations, we find the following differential equations for different orders of ε:𝑑2 𝑣0∗
𝑑 2 𝑣1∗𝑑2 𝑣2∗
∗
2
0 );
∗
1 );
= −σ , 𝒪(ε− 𝑣0 = 𝐸0 , 𝒪(ε= 𝑣1 ∗ , 𝒪(ε2 )𝑑𝑦 ∗ 2𝑑𝑦 ∗ 2𝑑𝑦 ∗ 2
With the following boundary conditions:

078-4

(11)

𝑑𝑣0 ∗ (0)𝑑𝑣1 ∗ (0) 𝑑𝑣2 ∗ (0)𝑦 = 0,= ασ(𝑢𝑖 − 1);
=
=⋯=0
𝑑𝑦 ∗
𝑑𝑦 ∗
𝑑𝑦 ∗
𝑦 ∗ = 1, 𝑣0 ∗ = 𝑣1 ∗ = 𝑣2 ∗ = ⋯ = 0
∗

(12)
(13)

Finally, we can write the velocity field as:𝑣𝑥∗ (𝑦 ∗ ) = 𝑣0 ∗ (𝑦 ∗ ) + 𝐻𝑎2 𝑣1∗ (𝑦 ∗ ) + 𝒪(𝐻𝑎4 ),(14)

Where, 𝑣0 ∗ is the solution for the purely hydrodynamic case and 𝑣1 ∗ is given by:

𝑣1 ∗ (𝑦 ∗ ) =𝐸0∗ ∗2
1
ασ ∗3
ασ
σ2
(𝑦 − 1) + 𝑢𝑖∗ [ (𝑦 ∗2 − 1) +(𝑦 − 1)] +(1 − 𝑦 ∗3 ) + (1 − 𝑦 ∗4 ).
2
2
6
6
24

(15)

If 𝐻𝑎 < 0, it means that ε ≪ 0, so the method gives a highly accurate solution in this regime. As 𝐻𝑎 increases, theasymptotic solution diverges from the exact solution of the Eq. (4), as we can verify in Fig. (4). This means that theterms of higher order cannot be neglected in this regime since the Hartmann number is no longer small. Using thisasymptotic solution for the velocity field Eq. (14), it is possible to calculate an approximate solution for the magnetic
field in the 𝑥 direction, in which the solution until 𝒪(𝐻𝑎4 ) is expressed by:(𝑦 ∗ − 𝑦 ∗2 )(𝑦 ∗2 − 𝑦 ∗ )(𝑦 ∗3 − 𝑦 ∗ )
+ ασ
+ σ2
]
2
2
6
(𝑦 ∗ − 𝑦 ∗3 )(𝑦 ∗ − 𝑦 ∗3 )(𝑦 ∗ − 𝑦 ∗4 )(𝑦 ∗4 − 𝑦 ∗ )+ 𝑅𝑒𝑚 𝐻𝑎2 {𝐸0∗+ 𝑢𝑖∗ [+ ασ
] + ασ
6
6
24
24
∗5
∗
(𝑦 − 𝑦 )
+ σ2
},
120

𝐵𝑥 ∗ (𝑦 ∗ ) = 𝑅𝑒𝑚 [𝑢𝑖∗ ασ(16)Note that 𝐸0∗ is given by:−1(1 − 𝑢𝑖∗ )
𝐻𝑎2
σ2
ασ σ2
1 ασ
∗
𝐸0 = ( + 1) { + ασ
+ 𝐻𝑎2 [ +− 𝑢𝑖∗ ( + )] + 𝒪(𝐻𝑎4 )}.

2

3

2

8

30

3

8

(17)

By Eq. (18), we can determine the flow rate in a purely hydrodynamic contribution and another MHD given by
Hartmann orders, as follows:
1

𝑄∗ =

1

𝑄 ∗ = ∫0 𝑣0 ∗ 𝑑𝑦 ∗ + 𝐻𝑎2 ∫0 𝑣1 ∗ 𝑑𝑦 ∗ + 𝒪(𝐻𝑎4 ).

(18)(32 + 7ασ)σ2 (4 + ασ)
ασ
−𝐸0∗
50ασ
+
+ 𝐻𝑎2 [
− σ2−
] + 𝒪(𝐻𝑎4 ),
12 (1 + ασ) 2(1 + ασ)3240(1 + ασ) 240(1 + ασ)(19)

6. Effective Viscosity
Due to the braking effect that the Lorentz force causes in the conducting fluid, it is possible to think of it as a nonionized fluid but with a higher viscosity responsible for hindering the flow, which is called effective viscosity. The

078-5

effective viscosity represents the viscosity that a non-conducting fluid would have in order to present the same
relationship between flow rate and pressure gradient as the conducting fluid for a certain Hartmann number, so theviscosity naturally must depend on 𝐻𝑎 . In this way, it is possible to express a purely hydrodynamic flow rate, even foran ionized fluid with an equivalent viscosity in the following form [7]:𝑄=𝐺ℎ3 ασ + 4 + 6α/σ[],
(1 + ασ)12η𝑒𝑓𝑓(20)

where η𝑒𝑓𝑓 is the effective viscosity given, in this flow, by:−1(32 + 7ασ + 50α/σ)
6 + 𝐻𝑎22η𝑒𝑓𝑓 = η [−𝐻+ 𝒪(𝐻𝑎4 )] .
𝑎(ασ + 4 + 6α/σ)3(2 + 𝐻𝑎2 )(21)

7. Results and Discussion
Fig. (3) shows plots of the non-dimensional velocity profile as a function of 𝑦 for different Hartmann numbers.
As the Hartmann number increases, we observe a deceleration of the flow. Additionally, we emphasize that when theFig. 4: Comparison between the analytical andasymptotic solution for the maximum velocity
profile for a Hartmann number variation from 0 to 1.
α = 1, σ = 1, 𝑦 ∗ = 0. Note that - represents theanalytical solution, and --- is the asymptotic solution.

Fig. 3: Non-dimensional flow velocity profiles. The −represents the velocity profile for a hydrodynamic flow.
The --- represents the velocity profile for flow with
MHD effects about different Hartmann numbers(differentiated by color). α = 1, σ = 1, η = 1.

Hartmann number approaches zero, the MHD flow reverts to the purely hydrodynamic flow (𝐻𝑎 = 0). We can see thisconvergence in Fig. (3), when the velocity profiles for 𝐻𝑎 = 0.01 and the hydrodynamic flow overlap. Fig. (4) showsthat the maximum velocity occurs in 𝑦 ∗ = 0 (i.e., bottom wall) because of the sliding condition. Fig. (4) depicts that
this maximum velocity decreases when the Hartmann number increases.

078-6

Numerous researchers have investigated the magnetic effects on various flows, significantly contributing to ourunderstanding of the phenomenon. For example, studies by [8] demonstrate that magnetic effects decrease flow velocity.
Consequently, as the Hartmann number increases, we can expect a reduction in flow rate for the flow analysed in the
problem.

Fig. 5: Profiles of the non-dimensional magnetic field
Fig. 6: Pressure profiles between bearing and shaft
induction in the longitudinal direction. Different
according to the geometry in que 𝛿𝑜 > 𝛿𝑙 , forcolors represent the profiles for the flow with several
different Hartmann values.
𝐻𝑎 numbers. 𝑅𝑒𝑚 = 1, α = 1, σ = 1. The insert
represents the location where the induced magnetic(5), thevalue
intrinsic
fieldFromhas itsFig.
maximumof 𝐻𝑎 .coupling between the induced magnetic flux density and the velocity can be seen. As

Hartmann is increased, the effects of the Lorentz force slow down the flow, making the modulus of the magnetic flux
density smaller.

Fig. 7: Flow rate profiles concerning G. Different
colors represent the magnetic field induction for theflow with several 𝐻𝑎 numbers. ℎ = 1, α = 1, σ = 1.

Fig. 8: Non-dimensional effective viscosity as afunction of Hartmann number. α = 1, σ = 1.

Fig. (7) shows the relation between the flow rate and the gradient pressure. If the Lorentz force effect is strong,
the flow rate diminishes. Despite this flow rate reduction, the relationship between this magnitude and the pressure
gradient remains linear in this Hartmann regime. Plotting the relation in Eq. (21), it is possible to verify by analysing

078-7

Fig. (8) that the effective viscosity increases with the Hartmann number as if the MHD effect made the fluid moreviscous. Because of this effect, the flow rate decays with the increase of Hartmann, as seen in Fig. (7).

8. Conclusion
In this work, the results have demonstrated how the effects of the MHD Lorentz force can modify the motion ofan electrically conducting fluid undergoing a pressure gradient. The magnetic effect can decelerate or accelerate (in thiscase, control of the intensity of the electric field is always required) the flow, depending on the intensities of magnetic
and electrical fields on the conductor fluid.
With the solution of the parallel walls problem, it was possible to obtain the expected results since the velocity
profile proved to be coherent with the boundary conditions and the assumptions made. The boundary condition of theporous medium (bottom wall) modifies the velocity profile by accelerating it locally.
The flow problem explored in this paper was similar to the one which occurs in the tiny pores of natural reservoir
during oil extraction by a pressure-driven flow. In conclusion, the application of a transverse magnetic field in the flowof an electrically conducting fluid in tiny pores can produce an effective effect like the flow deceleration produced as
the porous medium permeability is decreased. Therefore, it seems to be possible to produce such an effect by justmonitoring the magnetic field instead of changing the complex microstructure of the porous medium.
Finally, the use of a conductive fluid subject to a magnetic field in a lubrication problem does not help with thelift effect, but rather the opposite, inhibiting or reducing this phenomenon. And the main cause of this is the fact that
there is a dampening of the flow caused by the action of the Lorentz force.

Acknowledgements
The authors would like to thank the Brazilian Coordination of Superior Level Staff Improvement (CAPES) for
the financial support throughout the development of this work.

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