Electromagnetohydrodynamic flow through a periodically grooved channel

This paper harnesses the spectral approach to study electromagnetohydrodynamics (EMHD) flow through an undulating 2D microchannel for a Newtonian fluid. The present study aims to investigate EMHD flow through a periodically patterned channel for large values of corrugation amplitude (surface roughness) relative to the channel height, Hartmann number and a wide range of wavelengths. A mathematical model is developed for the hydraulic permeability and the velocity field using the Fourier approach. By imposing a small pattern amplitude constraint, asymptotic analysis was employed to investigate the model presented in the literature. In the present study, hydraulic permeability is shown to strongly depend on the Hartmann number, pattern amplitude and wavelength. The spectral method predicts a monotonic decrease in hydraulic permeability irrespective of pattern amplitude at a small dimensionless wavelength (λ = 1) and Hartmann number (Ha < 1). At large wavelengths (λ > 3), the hydraulic permeability demonstrates an augmentation corresponding to the pattern amplitude. At intermediate wavelength ( 2.5<λ<3 ), the permeability of the channel firstly decreases and then increases with pattern amplitude. For large Hartmann numbers (Ha≫1) , the prediction from the spectral model exhibits a similar trend as a function of wavelength. Across a range of wavelengths, the spectral model captured the permeability minima for various large Hartmann number values. The spectral model is significantly faster than the finite-element-based numerical simulation, with computational efficiency ranging from 150 to 250 times higher. The existence of a limited pattern amplitude and Hartmann number for small-large wavelength patterns, at which the permeability of the channel is minimized, is one example of a prediction from the spectral model that is beyond the resolution of currently accessible asymptotic theories.

One of the crucial microfluidic systems that produce a continuous flow without any moving parts is the EMHD micropump.EMHD refers to fluid flow and heat transfer effects from the interaction of electric and magnetic fields with a flowing fluid.
When an electric field is supplied to a fluid in an EMHD flow, the interaction of the electric field and magnetic field causes the formation of a Lorentz body force field.This Lorentz force acts as a pumping source in EMHD micropumps.EMHD devices have various applications, such as fluid pumping [3], flow control in fluidic networks, fluid stirring and mixing [1,5,6], and could also be used for actively controlled convection of liquid coolants in electronics cooling.The latter implementation would be beneficial in the context of waste heat recuperation from large-scale electronics systems such as data centers, where enhanced control over local convection coefficients could be used to maintain a high coolant outlet temperature under variable heat load conditions.The literature can be classified based on the orientation of the channel and driving force.At the microscale level, the fluid flow can be propelled by various forces, such as external pressure [7][8][9][10][11], surface tension [12,13], Lorentz force [14][15][16][17], and electroosmosis force [18][19][20].The present study focused on fluid flow through a sinusoidal grooved channel, and the Lorentz and viscous forces are considered as driving forces.
In the year 1937, Hartmann [21] experimentally investigated the flow behavior of a conducting liquid between two parallel plates under a constant magnetic field.The applied magnetic field results in the flattening of velocity profiles in the central channel region, induced by the development of a drag force.Additionally, a magnetic boundary layer was observed in the boundary region of the channel.Duwairi and Abdullah [22] studied MHD flow through a rectangular channel micropump and obtained the velocity and temperature distribution numerically and analytically.They also analyzed the effects of different parameters on the fluid transport.However, the study did not consider the effects of viscous dissipation.On the other hand, Chakraborty et al [23] analyzed the thermal characteristics of EMHD flows in a narrow channel while considering the electric double layer (EDL) effect.In this case, the combination of the Lorentz force and the electric body force propelled the flow.Several studies have been conducted to analyze the heat transfer characteristics of EMHD by considering the effects of Joule heating and viscous dissipation.For instance, Jian [24] studied the transient velocity distribution of MHD flow in microscale parallel slit plates, taking into account the electroosmotic effect.The steady part of the velocity distribution was utilized to analyze heat transfer and entropy generation graphically.In addition, many researchers have carried out heat transfer studies in conjunction with electroosmotic effects in the presence of a magnetic field [25].The EMHD micropump operates by harnessing the Lorenz force, which is generated through the interaction of electric currents imposed across a (horizontal) channel filled with an electrically conducting liquid and a perpendicular (vertical) magnetic field.Generally, the magnetic field can be generated using a permanent magnet or an electromagnet.The feasibility of EMHD micropumps has been demonstrated using both direct current and alternating current (AC) electric and magnetic fields [26,27].Tso and Sundaravadivelu [28] studied the influence of electromagnetic fields on surface tensiondriven flow in a parallel plate microchannel.After conducting both analytical and numerical calculations, it was determined that the application of an electromagnetic field can significantly increase the fluid flow rate between parallel plates, as compared to when there is no electromagnetic field present.The effects were thoroughly investigated and analyzed.Rivero and Cuevas [29] explored the consequences of considering fluid/wall slippage in micropumps under electromagnetic fields using both analytical and numerical methods.It was found that the slip condition resulted in a significant increase in the flow rate.Enright et al [30] examined the performance of an MHD pump and found apparent hydrodynamic and thermal slip on its major walls, which can be achieved utilizing a micro/nano-structured surface to reduce pressure losses from friction.Chakraborty and Paul [31] carried out a comprehensive study of the influences of EMHD forces in controlling fluid flow through parallel plate rectangular microchannels.These forces were induced by a combination of two electric fields in horizontal directions and an externally imposed transverse magnetic field.Numerous researchers have affirmed [32][33][34] that flows within small conduits can be more efficiently controlled by harnessing Lorentz forces through the application of low-magnetic fields.It has been demonstrated [31,35] that the 3-dimensional nature of electromagnetic forces can result in improved flow rates and enhanced energy conversion efficiency.Conversely, a limited number of studies on EMHD-driven capillary filling [28,36] have illustrated that the filling time can be precisely regulated by varying the strength of the applied magnetic field.
While the aforementioned studies have focused on smooth microchannels, it is essential to consider the presence of roughness on channel walls in practical applications.Wall roughness can occur during the fabrication process or due to the adsorption of other species, such as macromolecules.The examination of the impact of wall roughness on laminar flow has been a subject of interest since the 1970 s, and over the years, several analytical and numerical methods have been introduced for this purpose.References [37][38][39][40] investigated the stokes flow problem of viscous fluid through a corrugated channel.Chu [41] utilized the perturbation method to study the flow with a small Knudsen number inside a circular microtube with a corrugated wall.The effects of wavy roughness within a circular microtube, considering both slip and noslip cases, were explored in previous studies [42,43].Yoon et al [44] examined the interactions between wavy surface shape, magnetic field, and heat flux from the disc surface while studying magnetohydrodynamic flow over a rapidly rotating axisymmetric wavy disk.Over the past few years, researchers have extensively studied the flow behavior and transport characteristics of electroosmotic flow (EOF) in microchannels with various types of wall roughness, such as rectangular block shapes [45,46], wavy roughness [47], and complex wavy roughness [48,49].These studies revealed that the average velocity of EOF is lower in a rough microchannel than in a perfectly smooth one.However, in Shu's study (2010), where the channel had small amplitude sinusoidal wall corrugations and a cross-section that remained constant along the flow direction [50], it was found that for thick EDLs, the EOF could be enhanced by up to 30% for longer-wavelength corrugations when the phase difference between the wall corrugation was equal to π.
Additionally, previous literature has examined flows driven by pressure gradients in wavy microchannels and driven by the Lorentz force.The influence of wavy surface amplitude on heat transfer characteristics in MHD flow over a corrugated surface was investigated by Tashtoush and Al-Odat [51].Bujurke et al [52] analyzed the impact of surface roughness on the behavior of MHD squeeze films between two rectangular plates.The effects of surface roughness on couple stresses in MHD squeeze film lubrication between circular plates were studied theoretically [53].Buren et al [14] applied a perturbation method to analyze EMHD flow in a microscale parallel plate channel with slightly streamwise corrugated walls.They observed that the impact of corrugation on the flow diminishes with higher Hartmann numbers.In their subsequent study [15], they explored 2D EMHD flow in a similar channel with transversely corrugated walls using the perturbation method.The results revealed a consistent decrease in flow rate due to wall corrugation, regardless of the phase difference.Moreover, the phase difference became insignificant when the wave number exceeded 4. Buren et al [54] discussed the impact of combined EMHD flow in a corrugated microchannel and reported that the channel wall roughness distributes the flow and reduces the peak velocity.Li et al [16] performed three-dimensional surface corrugation on EMHD flow using two sinusoidal functions and obtained analytical solutions for the governing equations with the perturbation method.They highlighted that increasing the strength of a magnetic field can result in a maximum flow rate.In a recent investigation, Okechi et al [17] used the perturbation method to study the flow of an electrically conductive fluid through a curved channel with wavy boundaries.Their study highlighted that the influence of wavy boundaries on the flow diminishes as the flow Hartmann number increases.The transport of flow and heat in microchannels with waviness has been extensively investigated through numerical simulations [55,56] and experiments, as it has wide relevance in cooling systems.Through computational techniques [57], Cakir and Akturk [58] examined the heat transfer properties of nanofluids in a microchannel with waviness.Their study focused on the thermal conductivity and heat transfer coefficients of nanofluid flow at various Reynolds numbers and nanoparticle ratios.Meanwhile, Tiwari and Moharana [59] conducted a three-dimensional numerical investigation on two-phase boiling flow in a microchannel with waviness.They compared the behavior of boiling instability and bubbles in single-and twophase flows between microchannels with straight and wavy walls.
Recently, Ma et al [60] investigated the impact of small random transverse wall roughness (sawtooth ripple, rectangular ripple, triangular ripple) on EMHD flow in a microchannel, utilizing the perturbation method based upon stationary random function theory.Most researchers have used domain perturbation [14,16,17,61,62] and lubrication methods [19, 63,64] to study flow through a wavy channel, but these approaches have certain limitations.For example, the length of a channel should be greater than its height, and the ratio between pattern amplitude and channel length should be small.Buren et al [14,15,54] studied EMHD flow across a wavy surface using the domain perturbation method and demonstrated the flow rate and velocity profile up to second order.Okechi et al [17] examined the influence of surface roughness in the presence of an applied magnetic field on EMHD flow within a sinusoidal corrugated channel, employing the domain perturbation method.The findings of their study indicate that the flow rate diminishes as the wavenumber increases.Moreover, for a sufficiently large wavenumber and Hartmann number, the phase difference between the wavy curved boundaries becomes inconsequential to the flow, as concluded in the article.The existing models for EMHD flow are accurate in a small number of perturbation parameters.Dewangan and Datta [39] have recently studied pressure-driven flow in a sinusoidal microchannel using the grid-free spectral method.They have developed a mathematical model for the hydraulic permeability, which is accurate for large dimensionless pattern amplitude (α ≃ 1).Recently, Dewangan [40] studied the effect of roughness for a sinusoidal channel at the Stokes limit using the grid-free spectral method and developed a theoretical model for hydraulic permeability, which is accurate at large dimensionless pattern amplitude (α ≃ 1).
The aforementioned study motivates the use of the gridfree spectral method to investigate the effect of roughness on EMHD flow.Asymptotic-based mathematical models for EMHD flow are accurate within the limit of a small perturbation parameter [14].The work by Buren et al [14] gives some motivation to attempt this study.The primary objective of the present study is to develop a mathematical model for permeability that will be accurate beyond the limitations (dimensionless pattern amplitude ≃ 1 and Hartmann number ≫ 1) of the asymptotic theory.A mathematical model for permeability will be resolved, particularly when the crest of a sinusoidal wall (bottom wall) is positioned very close to the straight top wall, which has remained unresolved in the literature.The present study also aims to establish a quantitative correlation between the flow rate in a sinusoidal channel and that of a straight channel.Special cases such as the purely pressure-driven flow (where the Hartmann number tends to zero) and flow rate between hydrodynamic and electromagnetic flow limits will be analyzed to derive analytical expressions.Additionally, the influence of wall roughness or micropatterning effects will be examined.
In summary, the focus of the present study is to investigate EMHD flow through a periodically patterned topography using the spectral method.Section 2 elucidates the mathematical modeling including the governing boundary conditions and the physical paradigm.It follows that the hydraulic permeability depends on three dimensionless parameters: Hartmann number (Ha), pattern amplitude (α) and wavelength (λ).Additionally, the spectral method is explained and used to calculate the velocity profile and hydraulic permeability.Section 3 presents some special cases, such as the asymptotic prediction of EMHD flow (ϵ → 0), pressure-driven flow, and Ha → 0. Section 4 explains the numerical simulation methodology.Section 5 compares results from the present spectral model to numerical simulation and a model previously described in the literature [14].In comparison to asymptotic methods and the spectrally accurate grid-free semi-analytical approach discussed in the results section of the current manuscript, the numerical approaches require significant computational cost, which might have benefited from a closed-form approach.A comparison of the computational time between the numerical study and the spectral model is discussed in this section.However, computational predictions are essential for assessing the numerical accuracy of asymptotic predictions.Numerical validation based on finite-element simulations is considered in the present work.The prediction of the spectral model is found to agree well with the numerical results, whereas, the literature model is accurate for small values of pattern amplitude (α ≃ 0.1).Finally, the key conclusions and scope for future work are discussed in section 6.Some appendices are included with supplementary information for the interested reader.

Mathematical modeling
This study considers fluid flow in a grooved micro/nanochannel.The orientation of the channel is defined as the top and bottom walls being flat and corrugated surfaces, respectively.The flat wall is located at y = h and a sinusoidal bottom wall is considered as y = a cos , where the period-averaged (mean) height of the channel is of height equals h. a and L are the amplitude and wavelength of the topographic pattern.The configuration and orientation of the channel and coordinate system are shown in figure 1.In microfluidics, this type of microchannel is often preferred due to advantages such as its optical access and alignment [65].A noslip velocity boundary condition is assumed at both walls.The direction of the applied pressure gradient is along the z-axis (i.e.longitudinal direction), and the flow is considered fully developed.The following key assumptions are considered for the theoretical modeling: (i) the fluid properties are constant, including density (ρ), viscosity (µ), etc (ii) incompressible, steady-state, Stokes flow is assumed with Newtonian fluid properties throughout this grooved microchannel.(iii) The pressure gradient ( ∂p ∂z ) along the z-axis is constant (and negative), which propels the fluid in a positive direction.(iv) The magnetic field B is constant, which is applied along the y-axis as shown in figure 1.The next section explains the physical paradigm and key governing equations.

Physical paradigm and governing equations
In the presence of an electric current field in the direction, there are two propulsive forces driving this flow field: (i) the pressure gradient and (ii) the vector product J × B which results in a Lorentz body force, both of which are acting in the z-direction.The Lorenz force can be interpreted as the interaction between a y-directional magnetic field (B = Be y ) and an x-directional electric field, which results in an electric current density of J = σ(E + u × B).This creates the z-direction Lorentz force, as seen in figure 1.Here, a uniform electric field is denoted as E = Ee x , u = u(u, v, w) is the liquid velocity vector with components u, v, w in the x, y, z directions, respectively.The assumption that B = Be y is a uniform magnetic field is reasonable if the microchannel height h is small enough when compared to the size of the magnetic field source [66,67].After the initial entrance effects, the flow is fully developed in the z direction.Therefore, the streamlines are parallel, and the only non-zero velocity component is the axial velocity w in the z direction.The axial pressure-gradient p z is a prescribed negative-valued constant; p x and p y are zero.With the aforementioned assumptions, the Navier-Stokes equation is simplified, and the viscous force is balanced by applied pressure force and magnetic force at the Stokes flow limit.Because of periodicity, one periodic cell of the channel is sufficient for analysis.The dimensional governing momentum equation can be expressed as: where p is the pressure of the liquid.The no-slip boundary conditions at the top and bottom walls are, respectively: Enforcing the aforementioned non-dimensionalisation scheme onto the governing equations (i.e.equations ( 1), (2a) and (2b)), the following revised forms are obtained: where H = 2π h L = 2π λ , where λ = L/h is used to represent the relationship between the pattern wavelength and the mean channel height, ϵ = 2π a L is a function that calculates the pattern amplitude and wave number.The non-dimensional parameter which appears in the governing equation ( 3) is defined as Ha Ha is the Hartmann number indicating the strength of the applied magnetic field [14,28,31,68,69].In a simplified way, it represents the ratio of the Lorentz force to viscous force.The next section applies the spectral method to develop a mathematical formulation for the velocity field (W) and flow rate (Q).

The spectral approach
This study assumes a velocity profile in the form of a Fourier series expansion (equation ( 5)).Because of the periodicity of the patterns, the Fourier series will prove to be very useful in studying both confined and unconfined flows of interest.
Upon substituting equation ( 5) into equation ( 3), the coefficients of cos (nX) (n ⩾ 0) may be collected and equated to zero to find ordinary differential equations for F 0 (Y) and F n (Y).The no-slip boundary condition at Y = H given by equation (4a) requires that F 0 (H) = 0 and F n (H) = 0.The resultant equations are given as: where, N = √ n 2 + Ha 2 .After substituting equations (6a) and (6b) in equation ( 5), the velocity profile can be expressed as Equation ( 7) has to satisfy the no-slip boundary condition on the sinusoidal wavy surface (equation (4b)), which gives after trigonometric simplifications For m = 0, equation ( 8) is integrated between −π to π.
Here, m belongs to the set of non-negative integers.Using the property of a modified Bessel function that I n (z) = 1 π ´π 0 e z cos(θ) cos(nθ)dθ, the resultant equation after integration can be rewritten as: For m > 0, both sides of equation ( 8) are multiplied by cos(mX), and simplified with the trigonometric identities.The resultant expressions are integrated from −π to π.The equation from this set that corresponds to each non-negative integer m is outlined as: for m > 0 ) A numerical solution was obtained for the equation system governing the n + 1 constants {C 02 , C 1n , C 2n } by truncating each summation occurring in equations ( 8) and (10) to n terms.This was done through matrix inversion by truncating the number of equations to n + 1.It is worth noting that the truncation error in this approach decreases exponentially for reasonably large n [70], because the velocity field is endlessly differentiable.

Calculation for hydraulic permeability
The spectral model, as described in equation ( 5), has the ability to predict not only global quantities such as the permeability of a channel and effective slip length of a surface, but it also locally resolves flow features such as flow rate.Equation ( 5) is evaluated along the X and Y axes to obtain the dimensionless flow rate through one periodic cell of the channel, which is denoted by Q: which evaluates to: Expressions for Ω 1 and Ω 2 are included in appendix.In the absence of a topographic pattern (ϵ = 0), the flow rate through a straight channel (Q 0 ) can be written as: The hydraulic permeability is defined here as the ratio of flow rate in the patterned channel to that of the straight channel, which is denoted by Q/Q 0 : The flow rate ratio (hydraulic permeability) between the sinusoidal and straight channel can be defined as hydraulic permeability (Q/Q 0 ), which signifies the flow rate through a sinusoidal channel with reference to the straight channel.In the present study, the analysis quantifies the hydraulic permeability as a function of nondimensional independent parameters such as pattern amplitude (ϵ), wavelength (λ) and Hartmann number (Ha).The magnitude of the pressure gradient is always less than 1 and greater than 0, respectively, to obtain a flow field in the positive z-direction.In the present work, the pressure gradient ( ∂P ∂Z ) is assumed to be equal to 0.5.

Special cases: asymptotic prediction O(ϵ 2 )
This section discussed the evaluation of asymptotic predictions (limit for ϵ small) from the literature [9,14,39].The corrugation effect is investigated for an O(ϵ 2 ) solution using the domain perturbation method by Buren et al 2014 [14].The flow is considered between sinusoidal walls.In this particular instance, we are examining the unique scenario presented in [14], where the top wall of the channel is assumed straight, while the bottom wall is undulating.This type of channel orientation is the aim of the present study.The spectral method is used to investigate the effect of wall roughness (corrugation) and the effect of magnetic strength.The closed-form expression is obtained using a few terms of the Fourier approach.Interestingly, the final expression of flow quantities such as the velocity (equation ( 5)) and flow rate (equation ( 12)) from the present study are also useful for predicting O(ϵ 2 ) and O(ϵ 4 ) solutions.Additionally, this section explores a few special cases, such as (i) flow through an unperturbed straight channel (ϵ → 0) (ii) pure pressure-driven flow (Ha → 0) (iii) the permeability ratio between EMHD to hydrodynamics for ϵ → 0, which are obtained from the spectral method.An important benefit of using a spectral approach for the current problem is its ability to be analytically reduced to closed-form results in certain special limits, which is one of the key contributions of the current study.An alternate form of the dimensionless pattern amplitude can be introduced to study confined flows, denoted by α, which can be expressed as a/h or ϵ/H.Interestingly, a parameter with an identical name was used by [9,14,39] to carry out perturbation expansions through the domain perturbation procedure.

EMHD flow in a straight (unperturbed) channel (ϵ → 0)
. Equation (14) shows the flow rate behavior with different modes of the spectral approach.In this section, we drive the mathematical model for small-ϵ.In the small values of ϵ limit, only a few terms of equation ( 5) are expected to make a significant contribution to the flow rate given by equation (14).
The asymptotic form of a modified Bessel function I n (nϵ) in the small ϵ case is (nϵ/2) n /n!.Also, all coefficients should be O (1).To obtain an asymptotic power series using the spectral approach, the coefficients Cn that are obtained through matrix inversion need to be replaced with Taylor series approximations of the appropriate order of accuracy.If the equation system is truncated at n = m = 2 and n = m = 4, the permeability determined from equation ( 11) will be of O(ϵ 2 ) and O(ϵ 4 ) accuracy, respectively.Hence, calculating equations ( 9) and ( 10) for a 4 × 4 system will result in an approximate flow rate of o(ϵ 4 ) accurate, which is not shown here.The resultant coefficients for an O(ϵ 2 ) accurate solution, when expanded in the Taylor series lead to the following asymptotic forms: The expression of C 01 and C 02 are shown in appendix. Here, The first and second terms in equation (15a) represent O(ϵ 0 ) and O(ϵ 2 ) accuracy, respectively.By contrast, O(ϵ) accuracy is obtained for C 1 with two terms of the Fourier series.The corresponding asymptotic approximation for the hydraulic permeability is ) , (17) Equation ( 17) is similar to equation ( 25) of [14], if the top wall is flat and the bottom wall is sinusoidal.The flow rate was examined using the domain perturbation approach [14], and calculation was carried out up to O(ϵ 2 ).The literature model [14] is obtained using the two terms (n = 2) in the current spectral approach (equation ( 14)).
3.2.Pressure-driven flow limit (Ha → 0) In this section, a second special case of the spectral approach is discussed in detail.The driving force for a pressure-driven flow arises solely from the pressure gradient applied along the channel, represented by P z .In the absence of any externally imposed magnetic field, the fluid flows along the z-direction due to the applied pressure gradient.Therefore, the Hartmann number (Ha) is set to zero, and the absence of magnetic force and electric current simplifies equation (3).Hence, the nondimensional governing equation for pure pressure driven flow is given by: The boundary conditions at both walls are similar to equations (4b) and (4a).Equation (19a) shows the flow rate through the patterned microchannel.Equation (19b) represents the flow rate through the straight channel, where the pattern amplitude equals zero.
The behavior of the dimensionless flow rate is obtained from equation ( 14) for small Ha and ε.Stroock et al [9] investigated the flow through a grooved surface with a no-slip boundary condition at both walls using the domain perturbation approach.For a sinusoidal pattern and small Hartmann number (Ha = 0), an O(ϵ 2 )-accurate truncation of the spectral method recovers the results of [9,39].Equation (19a) is the same as equation ( 13) in [9], when the latter is expressed in the notation of the present work. (

Flow rate ratio between EMHD to pressure-driven flow
In the past, several researchers have studied pressure-driven flow through a wavy channel [9][10][11]63] at the Stokes flow limit.The spectral model can recover earlier predictions reported in the literature.The present study has an advantage in understanding the relation between the viscous, pressure and Lorentz forces.The analysis is carried out for two different situations, defined as follows: (i) If Ha = 0, this signifies that the applied pressure force balances the viscous forces.Equations ( 5) and ( 18) are the key governing and profile equations, respectively.The hydraulic permeability ratio (χ) is calculated to understand the difference in behavior between EMHD and HD flows.Equations ( 17) and ( 20) are used to calculate the permeability ratio.The analysis is carried out for O(α 2 )-accuracy.The resultant expression can be represented as follows: (ii) If Ha = 1, the viscous force is balanced with the combination of the body force (Lorentz force) and pressure gradient.In this case, the dimensionless governing equation can be represented as follows: Equations ( 23) and ( 5) are the key governing and profile equations, respectively.Equation ( 12) calculates the flow rate for Ha = 1.Two terms are considered in the Fourier series to obtain an O(α 2 )-accurate solution, and the hydraulic permeability can be written as: )) 2H tanh For Ha = 1, the comparison of hydraulic permeability between the spectral model (equation ( 12)) and numerical simulation is discussed further in the section 5.3.

Finite-element-based numerical simulation
A numerical study is employed to assess the validity of the asymptotic prediction [14] and the current prediction from the spectral model (equation ( 14)) at the continuum scale.Finiteelement-based numerical simulations are carried out using the software COMSOL Multiphysics ® .The mean channel height (h) is taken as 100 µm.The pattern wavelength L scales as and pattern amplitude (a) is taken as 19.89 µm.The geometry and meshing of the flow domain are generated in the same software.A single periodic cell of the channel is evaluated (the simulation domain was 1×λ in width), and the numerical study is obtained using the MUMPS (Multifrontal Massively Parallel Sparse) direct solver with a stationary solver and a relative tolerance of τ = 10 A 2D triangular mesh is used to discretize the computational domain.The simulations are conducted for various normalized pattern amplitudes (α = 0.8) and wavelengths (λ = 3, 10), and a grid sensitivity test is also carried out for each case, an example of which is shown in table 1.A grid sensitivity test is conducted up to four or five decimal places of the permeability and velocity to assess whether the solution is impact of grid size-independent.The size of the computational cells was determined based on factors such as the maximum and minimum element sizes, element size growth rate, and maximum allowed iterations.After conducting several simulations, it was observed that further refining the mesh did not significantly affect the accuracy of the results.The maximum and minimum element sizes were chosen based on the grid independence test.The computer used for numerical simulations is equipped with an Intel Core(TM) i7-1185G7 processor running at 2.80 GHz.For this configuration, the finite-elementbased simulation requires approximately 30-40 s, while the spectral method completes almost two orders of magnitude faster, ranging from 0.25 to 0.35 s.In the remainder of this paper, numerical results presented in figures are denoted as 'COMSOL' or 'NS'.Figure 2 illustrates the variation of velocity (W) with different Hartmann numbers (Ha) and pattern wavelength for a large pattern amplitude (α = 0.8).The computational domain with boundary conditions and grid is depicted in panels (a) and (b).Panels (c), (e), (g), (i) of figure 2 demonstrate the velocity for the grooved channel for λ = 3, different Hartmann numbers including 0.5, 1.0, 5.0, 10.0 at α = 0.8.Panels (d), (f), (h), (j) of figure 2 show velocity for a larger wavelength (λ = 10), and the same Hartmann numbers such as 0.5, 1.0, 5.0, 10.0.The comparison is conducted to understand the effect of the Hartmann number at two different wavelengths of the channel.For λ = 3 and Ha = 10, the maximum velocity is approximately 20×10 −4 , whereas, for λ = 10 and Ha = 10, the maximum velocity is around 7×10 −3 as depicted in panels (i) and (j) of figure 2. In comparison to a small wavelength with a large Hartmann number, the velocity is higher when the wavelength is larger with a large Hartmann number, as depicted in figure 2.

Results
To understand the flow rate through a grooved microchannel, an analytical investigation was performed.The governing equations of the current problem were evaluated analytically using the spectral method.After determining the flow rate and velocity distribution, a numerical simulation study was performed to assess the limitations of the theoretical model.The validity of the analytical solution was evaluated by comparing the results for velocity profiles and flow rate with those obtained from a sinusoidal channel in the analytical work of Buren et al [14].Buren et al [14] studied the magnetohydrodynamic flow through a sinusoidal wall channel using the domain perturbation theory and conducted a theoretical calculation up to an O(ϵ 2 ) accurate solution.The literature model is accurate for small values of pattern amplitude.As expected, it becomes inaccurate for large values of pattern amplitude.Therefore, the present study used the gird-free spectral method to obtain an analytical expression for the flow rate irrespective of pattern amplitude and wavelength.Key flow quantities, including the velocity profile and flow rate, were plotted in a dimensionless form in the present study.This section compares the theoretical results from [14] and the present model with finite-element-based numerical simulation.The finite element model results are labeled as 'COMSOL' or 'NS' throughout the paper.The behavior of hydraulic permeability is studied as a function of the pattern amplitude (0 to 1), wavelength (1 to 10) and Hartmann numbers (0 to 40).In this section, the results are discussed based on key parameters such as hydraulic permeability and velocity profile concerning the corrugation amplitude (α), wavelength (λ) of the channel, and different Hartmann numbers (Ha).The current results show that the spectral model is accurate up to quite a large pattern amplitude and across a wide range of wavelength and Hartmann numbers.The dimensionless amplitude can approximate unity, i.e. the undulations on the bottom wall almost reach the top wall, which is a unique contribution of the present study.The analysis is carried out for the effect of pattern amplitude (section 5.1), the transition between roughness-to-confined effect (section 5.2), the effect of Hartmann number (section 5.3), the effect of pattern wavelength (section 5.4), the magnetohydrodynamics velocity profile (section 5.5), the transition from hydrodynamics to EMHD flow (section 5.6).Some key constants are represented in the appendix.

Effect of corrugation amplitude
This section aims to understand the effect of surface roughness (expressed here as pattern amplitude, α) as a function of wavelength (λ) and Hartmann number (Ha) for the variation of hydraulic permeability (Q/Q 0 ).The prediction from the spectral model (equation ( 14)) is compared with the asymptotic-based model (O(α 2 )) by [14] and crossvalidated with the finite-element-based numerical simulation.17)) and spectral model (equation ( 14)).The red symbols show the prediction from a finite-element-based numerical simulation (COMSOL).
The investigation is performed for two cases:, Ha < 1 and Ha > 1. Figure 3 illustrate the variation of hydraulic permeability for a small Hartmann number (Ha = 0.5).The six panels of figure 3 show the variation of the dimensionless hydraulic permeability with the dimensionless amplitude α for various pattern wavelengths.In this low Hartmann number case (Ha = 0.5), the viscous force dominates the electromagnetic force.For a small wavelength (λ = 1.0), the hydraulic permeability decreases with increasing amplitude (α).The reason for the decreasing permeability is discussed later in the section.Numerical simulations and the asymptotic model exhibit a similar behavior.This comparison shows the asymptotic model is accurate at a small value of pattern amplitude, up to α ∼ 0.1, while the prediction from the spectral model is found to be in good agreement with the numerical simulation results across a much wider range of pattern amplitude values, as shown in panel (a) of figure 3.For an intermediate wavelength (2.5 ⩽ λ ⩽ 3), the behavior of hydraulic permeability initially decreases and then increases with pattern amplitude.This phenomenon is captured by both the spectral method and the numerical simulations.By contrast, the asymptotic model predicts a monotonically decreasing trend, as illustrated in panel (b).At Ha = 0.5, this transition is only possible with 2.3 < λ < 3.22.
For large wavelength (λ = 10), the hydraulic permeability increases with patterned amplitude, which is observed by all  17)) and spectral model (equation ( 14)).The prediction from a finite-element-based numerical simulation (COMSOL) is represented in the red symbols.methods as depicted in panel (e) of figure 3. Similar behavior is also noticed for λ = 5 and λ = 7 as shown in panels (c) and (d) of figure 3.At any wavelength, the O(α 2 )-solution by [14] is accurate for small amplitude.Panel (f) shows the variation of hydraulic permeability at the different intermediate wavelengths, such as λ = 2.5, 2.75, 3.0, where the minima in permeability are captured.
For a larger Hartmann case (Ha > 1), figure 4 illustrates the variation of nondimensional flow rate with pattern amplitude for different wavelengths.Here, the magnetic force dominates over the viscous force.For a small wavelength (λ = 1) and Ha = 1.2, the permeability decreases with pattern amplitude, as shown in panel (a).A similar trend is also captured for Ha = 0.5, as shown in panel (a) of figure 3.For an intermediate wavelength (λ = 3), the flow rate initially decreases and then increases with pattern amplitude.These permeability minima are demonstrated in panels (b) and (f) of figure 4 for Ha = 1.2.Buren et al [14] does not show the existence of a minimum flow rate as α is decreased through asymptotic prediction.Instead, the asymptotic method wrongly predicts a monotonic decrease in permeability as α increases, as shown in panel (b) of figures 3 and 4. At Ha = 1.2, this inflection behavior is only observed for 2.3 < λ < 3.64.By contrast, in the case of large wavelength (λ = 5, 7, 10), permeability always increases with pattern amplitude as shown in panels (c)-(e) of figure 4. Across the range from small to large wavelength, the results from the finite element model and the spectral model are in good agreement for all values of pattern amplitude.However, as demonstrated here, the asymptotic model is accurate only for smaller amplitude (up to approximately α = 0.06).
The variation of permeability with pattern amplitude for small and large Hartmann numbers can be summarized as follows: (i) For a small wavelength (λ = 1), the hydraulic permeability decreases monotonically with pattern amplitude for Ha = 0.5 and Ha = 1.2 as shown in panel (a) of figures 3 and 4. In comparison to a channel with a planar bottom wall, the additional surface on the lower wall in the presence of undulations (non-zero amplitude) across which viscous forces can act to enforce the no-slip condition is greater, the rougher the patterns are (i.e. the larger the value of alpha is).The velocity gradients near the bottom wall for short pattern wavelengths are the most significant, and these scale inversely with wavelength, emphasizing the importance of the viscous forces acting on the lower surface.At short pattern wavelengths, this effect decreases permeability (i.e.decreases flow rate at a given pressure gradient) with pattern amplitude.Therefore, the grooves serve as drag-enhancing roughness components for short wavelengths.(ii) For long wavelengths (λ > 3), the flow rate increases with increasing pattern amplitude for Ha = 0.5 and Ha = 1.2 as shown in panels (c)-(e) of figures 3 and 4. At longer pattern wavelengths, this effect increases permeability (i.e.increases flow rate at a given pressure gradient) with pattern amplitude.Therefore, the grooves serve as drag-reducing roughness components for longer wavelengths.The x-gradients in the flow, which are influenced by wavelength in the long-wave limit, are far less important than the y-gradients, which are influenced by local channel thickness.As a result, at sufficiently long wavelengths, the permeability increases monotonically with α as the size of the domain increases.The aforementioned confinement effect becomes predominant for long waves.(iii) For intermediate wavelengths (λ = 2.5 to 3), the permeability predicted by the spectral model (equation ( 14)) passes through a minimum with increasing amplitude.This trend is observed for Ha = 0.5 and Ha = 1.2 as shown in panel (b) of figures 3 and 4, and confirmed by finite element simulation results.As the wavelength decreases, the minimum in panel (f) becomes shallower but shifts towards smaller values of α.Additionally, within this range, the position of the minimum also moves from higher to lower alpha values.Panel (f) of figure 4 also provides a numerical estimate of the dimension for the curve with the minimum, where λ = 3.0.
The present study illustrates the importance of micropatterning with regard to drag-enhancing and drag-reducing roughness elements in the grooved wall.Assuming a channel thickness of 100 µm, for Ha = 1.2, if we choose a pattern wavelength of 300 µm with an amplitude of 64 µm, the flow rate reaches a minimum of 94.17% of the flow rate of a straight channel with the same surface-tosurface spacing.Interestingly, in such a microchannel, either increasing or decreasing the pattern amplitude from 64 µm is expected to result in a larger flow rate.Furthermore, it is of particular interest to utilize the current spectral model with real-world parameter values.For example, experiments have been described in magnetohydrodynamics for a wide range of Hartmann numbers, i.e.Ha ∼ 0.1-1000 [71].Similarly, references [33,71,72] adopted a channel height H ∼ O(10-100) µm, and [15,23,73,74] to investigate the effect of surface roughness on electromagnetically driven flow with the following parameters: The aforementioned parameter values can be utilized with the current spectral model, which will help to understand the effect of large pattern amplitude and the transition from wall roughness to the confinement effect.The spectral method has the advantage of accurately predicting the flow rate behavior as a function of a wide range of pattern amplitude, Hartmann numbers and wavelengths.Moreover, it does so at a minimal computational effort when compared to finite element simulations.Beyond accuracy, computational efficiency sets the spectral method apart.Completing in a fraction of the time required by finite-element simulations (an improvement by just over two orders of magnitude, or 0.25-0.35s compared to 30-40 s for a representative case described in this study), the spectral method proves to be exceptionally computationally efficient.For instance, in obtaining permeability predictions for 100 points between 0 to 1, the spectral solution outpaces finite-element simulation by approximately 150-250 times.This computational advantage enables predictions beyond the reach of current asymptotic theories, exemplified by identifying limited pattern amplitude and Hartmann numbers.

Transition between roughness-to-confined effect
Section 5.1 demonstrates the variation of hydraulic permeability as a function of pattern amplitude for different wavelengths and Hartmann numbers.For intermediate wavelengths (λ = 2.5 to 3.0), the behavior of the flow rate is non-monotonic with increasing for Ha = 0.5 and 1.2.Panels (a)-(d) of figure 5 show how flow rate changes with pattern amplitude for Ha ≫ 1 and different wavelengths.The prediction from the spectral model (equation ( 14)) is used to analyze the behavior.With increasing pattern amplitude, the flow rate first exhibits a decreasing trend followed by an increasing trend at each investigated wavelength.This minimum flow rate as illustrated in panels (a)-(d) of figure 5, is also captured by the numerical simulations.The hydraulic permeability exhibits a gradual variation from a decreasing to an increasing effect, indicating a transition from wall roughness (i.e.small-to-moderate values of pattern amplitude) to confinement effect (i.e.pattern amplitude approximating unity).The reason behind this transition phenomenon is the increasing dominance of the Lorentz force, which plays a crucial role near the wavy wall.The viscous force can be dominated by a high strength magnetic field near a wavy wall, and this balance depends on the amplitude of the waviness.This minimum permeability depends on the Hartmann number.For instance, when the wavelength of the channel is λ = 3 (panel (a)), the minima occur at small Hartmann numbers such as 0.5, 1, 2, and 3.However, the transition point of permeability is noticed at larger Hartmann numbers, such as 20, 25, and 30, at λ = 10 (panel (d)).This observation suggests that if the channel is short in length, the transition from roughness to confinement effect can be identified with small values of Hartmann number.In contrast, a larger Hartmann number is required to capture the transition behavior of the flow rate in the case of a longer channel.The prediction of the spectral model is accurate at large pattern amplitude (α ∼ 1).It is observed that the theoretical and numerical results are in good agreement at various wavelengths, pattern amplitudes and Hartmann numbers.3), then the new governing equation will be equation (23).Similarly, in equation (12) for the flow rate, the Hartmann number is taken as 1.In this case, the Lorentz force is equal to the viscous force.The prediction from the spectral model of flow rate ratio (Q Ha̸ =0 to Q Ha=1 ) is compared with numerical results.At different normalized wavelengths, the flow rate ratio decreases monotonically with increasing the Hartmann number, as depicted in panels (a) and (b) of figure 6.The flow rate for Ha = 1 always dominates the Q Ha̸ =0 flow rate.A good agreement is noticed between the theoretical and numerical results at different wavelengths and pattern amplitude.

Effect of the wavelength
Figure 7 shows the variation of hydraulic permeability (Q/Q 0 ) with varying channel wavelengths (λ) at reasonably large pattern amplitude (α = 0.8).From small (Ha = 0.5) to large (Ha = 2) Hartmann numbers, α = 0.8 represents an extreme situation of confined microchannel flow where the crest of the bottom undulating wall nearly coincides with the upper planner wall.The thickest part of the channel is 9 times greater than the thinnest portion.At smaller wavelengths, the prediction of permeability based on the spectral model accentuates the roughness effect, whereas, at large wavelengths, the confinement behavior is observed for both small and large Hartmann numbers as depicted in panels (a) and (b) of figure 7.An O(α 2 )   17)) and spectral model (equation ( 14)), respectively.The red symbols show the prediction from a finite-element-based numerical simulation (COMSOL).solution of the asymptotic model is accurate only for a small pattern amplitude; by contrast, it diverges at a large amplitude.The actual perturbation parameter is ϵ, which is equivalent to α × H.The O(H 2 ) solution starts to diverge at a large H.However, the spectral method and the finite element results are in good agreement for both small and large Hartmann numbers across a wide range of wavelengths.

Magnetohydrodynamic velocity profile
To gain a better understanding of the role played by the Hartmann number in the current configuration, it is important to analyze flow parameter quantities such as the velocity profile at different locations of the patterned channel.The model described by equation ( 5 5) is used to investigate the velocity profile and is represented as a solid line.The prediction from the numerical simulation is shown symbolically.By contrast, panel (c) of both figures shows the profile for Ha = 5.At the largest width of the channel (X = −π) for λ = 3, 7, the parabolic behavior of the velocity profile is obtained using the grid-free spectral method (equation (5)) and the numerical approach as depicted in panels (a) and (b) of figure 8.A similar trend is also captured for the smallest width (X = 0) for λ = 3, 7 as illustrated in panels (a) and (b) of figure 9.At X = −π, for a large Hartmann number (Ha = 5) with λ = 7, the maximum velocity is always observed at the center of the channel with a parabolic shape as shown in panel (c) of figure 8.However, the velocity profile is not parabolic in the case of λ = 3.This signifies that the Lorentz forces dominate over viscous forces in the largest gaps between both walls.Therefore, the flow profile will be flattened at a significantly large Hartmann number (Ha = 5) and moderate wavelength (λ = 3) as demonstrated in panel (c) of figure 8.By contrast, at the thinnest section (X = 0), the velocity profile remains quasi parabolic for a large wavelength (λ = 3).In the case of λ = 7, the profile is slightly flattened at the center of the channel, as shown in panel (c) of figure 8.The Lorentz force is more significant in the thinnest section compared to the thickest section of the channel.Overall, a good agreement is found between the spectral model (SM), and the numerical simulation (COMSOL) results for different values of wavelengths and Hartmann  numbers at different channel locations, as depicted in figures 8 and 9.

Transition from Hydrodynamic flow to EMHD flow
In order to obtain more insight into the effect on hydraulic permeability ratio (χ) between electromagnetohydrodynamic (EMHD) and hydrodynamic (HD) flow as a function of Ha, panels (a) and (b) of figure 10 are plotted for the range of Ha over which substantial flow rate (i.e.permeability) augmentations can effectively be realized.The behavior is characterized by a new quantity, defined as the parameter χ = (Q/Q 0 ) EMHD /(Q/Q 0 ) HD , whereas the amplitude α is kept constant as 0.1, 0.2, and 0.3.Figure 10(a) shows that for a small wavelength (λ = 1), the permeability ratio χ increases with Ha initially and then reaches a plateau at larger values of Ha.This transition behavior is captured at a critical Hartmann number.Similar trends are observed for different values of pattern amplitude.In the case of λ > 3, the permeability decreases with increasing Ha initially and then plateaus at larger values of Ha as shown in figure 10(b).
For any values of α = 0.1, 0.2, 0.3, figure 11 reveals that the transition from hydrodynamics to EMHD flow occurs beyond Ha ≃ 2.27 at intermediate wavelength (λ = 3).A crucial difference between EMHD and HD flow is that, as Ha increases, the Lorentz forces become increasingly dominant over the viscous forces.This transition is due to the progressive strengthening of viscous force and transverse directional Lorentz force, as the magnetic field is acting more effectively on the grooved surface.The transition behavior of χ can be identified through a critical Hartmann number (Ha crit ).The viscous forces are more significant for low Hartmann number (Ha ⩽ 2.27), whereas the Lorentz forces are more dominant for Ha ⩾ 2.27.As α increases (0.1 to 0.3), Ha crit remains always constant (Ha ≃ 2.27) for λ = 3.This observation suggests that the critical Hartmann number is independent of pattern amplitude.The variation of χ with different pattern amplitude passes a unity crossing at Ha ≃ 4, as shown in figure 11.
In other words, when Ha ≃ 4 and thus χ = 1, the permeability ratio from the EMHD to HD flow through the grooved channel is equivalent to the permeability ratio for a flat channel (i.e. a channel with the same height but no topographic patterns).

Conclusions
The present study analyzed electromagnetohydrodynamic flow through a grooved microchannel with no-slip boundary conditions at the walls.The effect of wall roughness (i.e.pattern amplitude and wavelength) was investigated using the meshless spectral method.Analytical calculations are carried out to obtain the velocity profile (equation ( 5)) and hydraulic permeability (equation ( 14)).In the present study, the hydraulic permeability strongly depends on Hartmann number (Ha), pattern amplitude (ϵ or α) and wavelength (λ).We demonstrated the impact of these nondimensional parameters on flow rate and velocity profiles.Previous model, such as the asymptotic model used by Buren et al (2014) [14], is valid only for small pattern amplitude.To the best of the author's knowledge, this paper provides the first analytical solution of hydraulic permeability, which is valid from small to large pattern amplitude, different wavelengths and Hartmann numbers.
The following conclusions are drawn from the theoretical and numerical results for two cases, one for Ha < 1 and one for Ha > 1.Some of the most important observations from this study are that (i) the hydraulic permeability decreases with increasing wall roughness for small wavelengths (λ = 1).At the same time, (ii) the permeability increases with wall roughness for large wavelengths (λ > 3), such as λ = 5, 7, 10.This behavior is observed when the Hartmann number is less than unity.A similar phenomenon, albeit less pronounced, is also noticed when the Hartmann number exceeds unity.This increasing (at large wavelength) and decreasing (at small wavelength) behavior of hydraulic permeability as a function of pattern amplitude shows the confinement and roughness effects, respectively.Importantly, (iii) for intermediate wavelength (λ = 2.5 to 3.0), the permeability initially decreases and then increases with wall roughness for both cases (Ha < 1, and Ha > 1).At this point, a minimum permeability (i.e. the minimum flow rate for a given pressure gradient) is observed for a critical pattern amplitude (α critical ).The value of α critical varies with the pattern wavelength.For example, when λ = 2.5 then α critical is approximately equal to 0.8.In the case of λ = 3.0, α critical equals 0.65.
Additionally, the transition between roughness and confinement effect strongly depends on the Hartmann number for different pattern wavelengths.Before reaching a critical pattern amplitude, the resistance near a sinusoidal wall surpasses the magnetic strength.As a result, hydraulic permeability decreases, which is expected to be influenced by surface roughness.In contrast, after surpassing the critical pattern amplitude, the resistance near a sinusoidal wall becomes lower than the magnetic strength.Consequently, hydraulic permeability increases, which is anticipated to be associated with the confinement effect.Similarly, if Ha ≫ 1 is large, the transition is observed across the entire wavelength range investigated.In this context, the dominance of the Lorentz force over the viscous force is observed in the vicinity of the corrugated surface.As a result, the interaction between roughness and confinement effects is evident at various wavelengths.When the Hartmann number is small (Ha < 1), the velocity profile of EMHD flow is comparable to the well-known parabolic Poiseuille profile.In the case of a large Hartmann number Ha > 5, the velocity profile is not parabolic in nature.
A close agreement is found between the prediction from the spectral model and numerical results regardless of pattern amplitude.A noteworthy aspect of our work is the application of the spectral method, capturing the transition behavior in the permeability.The spectral model identifies a minimum permeability not reported before in the literature.Importantly, the spectral model remains accurate for significantly larger dimensionless pattern amplitudes (approximating ~1), dimensionless wavelengths (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15), and Hartmann numbers (0-40).Specifically, when the pattern amplitude approaches unity, and the peaks from the bottom (sinusoidal) wall almost reach the top wall, the spectral model still accurately represents the flow rate behavior.This comparative analysis highlights the unique contributions of the present study.The present work can be used to optimize the flow of coolant or air around electronic components and microfluidics pumps.By designing microchannels or microstructures in specific configurations, it is possible to control fluid flow patterns, local flow velocities, convective heat transfer rates, and pressure drop.This additional level of flow control could result in the development of novel heat transfer and cooling devices.In the future, a flow configuration where the flow occurs along and across the corrugation with an applied electric field may be studied.The findings of this study will provide a relation between electroosmotic force and Lorentz force.
Velocity of fluid in z-direction (m s −1 ) Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.− → B Magnetic field strength vector (T) − → E Electric field strength vector (V/m) x Transverse coordinate (m) y Depth-wise coordinate (m) z Stream-wise coordinate (m) pz Axial pressure gradient (Pa) X Dimensionless transverse coordinate Y Dimensionless depth-wise coordinate Z Dimensionless stream-wise coordinate W Dimensionless velocity of fluid in z-direction Q Dimensionless flow rate through the corrugated channel Q 0 Dimensionless flow rate through parallel plate channel with spacing h λ Ratio of pattern wavelength to mean channel height ϵ Ratio of pattern amplitude to mean channel height α Ratio of ϵ to 2π /λ µ Dynamics viscosity of the fluid (kg/m s −1 )

Figure 1 .
Figure 1.(a) Demonstrate the fluid flow through the grooved microchannel, where the bottom wall is a sinusoidal curve and the top wall is flat.(b) Schematic illustration of the flow through a grooved narrow channel of height (h) and pattern wavelength (L).A sinusoidal surface is defined as y = a cos ( 2π x L) , here a is pattern amplitude.The pressure gradient is applied along the z-axis, while a constant magnetic field of strength B acts along the y-axis.

− 4 .
The Partial Differential equation module of the software is used to analyze the present study.No-slip boundary conditions at both walls are considered.Periodic boundary conditions along the Y-edges are chosen.

Figure 2 .
Figure 2. panel (a) illustrates a schematic diagram of a 2D computational domain with boundary conditions, and panel (b) shows the triangular mesh employed for the numerical study.At a large pattern amplitude (α = 0.8), pans (c)-(j) show the comparison between a variation of dimensionless velocity (W) with different Hartmann numbers (Ha) for two different wavelengths, such as λ = 3 and 10.

Figure 3 .
Figure 3. Variation of the scaled permeability ( Q Q0 ) of the channel with dimensionless pattern amplitude (α) at different wavelengths of the channel (λ).Dashed and solid lines show the prediction of the theoretical model by the asymptotic model (equation (17)) and spectral model (equation (14)).The red symbols show the prediction from a finite-element-based numerical simulation (COMSOL).

Figure 4 .
Figure 4. Variation of the scaled permeability ( Q Q0 ) of the channel with dimensionless pattern amplitude (α) at different wavelengths of the channel (λ).Dashed and solid lines show the prediction of the theoretical model by the asymptotic model (equation (17)) and spectral model (equation (14)).The prediction from a finite-element-based numerical simulation (COMSOL) is represented in the red symbols.

Figure 5 .
Figure 5. Variation of the scaled permeability ( Q Q0 ) of the channel with dimensionless pattern amplitude (α) at different wavelengths of the channel (λ) with different Hartmann number (Ha).Lines and symbols show the prediction of the theoretical model by the spectral model (SM, equation (14)) and the prediction of numerical based on a finite-element-based numerical simulation (COMSOL), respectively.

Figure 6
Figure 6 demonstrates the variation of hydraulic permeability as a function of the Hartmann number for two dimensionless pattern amplitudes (α = 0.3, 0.7).The analysis is carried out for flow rate at Ha ̸ = 0 and Ha = 1, respectively.If the Hartmann number is considered equal to unity in equation (3), then the new governing equation will be equation(23).Similarly, in equation (12) for the flow rate, the

Figure 6 .
Figure 6.Variation of the scaled permeability of the channel with dimensionless Hartmann number at different wavelengths of the channel.Panels (a) and (b) illustrate the behavior of hydraulic permeability when the Hartmann number is equal to 1 at pattern amplitudes that are significantly small and large, such as 0.3 and 0.7, respectively.The solid lines and symbols represent the prediction from the spectral model (SM, equation (14)) and numerical simulation (NS).

Figure 7 .
Figure 7. Panels (a) and (b) show the variation of dimensionless hydraulic permeability ( Q Q0 ) for smaller to large Hartmaan numbers (Ha = 0.5, 2.0) concerning the wavelength of the channel (λ) at significantly large pattern amplitude (α = 0.8).Lines blue and black show the prediction of the theoretical model by the asymptotic model (equation (17)) and spectral model (equation (14)), respectively.The red symbols show the prediction from a finite-element-based numerical simulation (COMSOL).
) can predict global quantities, such as the permeability as well as local flow features.Variations in velocity at different transverse locations and depths are represented in figures 8 and 9.The theoretical velocity distribution using the spectral model (SM) is calculated with a few terms of the Fourier series and compared with corresponding numerical results as shown in figures 8 and 9. Velocity profiles have been plotted for α = 0.6, λ = 3, 7 and Ha = 0.5, 1.0, 5.0 at the constant-Y sections with the largest surface-to-surface distance (at X = −π) as illustrated in panels (a)-(c) of figure 8 .Similarly, the velocity profile is also investigated at the smallest surface-to-surface distance (at X = 0) in the y direction, as shown in figure 9.A unified representation of velocity profiles has been achieved by utilizing a normalized Y-coordinate.The velocity profile is plotted for three cases: (i) viscous force dominating over the Lorentz force (Ha < 1) as shown in panel (a) of figures 8 and 9. (ii) viscous force comparable to the Lorentz force (Ha = 1) as shown in panel (b) of figures 8 and 9, and (iii) the Lorentz force dominating the viscous force (Ha > 1) as shown in panel (c) of figures 8 and 9.

Figure 8 .
Figure 8. Panels (a), (b) and (c) show the variation of velocity profile along Y-axis for moderately large pattern amplitude (ϵ = 0.6) at different Hartmann numbers (Ha) and wavelength (λ).The velocity profile is calculated at the largest gap between walls.The location X = −π corresponds to the largest gap between walls.Equation (5) is used to investigate the velocity profile and is represented as a solid line.The prediction from the numerical simulation is shown symbolically.

Figure 9 .
Figure 9. Panels (a), (b) and (c) show the variation of velocity profile along Y-axis for moderately large pattern amplitude (ϵ = 0.6) at different Hartmann numbers (Ha) and wavelength (λ).The velocity profile is calculated at the largest gap between walls.The location X = 0 corresponds to the smallest gap between walls.Equation (5) is used to investigate the velocity profile and is represented as a solid line.The prediction from the numerical simulation is shown symbolically.

Figure 10 .
Figure 10.Demonstrate the ratio of scaled permeability (χ) between electromagnetohydrodynamics to pure hydrodynamics flow with reference to different Hartmann numbers, panel (a) is for λ = 1, and panel (b) is forλ = 5.The solid curves such as black, red, and blue indicate predictions from an asymptotic approach obtained using equation (22).

Figure 11 .
Figure 11.Variation of the ratio of the scaled permeability (χ) between EMHD to HD flow with reference to the Hartmann number (Ha) at different nondimensional pattern amplitude (α) and wavelength (λ = 3).The solid curves such as black, red, and blue indicate predictions from an asymptotic approach obtained using equation(22).

Table 1 .
Variation of hydraulic permeability at λ = 10, ϵ = 0.5, Ha = 5 with respect to different mesh size or grid sensitivity test.