Effects of variable slip length over induced electric potentials in microchannels

The exploration of new alternatives in renewable energy has embraced several branches: from solar, hydroelectric, and wind energy on the macro scale, up to those more recently such as to harvest power from some electrokinetic phenomenon, for instance, the well-known electric generation by means of the induced streaming potential process. This electric potential is found in micro and nano scales, where the physicochemical properties of surfaces have fundamental importance, such as the slippage effect. Most works oriented to determine induced electric potential in micro and nanochannels assume constant Navier slip length, which quantifies the hydrophobic degree of the surface. This assumption could overestimate the performance of the so-called electrokinetic batteries. For this reason, it is necessary to enlarge the analysis of variable slip over electrokinetics in micro and nanoscale. In the present analysis, we develop an analytical procedure to estimate the electric potentials induced in a microchannel by a flowing laminar solution due to osmotic gradients, whose configuration is better known as an electrokinetic forward osmosis battery. Assuming a non-linear dependence of Navier slip as a function of the pressure, we obtain regular perturbative series solutions for velocity, pressure, and the streaming potential. For this purpose, we use the dimensionless decay parameter ϵ̃, which measures the decrease of slippage due to the pressure at the wall as the perturbation parameter. We conclude that the velocity field has not only a longitudinal component if not also a transverse component normal to the surfaces as a consequence of the variable slippage, showing that the classical assumption of fully developed flow is no longer possible. In addition, our data shows that constant slip overpredicts the electric potentials up to 5% in comparison with the case of exponential slip. Finally, the results show that induced electric potentials are benefited from variable slippage; however, the influence of the decay parameter that appears in the boundary condition (32) associated with the variable slip decreases the induced potentials.


Introduction
The crescent knowledge about electrokinetic phenomena has fostered several branches of interest such as those focused on microdevices become more and more specialized.An example of these disciplines is oriented to renewal energies, whose main interest is harvesting electric potential through the well-known streaming potential, that converts the flow energy into electric energy.In view of this phenomenon being intricately linked to the field of velocity, boundary conditions play a relevant role, especially those that are concerned with slippage.It is well known that in the micro and nanoscale, the non-slip condition may no longer be sustained as a consequence of the properties of the surface since the fluid interacts with the surface giving rise to chemical activity.For this reason, several studies have been primarily focused on this topic.Experimental and theoretical works such as those developed by Tretheway and Meinhart [1] examined the possible mechanism for the measured fluid slip for water over hydrophobic surfaces, considering a two-phase model at the surfaces of two parallel plates.Their results suggested that the apparent fluid slip could surge from the presence of nanobubbles or low-density layers at surfaces in contact with the fluid.Ramos [2] carried out an asymptotic analysis for laminar flows in a rectangular microchannel where the slippage depends on the axial pressure and its respective gradient, altogether with an analytic thermal analysis considering a regime fully developed in microchannels for the isothermal case.This study determined that pressure drop along the channel is nonlinear and depends on the values of the pressures at the beginning and to the end of the microchannel for slip values that decrease progressively with the pressure.
Panaseti et al [3] studied the effect of pressure-dependent slip at the wall for Poiseuille flows over a channel and a tube.They obtained analytical results by applying perturbation series through a linearized exponential slip function to obtain the velocity field and the pressure.These results showed that by increasing the slip coefficient, the effect of the pressure decreases, and the slip velocity becomes uniform along the surfaces.Papanikolaou et al [4] employed a fractal model for roughness to study liquid argon confined in solid walls using numerical simulations.Their findings showed that the roughness disturbs the flow making it three-dimensional and that the slip length decays slowly as a function of the depth of roughness in the two-dimensional model; however, it is not well accurate in rough nanochannels.Besides their simulations showed that the slip length decreases exponentially by increasing the average roughness depth.
Wang and Wu [5] presented an analytic study for the influence of slip and viscoelectric effect on a microchannel with a pressure-driven flow.They developed their study in a two-dimensional microchannel made of PDMS (Polydimethylsiloxane) and glass, where it was found that the wall slip enhances the induced streaming potential and increases the presence of the viscoelectric effect.Malekidelarestaqi et al [6] studied numerically a pressure-driven flow in a finite microchannel with charged superhydrophobic walls.The radii of the microchannel were 1 mm and m 0.5 m.The study was carried out using finite element analysis through COMSOL 5.2a.They demonstrated that slip on walls affect streaming potential, and streaming current among other electrokinetic phenomena.At increasing the slip length from 0 to 144 nm the energy conversion increased up to a factor of 3.4 in comparison to a microchannel with no-slip conditions.Similarly, Sánchez and Méndez [7] conducted a theoretical study for power generation in microchannels employing osmotic gradients.They conclude that hydrophobic surfaces promote increasing electric potentials.Nonetheless, their results showed a saturation phenomenon that limits the magnitude of the induced potentials for high wall potentials as a consequence of the viscoelectric effect.In parallel, a complementary analysis of the previous work using the Debye-Hückel approximation with asymmetric wall potentials and slippage exhibited similar results [8].The work developed by Liu et al (2023) explored the energy conversion subject to slip effects and surface charge in nanochannels where it was obtained that there is a link between the boundary slip and the surface charge density: if the slippage increases the charge density decreases and vice versa.Besides, the slippage proved to enhance the power output efficiency by up to 30%.Such a study suggests that constant boundary slip conditions could overestimate the performance of the energy conversion.A similar study carried out by Li et al [9] for energy conversion in nanochannels compared three different cases: classical non-slip, independent slip, and surface charge-dependent slip.Again, the results indicated that slippage enhances power generation, reaching up to 13 times higher maximum power output than in non-slip conditions.However, there were significant differences between independent slippage and dependent charge slippage, showing that the first one is not suitable for practical applications.
Recently, novel alternative cells batteries have been studied based on different mechanisms.In this direction, the clearest contributions come from the analysis, Husam et al [10,11] studied numerically the performance of a ingenious design of an efficient air-cooling system for lithium-ion batteries.The study shows the reduction of the operational temperatures of the batteries under different coolant flow rates, meanwhile in a posterior analysis, the previous authors extended a CFD simulation to study the influence of a novel design of an efficient Air-cooling system to improve the performance of lithium-ion batteries.Banerjee et al [12] carried out a study for the heat transfer in microchannels for both electroosmotic and pressure-driven flow where it was considered the interfacial slip and the slip-dependent wall potential, where it was assumed a fully developed hydrodynamic and thermally flow.Also, they obtained analytical expressions for the induced electric potentials inside the electric double layer (EDL).Their results revealed that the slip dependence on wall surface potential has a great influence on Joule heating, slip length and viscous dissipation, mainly.In addition, the Nusselt number on slippage is very sensitive to changes in the Debye length, in other words, in the size of the EDL.Independently, Siva et al [13] developed a theoretical study on electroosmotic and electromagnetic flow focused on the hydrodynamic, thermal transport, and entropy generation in a rectangular microchannel considering a slip length as a function of the zeta potential (wall potential).Analytical expressions for velocity and temperature field and entropy generation were obtained.The results showed that slip-dependent zeta potential has a strong influence on velocity field, heat transfer, and entropy generation, in addition, high Brinkman numbers tend to increase the temperature for variable slip than constant slip.
The present work aims to develop a theoretical analysis of the induced electric potential between the extremes of a rectangular single microchannel when the electrolyte is prompted to flow due to osmotic gradients (that is, forward osmosis) in conjunction with an exponential relation for slippage that depends on the pressure inside the microchannel.In addition, the study makes use of the Debye-Hückel approximation for EDL potential for regimes of low wall potentials.Finally, the mathematical model was solved with a regular perturbation method due to the boundary conditions for slippage, which in turn allowed us to calculate all the variables of interest, such as velocity, pressure, and streaming potential.The results concerning streaming potential are shown in terms of average values for practical purposes.In addition, we provide information about the overestimation of the constant slip in comparison with the nonlinear slip.

Physical model
Figure 1 shows a scheme of a rectangular microchannel constructed with two parallel plates made of silica surfaces.The entire system is filled with an aqueous solution (water with sodium chloride), that promotes that the surfaces acquire a charge (assumed as negative) and therefore attract counterions, giving as a result a normal distribution with respect to the planar walls, namely, the electric double layer (EDL).At the same time, an asymmetric porous membrane is placed horizontally whose fundamental function is working as a driving mechanism for a flow parallel to the microchannel because between both sides of the membrane exists a finite concentration gradient of the electrolyte, which in turn provokes a depletion of counter ions upstream and an accumulation downstream, originating the well-known streaming potential [14].In addition, the surfaces of the plates are such that their respective slip coefficients are functions of the induced pressure.This last condition will be written as a boundary condition, in the following section 2.3.The domain of the microchannel is considered bidimensional due to the dimensions proposed, where the depth is much larger than the width and the length; that is,  w l and  w H, being the width lower than the length  ( ) l H .

Electric double-layer
The EDL potential y is mathematically described by the Poisson-Boltzmann equation for a symmetric electrolyte as follows [15]: where e, r , f C osD,b and z are the permittivity, the free charge density, the molar concentration, and the valence of the suspended in the electrolyte solution, respectively.Besides F is the Faraday constant, R is the universal gas constant, and T is the absolute temperature.For low surface potentials, that is,  y / RT zF equation (1) can be linearized as follows: y r e where is defined the Debye length defined as this length quantifies the span of the EDL near the surfaces.The linearization given in equation ( 2) is better known as the Debye-Hückel approximation.Solving equation (2) with the boundary conditions for a parallel flat surface y z  = ( ) l , the EDL potential is given as follows [14],

Nernst-Planck equation
The electric potential in stationary conditions inside the microchannel is compound by the superposition of the streaming potential and the potential of the EDL, that is, x y , [14,16].Both potentials are related through the Nernst-Planck equations for the longitudinal and transversal components [17]:

Where j i x
, and j i y , are the longitudinal and the transversal components of the ionic flux  j i of the i th specie, respectively; besides D , i z i and C i are the corresponding coefficient of mass diffusion, the valence, and the concentration of the i th specie diluted in the electrolyte.Finally, u and v are the longitudinal and transversal velocity components, respectively.The components of the ionic flux produce an electric current vector I given by   Since  H l, we neglect the longitudinal concentration variations.Also, in steady conditions, the net current is zero due to the electroneutrality properties at the extremes of the electrokinetic region, giving as a result, The molar concentration C i is computed from the differential equation obtained from equation (9) for the i th component, where we make use of the conditions just in the centre of the microchannel , and For a symmetric electrolyte, we have = -= z z z, Consequently, through direct integration of equation (8) along the transversal direction we obtain: where r = å = Fz C .
Invoking the Debye-Hückel approximation equation (11) adopts the following form: In the last expression, s is the electrolyte conductivity in the bulk.Besides, from equation (2) we rewrite the free charge density as a function of the electric double layer potential, that is, Equation ( 13) is fundamental to connecting the velocity field inside the microchannel and the induced potentials.The first term in equation ( 13) represents the electromigration due to the flow dragging the free ions in the EDL, whereas the second term is the conduction current that arises from the downstream accumulation of charges which induces an electric field in the upstream direction.In the following section, we derive the Navier-Stokes equations for the electric body force that accounts for the streaming potential and the EDL potential.

Navier-Stokes equation and mass conservation
With the aid of equations (4, 13) has two unknowns: the velocity component u and the streaming potential f, therefore, we must provide additional information that can be obtained from the hydrodynamic equations for the electrolyte solution.Thus, these equations that describe the behaviour of a non-compressible Newtonian fluid in a cartesian system are and Where r and m are the density and the dynamic viscosity of the electrolyte, respectively; and P is the pressure field in the microchannel.If the electric field is given by  [18], equations (15) and ( 16) have the corresponding equations and Since we obtain a flow provided by the asymmetric membrane, the following volumetric flow rate equation must hold inside the microchannel, i.e.: where A is the longitudinal area of the microchannel and R is the region to integrate normal to the flux.Therefore, the water flux provided by the osmotic membrane is Solving the set of equations compound by equations ( 13), ( 14), ( 17), ( 18) and (20) require the following boundary conditions: where b are the Navier slip lengths associated with lower and upper surfaces [5,19].The boundary conditions for the velocity field are directly derived from the equation for tangential velocity at the plates and n is the respective normal vector to the surface; being = -ˆn j for the upper plate and = ˆn j for the lower plate [20].

Slip conditions dependence on pressure
Several works suggest that velocity depends not only on shear stress but also on normal stress and several experiments indicate that the slip velocity depends on the pressure for lower pressures regimes [19], which suggests a dependence on the slip coefficient b with the pressure P at the surfaces [3, 21] as follows: where b 0 is of the order of - 10 9 to - 10 6 meters [19,22,23], and c is called the exponential decay parameter which measures the decrease of slippage due to the pressure at the wall.For the present work, we must rewrite equation (22) using the aforementioned change of variable proposed by Ajdari [18], giving as a result:

Dimensionless equations
We propose the following dimensionless variables, based on the geometric and physical parameters in the Electrokinetic-Forward Osmosis Cell (EKFOC): From an analysis of the order of magnitude in equation ( 14), we determine the characteristic value for the transversal velocity, that is, Similarly, from equation (17) viscous and electric terms are assumed as similar, therefore, comparing such terms m ezj l and since  H lthe characteristic value for the streaming potential is given by j m ez l ~( 2 Substituting the dimensionless variables given in equation (24) into equations ( 13), ( 14), ( 17), ( 18), ( 19) and (20 In addition, from Ajdari's [18] dimensionless variable we obtain the dimensionless original pressure as The dimensionless parameters that appear in the above equations are a k x d Besides, the boundary conditions that were written by equation ( 21) are rewritten as , 1 0, 0 0 and 0 1, 32 Solving equations (25)-(30) requires calculating the dimensionless values given in equations (32) and (33).Table 1 shows a set of values that have been reported by Sánchez et al [8], Jimenez Bolaños and Vernescu [19], and Henry et al [23].With this information and the Re previously reported by Sánchez and Méndez [7] and Sánchez et al [8] for an asymmetric membrane, the dimensionless parameters given in equations (31) and (33) are shown in table 2.
Making use of the values in table 2, equations ( 25) and (29) are simplified as follows , 1 0, 0 0 and 0 1. 39 Since    1 we can use a truncated Taylor series for each boundary condition for U, that is, , 1.

X, 1
As a consequence, the boundary conditions are finally written as

Perturbation series solution
The dimensionless decay parameter   defined in equation (33), regularly reaches values from - 10 2 up to - 10 1 [3,21].These characteristic permits the application of a regular perturbation method based on   .Considering that the boundary conditions for the longitudinal velocity U depend on the small parameter  ~-  10 2 we propose the following series to solve the regular perturbation problem, Substituting the perturbation series above mentioned into equations (34), ( 35), (37), and (38) together with the boundary conditions given in equation (40) we obtain the following systems of equations for the zeroth and first order: Zeroth order , 1 0, 0 1, and 0 0. 46 , 1 0, 0 0, and 0 0. 51

Zeroth-order solution
The solution of the equations (42)-( 46) has been already reported in the work developed by Sánchez et al [8] for both symmetric potentials and slip lengths in the surfaces and low wall potentials.Such solutions were computed from the change of variable d dX 0 0 which permits obtaining an ordinary Fredholm integrodifferential equation for ( ) F Y , which in turn, allows obtain U , 0 P 0 and F̅ .0 These results are properly rewritten for our study case as follows:

D
and Besides we define ax P = D 2 for simplicity.These solutions allow to convey the information from the zeroth order to the solution of the first order specifically in the boundary conditions for U 1 as was stated in equation (51).

First-order solution
The solution of the equations (47)-( 51) is developed in the following way.Let us consider equation (48), which is a Fredholm integrodifferential partial equation, where the integral that appears can be expressed as After integrating twice in the transverse direction, equation (61) turns into with the respective boundary conditions given in equation (51) for U , 1 we determine that Then, the first-order longitudinal velocity is determined by substituting equations (63) and (64) into equation (62) and after rearranging terms we obtain,

D
With equations (66) and (67) the first-order longitudinal velocity (equation ( 65)) is completely determined.Besides, after using the boundary condition P

D D D D D
The first-order transverse velocity is computed from equation (47), which is differentiated with respect to Y once, thus, Making use of the respective boundary conditions given in equation (51 We must point out that V 1 has no dependence on the longitudinal direction because of the partial differentiation with respect to X carried out in equation (72).Finally, we must notice that the integral that appears in equation (50) is indeed ( ) f X , previously given in equation (65).Therefore, equation (50) is rewritten as follows,

D D D D D
Such a differential equation for F̅ 1 is solved using the boundary condition F = ̅ ( ) 0 0, 1 therefore, its solution is given as:

Results
The solutions of the zeroth and first order finally provide the overall solution using the perturbation series given in equation (41).Nonetheless, before proceeding with the evaluation of thesesolutions, we must highlight that the perturbation series for the P is used to derive the dimensionless original pressure P R from equation (30), which gives the result: 6 In addition, we compute the average dimensionless streaming potential by integrating its respective perturbation series, thus, Av 0 1 0 1 in this way, we obtain a more useful expression for quantifying the performance of the micro electrokinetic battery.It is important to mention the implicit dependence of U, V , P R and F̅ Av solutions in terms k, d, b, P D and P E besides, of course, the coordinate variables X and Y .
Figure 2 shows the longitudinal velocity inside the microchannel.Notice that the maximum velocity is reached in the centre of the microchannel whereas the flow near the plates differs from the non-slip conditions.We must remark, that despite the apparent symmetry on the component U, this is indeed a non-full developed flow due to slight variations in the transverse direction which are not appreciated in the plot.This is noticeable in figure 3 where higher the magnitude of   , themore the presence of the transverse velocity V .These findings are fundamental to understanding the flow on the microscale, which is far from the classical assumption of developed flow.The latter suggests that modelling microdevices with variable surface properties should take into account not only the longitudinal velocity but also the transverse component.
Figure 4 shows the average streaming potential as a function of the   parameter, which quantifies the tendency of the surface to decrease the slippage.This correlation exhibits a decreasing tendency for progressively higher values of   .The variation of the slenderness ratio between the width of the microchannel and the EDL, i.e., d, suggests that microchannels with lower sizes of EDL tend to increase the power output altogether with low decay slip parameters   .This can be easier appreciated in figure 5 where this behaviour is more evident for d > 10, which implies that microchannels with sizes several times larger than EDL promote better performance at inducing electric potential; on the contrary, d ~1 makes visible the independence with respect to the parameter   .This is critical for microdevices, especially for electrokinetic forward osmosis batteries, where the yield is increased through the superposition of several microchannels, for example, the energy conversion in   porous glass driven byforward osmosis and proved experimentally by Jiao et al [24].The last results suggest whereas exponential slippage affects power generation on the microscale, the nanoscale is unaffected by this decreasing slippage.Figures 6 and 7 illustrate that streaming potential is enhanced by greater values of the dimensionless slippage b.Conversely, the dimensionless decay parameter counteracts the induced electric potential, making this performance more markedly for surfaces with higher dimensionless slippage.These results are transcendental for microdevices, where hydrophobic surfaces are related to low chemical activity, such as those made of fumed silica or endowed with physicochemical treatments [25,26].
Figure 8 depicts the dependency of the dimensionless streaming potential as a function of the parameter k, which quantifies the induction of the electric potentials.Figures 8(a) and (b) show an increasing tendency of the voltages for k < 1, where higher magnitudes of   provoke a reduction of the induced potentials.Nonetheless, for certain values of k (about 25.6), this behaviour is reverted, which may indicate that significant values of this parameter could offset the not desired effect of the decreasing slip condition along the surfaces.
Figure 9 shows the dimensionless streaming potential as a function of the parameter P .D It is appreciated a decreasing tendency for P > 1 D as   adopts greater values, that is, for surface conditions where the slippage decreases along the microchannel.Nonetheless, it arises a surprising behaviour for P < 1 D which are related to low concentration regimes in the bulk, where it may be desirable to have conditions where slippage goes down rapidly.These conditions are relatively easy to reach due to some process of fabrication, which inherently endows the surface of irregularities.Such surfaces are found in microdevices made of silica, which is normallyhydrophilic if it has not undergone physicochemical treatments.
Figure 10 depicts the percentage of overestimation F OE Av at calculating F Av when using constant slip versus variable slip.For all values of the decay parameter   , the constant slip always overpredicts the induced potentials in the microchannel ranging such a deviation from 0.594 to 5.002%.These results agree with those reported by Liu et al [27], showing that a variable slip is more suitable for forecasting electric potentials in microchannels.
Finally, figure 11 shows the dimensionless physical pressure P R in the entire domain of the electrokinetic region inside the microchannel.As was previously reported by Sánchez et al [8], the physical pressure remains    quasi-constant, with a scarce longitudinal pressure loss and a surge of the electrostatic pressure at the surfaces due to the counterions accumulation in the EDL.Although the parameter   affects the longitudinal pressure, these are several orders of magnitude lower than the pressure on the zeroth order, giving as a result an apparent imperceptible dependence on   .

Conclusions
In the present study, the principal objective was to determine the streaming potential induced in a microchannel prompted by the flow provided by an asymmetric membrane due to the forward osmosis phenomena.The surfaces that compound the microchannel have a slippage that depends exponentially on the pressure inside the microchannel.Therefore, it is developed an analytical model that predicts the velocity and pressure fields, besides the induced electric potential over all the electrokinetic region, whose solutions were obtained through regular perturbation series using as perturbation parameter the dimensionless decay parameter   .The zeroth and first-order solutions were obtained as solutions of Fredholm integrodifferential equations, each one of them being exact.
In summary, the results derived from the analytical solutions are the following: • The velocity field has longitudinal and transverse components because of the variable slippage along all the surfaces of the microchannel, showing that the classical assumption of fully developed flow is not appropriate.
• Constant slip always overpredicts the average induced potentials in the microchannel ranging from 0.594 to 5.002%.
• High values of b and d, that is, surfaces with high slip conditions (hydrophobic) and small sizes of EDL promote an increment in the induced streaming potential.
• The streaming potential harvested mainly decreases for progressive higher values of the exponential decay   parameter present in the slippage conditions, being this behaviour reverted for values of k 25.6  and P < 1.  • The pressure field inside the electrokinetic region of the microchannel is compound by the superposition of the longitudinal pressure (produced by the flowing electrolyte driven by the forward osmosis asymmetric membrane) and the electrostatic pressure (originated by the charge accumulation in the EDL).The effect of the nonlinear slip over the pressure field is negligible.
The results here described can be used to improve the design of microelectromechanical systems, especially those based on power generation via streaming potential, namely those that take advantage of saline gradients as in the case of electrokinetic forward osmosis batteries, among others.Besides, additional theoretical and numerical research is necessary for exploring other types of dependencies on slip lengths, high wall potential surfaces (avoiding the Debye-Hückel approximation of the EDL) and reducing the size of microchannels to nanochannels.

Figure 1 .
Figure 1.Microchannel with an electrokinetic region compound of parallel plates made of silica (yellow surfaces) and an asymmetric porous membrane (green) which prompts a downstream flow arising the streaming potential at the beginning and the end of the microchannel.

1 2 2
where the dimensionless Navier length for both plates is b = /

2 1 1
Notice that equation (36) declares that ¶P ¶ = / Y 0, then ¶P ¶ » P / / X d dX, therefore, all the variations of P are considered only in the longitudinal direction.Consequently, the boundary conditions in equation (32) are modified as follows,
f X are unknown at this stage.Nonetheless, both are simultaneously computed by substituting equation (65) into equations (49) and (60), giving as a result,

Figure 3 .
Figure 3. Dimensionless transverse velocity V as a function of Y for several values of the parameter   , with = k 0.00285207, d = 23.2419,P = 320D

Figure 4 .
Figure 4. Dimensionless average streaming potential as a function of the parameter   , for several values of the parameter d, with = k 0.00285207, P = 320D

Figure 5 .
Figure 5. Dimensionless average streaming potential as a function of the parameter   .The profiles show several values of the parameter d, with = k 0.00285207, P = 320 D

Figure 6 .
Figure 6.Dimensionless average streaming potential as a function of the parameter   , the profiles show several values of the parameter b, with = k 0.00285207, P = 320 D

Figure 7 .
Figure 7. Dimensionless average streaming potential as a function of the parameter b.The profiles show several values of the parameter   , with = k 0.00285207, P = 320 D

Figure 8 .
Figure 8. Dimensionless average streaming potential as a function of the parameter b the profiles show several values of the parameter   .with = k 0.00285207, P = 320 D and d = 23.24.Subfigure (a)show the range of values for k up to 1, subfigure (b) provides a zoom in to make more evident the variation between the different solutions.Finally, subfigure (c) depicts that the tendency of the dimensionless streaming potential reverts the increasing tendency respect to the parameter   .

Figure 9 .
Figure 9. Dimensionless average streaming potential as a function of the parameter P D for several values of the parameter   , with = k 0.00285207, d = 23.24 and b = 0.1.

Figure 10 .
Figure 10.Percentage of overestimation OE at calculating F Av as a function of the parameter b.The profiles show several values of the parameter   , with = k 0.00285207, P = 320 D

Table 1 .
Physical and geometrical parameters.