Unlike the conventional electroosmotic flow (EOF) driven by direct current and alternating current electric fields, this study investigates the pulse EOF of Newtonian fluids through a parallel plate microchannel actuated by pulse electric fields. Specifically, the pulses considered encompass triangular and half-sinusoidal pulse waves. By applying the Laplace transform method and the residual theorem, the analytical solutions for the velocity and volumetric flow rate of the pulse EOF associated with these two pulse waves are derived, respectively. The influence of pulse width a ¯ and electrokinetic width K on velocity is further considered, while the volumetric flow rate as a function of time t ¯ and electrokinetic width K is examined separately. A comparison of the volumetric flow rates related to these two pulse waves under varying parameters is also conducted. The research findings indicate that irrespective of the pulse wave, a broader pulse width results in a prolonged period and increased amplitude of the velocity profile. Elevating the electrokinetic width yields higher near-wall velocities, with negligible effect on near-center velocities. It is noteworthy that regardless of the electrokinetic width, the near-wall velocity exceeds that of the near-center during the first half-cycle, while the situation reverses during the second half-cycle. The volumetric flow rate varies periodically with time, initially surging rapidly with electrokinetic width before gradually stabilizing at a constant level. More interestingly, independent of pulse width and electrokinetic width, the volumetric flow rates linked to the half-sinusoidal pulse wave consistently surpass those of the triangular pulse wave. For any pulse width, the volumetric flow rates corresponding to the two pulse waves grow with higher electrokinetic widths, especially prominent at alternating intervals of the two half-cycles within a complete cycle. These findings have important implications for improving the design and optimization of microfluidic devices in engineering and biomedical applications utilizing pulse EOF.

Topics
Electric-field effects, Integral transforms, Volumetric flow rates, Flow velocity, Fluid dynamics, Newtonian fluids, Microchannel, Electroosmosis, Microfluidic devices

As micro-manufacturing technology advances, microfluidic systems are increasingly being deployed across biology, chemistry, and national defense sectors. Although the manufacturing of micro-devices is now easily achievable, the limited understanding of fluid transport phenomena poses significant challenges in achieving comprehensive design and precise control of these devices. Hence, there is an urgent requirement for theoretical research on microfluidic systems. Microfluidics is a technology dedicated to exploring fluid dynamics within microscale channels. In recent years, it has emerged as one of the leading frontiers of scientific inquiry owing to its broad applications in fields like chemical separation,1 energy harvesting,2 material synthesis,3 and medical diagnostics.4 

The key difference between microfluidic and macroscopic systems lies in the impact of reduced device size on microfluidic channels. This impact is characterized by various effects, such as the microscale, laminar flow, surface tension, capillary action, surface slip, rapid heat conduction, and diffusion, all of which significantly influence microfluidic systems.5 The driving and control techniques used in microfluidic systems are essential for optimizing system design and ensuring accurate operation. These techniques rely heavily on both experimental and theoretical support. Notably, the advancement of Lab on a Chip (LOC) technology has sparked significant interest in the driving and control techniques within microfluidics. Microfluidics employs a variety of driving mechanisms, including pressure gradients,6 surface acoustic waves,7 electric fields,8 magnetic fields,9 and their appropriate combinations.10,11 Among them, external electric fields are presently the predominant driving force in microfluidic systems, attributed to the remarkable efficiency, control convenience, and integration facilitated by electroosmotic actuation.12 The utilization of external electric fields to directly drive and control fluid flow within microchannels offers a non-damaging approach that avoids potential harm to mechanical components like micropumps and microvalves. This advantageous feature makes it particularly well suited for biomedical applications, establishing it as the leading driving technology in the field.

In general, when a charged surface comes into contact with an electrolyte solution within microchannels, oppositely charged ions tend to move toward the surface, creating an electric double layer (EDL) characterized by a high concentration of these ions. Upon the application of an external electric field at the ends of the microchannel, the ions within the EDL are mobilized by the Coulomb force, propelling nearby liquid microclusters due to the influence of liquid viscosity, thus generating an electroosmotic flow (EOF).13 Considerable research has been dedicated to investigating the steady EOF induced by direct current (DC) fields, leading to a wealth of established outcomes spanning numerical simulations,14 numerical computations,15 experimental investigations16 and theoretical analyses.17 Nevertheless, the necessity of high voltages and substantial field strengths to maintain steady electroosmotic flow presents challenges in experimental setups. As a result, there has been notable interest in studying periodic EOF induced by alternating current (AC) fields.18,19 Compared to steady EOF, periodic EOF offers the advantage of requiring a lower driving voltage, thereby reducing the generation of Joule heat and bubbles in the flow field. Furthermore, periodic EOF within microchannels not only overcomes the limitations of steady EOF under a stable electric field but also enhances the mixing efficiency of solutions. Importantly, periodic EOF can be utilized to measure particle electrophoretic mobility,20 concentrate nanoparticles from continuous flow,21 and investigate the effects of mobile phone magnetic fields on the brain.22 Over the past few decades, a multitude of researchers have engaged in investigating periodic EOF within microchannels of diverse geometries, employing a combination of theoretical and numerical calculations. Moghadam23 explored the dynamics of EOF of a Newtonian fluid within a microannulus, underscoring the impact of various unsteady electric fields. Gheshlaghi et al.24 developed an analytical solution to describe the unsteady flow of fluid in a microchannel characterized by parallel rotating plates, incorporating the effects of electrokinetic forces using the Debye–Hückel (DH) approximation. The time periodic EOF of a Newtonian fluid in a rectangular microannulus was analyzed by Moghadam,25 who delivered an extensive discussion on the influence of fundamental sinusoidal waveforms and geometric parameters on the flow behavior. The EOF of a viscoelastic fluid through short 10:1:10 constricted microchannels was numerically examined by Ji et al.,26 with particular emphasis on the dependence of the flow characteristics on the frequency of the pulsating electric field. Yang et al.27 studied the EOF of a second-grade fluid in a circular microchannel driven by an AC electric field. Utilizing the integral transform method, they derived analytical expressions for the electric potential and velocity by solving the linearized Poisson–Boltzmann (PB) and Navier–Stokes (NS) equations.

It is important to note that the applied electric fields used in the aforementioned studies are restricted to DC and AC fields. However, both of these fields have inherent limitations in practical applications, such as the unidirectional flow of DC field and the continuity and high loss of AC field. Therefore, it is crucial to urgently explore alternative and more promising applications of electric fields to overcome these drawbacks. Fortunately, recent findings reveal that Li et al. have conducted extensive research on pulse EOF in various microchannels, encompassing parallel,28 circular,29 and annular30 configurations. Their investigations predominantly focused on the impact of simple rectangular pulses. Remarkably, despite these strides, the breadth of pulse EOF research remains somewhat constrained, especially concerning the examination of more frequently employed pulse types. Building on the current research landscape and considering the simplicity, periodicity, frequency control, and wide applicability31,32 of triangular and half-sinusoidal pulses, the present study aims to examine the pulse EOF of Newtonian fluids in parallel plate microchannels driven by these pulse electric field forces. Analytical solutions for the velocity and volumetric flow rate of the pulse EOF associated with these two pulses are obtained by using the Laplace transform method and the residual theorem, respectively. Moreover, the impacts of various parameters on these solutions are discussed. An extensive analysis is performed to compare the volumetric flow rates of pulse EOF linked with these two pulses under diverse sets of parameters. The primary innovation of this study lies in the systematic examination of the effects of triangular and half-sinusoidal pulses on pulse EOF, a topic that has not been previously investigated in the literature. By addressing this significant gap, it is anticipated that fresh insights and pathways for future research will emerge. This contribution aspires to enhance the understanding of pulse EOF phenomena and to establish a robust foundation for practical applications in microfluidic technology.

This work explores the pulse electroosmotic flow (EOF) of Newtonian fluids through a parallel plate microchannel with dimensions of height 2 H⁠, length L, and width W (⁠ L ≫ 2 H and L ≫ W⁠). A two-dimensional (2D) Cartesian coordinate system ( x , y ) is established, positioning the origin at the middle of the microchannel, as illustrated in Fig. 1. The flow is considered to be laminar and is induced by a pulse electric field with strength E 0 applied along the x-axis. The pulse is depicted as either a triangular pulse wave or a half-sinusoidal pulse wave, with both having a pulse amplitude and pulse width equal to a, as well as a pulse repetition period of 2 a (see Fig. 2). It should be noted here that due to the symmetry of the channel, only the upper half of the channel ( 0 ≤ y ≤ H ) is considered. Assuming that the non-overlapping of electric double layer (EDL) on the channel walls ( κ − 1 ≪ H )⁠, with κ − 1 representing the EDL characteristic thickness (also known as the Debye length).28 This investigation utilizes the following continuity and Navier–Stokes (NS) equations to elucidate the dynamics of flow behavior,33,34
(1)
(2)
where u → = ( u , v ) is the velocity vector, ρ is the fluid density, t is the time, p is the pressure, and μ is the fluid viscosity. The vector F → = ρ e E → indicates the net body force per unit volume,35 where ρ e signifies the net charge density and E → = ( E 0 f ( t ) , 0 ) represents the applied pulse electric field. Detailed expressions for the pulse waves f ( t ) are provided in Eqs. (11) and (29).
FIG. 1.
Schematic diagram of the pulse electroosmotic flow in a parallel plate microchannel.28
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Schematic diagram of the pulse electroosmotic flow in a parallel plate microchannel.28 

FIG. 1.
Schematic diagram of the pulse electroosmotic flow in a parallel plate microchannel.28
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Schematic diagram of the pulse electroosmotic flow in a parallel plate microchannel.28 

Close modal
FIG. 2.
Schematic of the pulse wave. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.
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Schematic of the pulse wave. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.

FIG. 2.
Schematic of the pulse wave. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.
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Schematic of the pulse wave. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.

Close modal

It is essential to highlight that numerous prior research efforts on electroosmotic flow have frequently disregarded transient behavior, failing to account for the dynamic changes in flow over time. This simplification can result in considerable misinterpretations regarding rapid variations and the characteristics of unsteady flow. In many practical applications, especially in scenarios involving complex flows and significant transient phenomena, conducting a transient analysis becomes crucial. This study focuses on unsteady pulse EOF, thoroughly considering transient behavior to ensure a profound understanding and accurate forecasting of flow characteristics.

If the pressure gradient along the x-axis direction is ignored, the one-dimensional NS equation can be formulated as
(3)
where u ( y , t ) is the velocity component along the positive x-axis direction.
Subject to the below imposed no-slip boundary conditions and initial condition,36 
(4)
(5)
For a symmetrical ( z v = z v + = z v − ) low-concentration binary electrolyte solution and a thin electric double layer (EDL), the net charge density ρ e ( y ) is given by the Poisson–Boltzmann (PB) equations as follows:37 
(6)
(7)
where sinh ⁡ ( ⋅ ) represents the hyperbolic sine function, ψ ( y ) is the electric potential of the EDL, ε⁠, n 0⁠, e 0⁠, z v⁠, k B⁠, and T are the dielectric permittivity of the medium, the ionic number concentration, the electron charge, the valence of ions, the Boltzmann constant, and the absolute temperature, respectively.
If the electric potential ψ ( y ) is considered to be axially invariant and significantly smaller than 25 mV⁠, then the Debye–Hückel (DH) linearization approximation can be effectively applied, that is, sinh ⁡ ( ϑ ) ≈ ϑ⁠, with ϑ = z v e 0 ψ ( y ) / ( k B T )⁠. Hence, the linearized PB equation and its corresponding boundary conditions can be expressed by
(8)
(9)
where φ 0 is the wall zeta potential.
The ultimate net charge density ρ e ( y ) can be derived by solving equations (7)–(9) as
(10)
where cos h ( ⋅ ) signifies the hyperbolic cosine function.

1. Triangular pulse wave

The triangular pulse wave can be represented by the following expression:
(11)
To streamline the calculation, certain dimensionless variables are defined as28 
(12)
where u H S denotes the Helmholtz–Smoluchowshi EOF velocity of Newtonian fluids, K is called the electrokinetic width representing the ratio of the half-height of microchannel to the EDL characteristic thickness.
The dimensionless equation (3) and its associated conditions (4) and (5) can be derived from the utilization of Eq. (12),
(13)
(14)
(15)
To remove the function f ( t ¯ ) in Eq. (13), the Laplace transform method is employed as defined below:
(16)
Noting that the periodic function f ( t ) in Eq. (11) satisfies piecewise continuity, we can apply the Laplace transform of periodic functions to its dimensionless form, thereby yielding
(17)
where tanh ⁡ ( ⋅ ) stands for the hyperbolic tangent function, which can be expressed as the ratio of the hyperbolic sine function to the hyperbolic cosine function.
Utilizing Eq. (17) and the initial condition (15), the Laplace transform of Eq. (13) along with its corresponding boundary conditions (14) can be formulated as
(18)
(19)
It is important to emphasize that Eq. (18) represents a second-order nonhomogeneous ordinary differential equation (ODE). Its solution can be constructed by combining the general solution U h ( y ¯ , s ) of the related homogeneous equation with the special solution U s ( y ¯ , s ) of the nonhomogeneous equation, as demonstrated below:
(20)
where C j ( j = 1 , 2 , 3 ) are constants and the coefficient C 3 can be straightforwardly identified by incorporating the particular solution U s ( y ¯ , s ) from Eq. (20) into Eq. (18),
(21)
By combining equations (18), (20), and (21) and using the boundary conditions (19), we can efficiently deduce the coefficients C 1 and C 2 as follows:
(22)

Therefore, the velocity U ( y ¯ , s ) can be reformulated as

(23)
The definition of the inverse Laplace transform is shown below:
(24)
where Γ denotes the vertical line to the right of all singularities in the complex s-plane. Applying the residue theorem allows us to express the solution of Eq. (24) as the sum of the residues at singular points within the contour where U ( y ¯ , s ) is involved,
(25)
where s n 1 = − ( 2 n − 1 ) 2 π 2 / 4 , s n 2 = ( 2 n − 1 ) π i / a ¯ , n = 0 , ± 1 , ± 2 , … .⁠.
Furthermore, in accordance with the definition of volumetric flow rate, we acquire38 
(26)
The following dimensionless variables are introduced for ease of computation:
(27)
Subsequently, the dimensionless volumetric flow rate can be achieved by substituting Eq. (26) into Eq. (27),
(28)

2. Half-sinusoidal pulse wave

The half-sinusoidal pulse wave can be described by the following form:
(29)
Using Eq. (12) and the matlab software, we can obtain the Laplace transform of the dimensionless representation of Eq. (29),
(30)
where coth ⁡ ( ⋅ ) indicates the hyperbolic cotangent function, which is the inverse of the hyperbolic tangent function.
Consequently, with the aid of equations (15), (16), and (30), Eq. (13) along with its relevant boundary conditions (14) can be rephrased as
(31)
(32)
Similarly, the coefficients C j ( j = 1 , 2 , 3 ) can be determined as
(33)
(34)
Thus, the velocity U ( y ¯ , s ) can be rewritten as
(35)
Inserting Eq. (35) into Eq. (24) and utilizing the residue theorem, we get
(36)
where s 1 = π i / ( 2 a ¯ ) , s 2 = − π i / ( 2 a ¯ ) , s n 1 = − ( 2 n − 1 ) 2 π 2 / 4 , s n 2 = n π i / a ¯ , n = 0 , ± 1 , ± 2 , … .⁠.
Afterward, the dimensionless volumetric flow rate can be further derived with the help of Eqs. (26) and (27),
(37)

In Sec. III, we have derived analytical solutions for the velocity and volumetric flow rate of pulse electroosmotic flow (EOF) of Newtonian fluids in a parallel plate microchannel propelled by triangular and half-sinusoidal pulse electric fields, respectively. It is evident from Eqs. (25), (28), (36), and (37) that these solutions are largely dependent on the pulse width a ¯ and the electrokinetic width K, which will serve as crucial references for analyzing these solutions in this section. To thoroughly understand the comprehensive characteristics of fluid flow and address the requirements of practical engineering applications, it is essential to introduce the relevant physical parameters along with their respective values or ranges, as detailed below:36,39 H = 100 μ m⁠, ρ = 10 3 kg m − 3⁠, η 0 = 10 − 3 kg m − 1 s − 1⁠, u H S = 100 μ m s − 1⁠, 0 ≤ K ≤ 100⁠, 2 ≤ a ¯ ≤ 5⁠. A detailed compilation of definitions and units for the various key symbols utilized in this study is provided in the Nomenclature section. Additionally, the periodicity of these pulse waves is being evaluated in this section. For instance, with a pulse width a ¯ = 2⁠, the time spans from 0 to 2 ( 0 ≤ t ¯ < 2 ) for the first half-cycle and from 2 to 4 ( 2 ≤ t ¯ < 4 ) for the second half-cycle, as shown in Fig. 2(b). Finally, it is noteworthy that the solution process in this paper yields results based on dimensionless parameters.

The effect of pulse width a ¯ on the velocities associated with the triangular and half-sinusoidal pulse waves is illustrated in Figs. 3 and 4, respectively. As anticipated, the velocity of each pulse wave undergoes periodic changes over time. As the parameter a ¯ value increases, the period of velocity profile variation elongates and the velocity amplitude escalates. This phenomenon stems from the fact that the pulse width, pulse amplitude, and half of the pulse repetition period have equal values (excluding units), as depicted in Fig. 2. Moreover, it can be observed from Figs. 3 and 4 that the evolving shape of the velocity profile over time resembles the respective pulse waves, presenting triangular (refer to Fig. 3) and half-sinusoidal (see Fig. 4) shapes, respectively. This observation is in accord with earlier research findings,28 thereby providing indirect support for the reliability of the results obtained in this study.

FIG. 3.
3D velocity distributions corresponding to triangular pulse wave for different pulse widths a ¯ ( K = 20 ). (a) a ¯ = 2; (b) a ¯ = 3; (c) a ¯ = 4; (d) a ¯ = 5.
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3D velocity distributions corresponding to triangular pulse wave for different pulse widths a ¯ ( K = 20 )⁠. (a) a ¯ = 2⁠; (b) a ¯ = 3⁠; (c) a ¯ = 4⁠; (d) a ¯ = 5⁠.

FIG. 3.
3D velocity distributions corresponding to triangular pulse wave for different pulse widths a ¯ ( K = 20 ). (a) a ¯ = 2; (b) a ¯ = 3; (c) a ¯ = 4; (d) a ¯ = 5.
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3D velocity distributions corresponding to triangular pulse wave for different pulse widths a ¯ ( K = 20 )⁠. (a) a ¯ = 2⁠; (b) a ¯ = 3⁠; (c) a ¯ = 4⁠; (d) a ¯ = 5⁠.

Close modal
FIG. 4.
3D velocity distributions corresponding to half-sinusoidal pulse wave for different pulse widths a ¯ ( K = 20 ). (a) a ¯ = 2; (b) a ¯ = 3; (c) a ¯ = 4; (d) a ¯ = 5.
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3D velocity distributions corresponding to half-sinusoidal pulse wave for different pulse widths a ¯ ( K = 20 )⁠. (a) a ¯ = 2⁠; (b) a ¯ = 3⁠; (c) a ¯ = 4⁠; (d) a ¯ = 5⁠.

FIG. 4.
3D velocity distributions corresponding to half-sinusoidal pulse wave for different pulse widths a ¯ ( K = 20 ). (a) a ¯ = 2; (b) a ¯ = 3; (c) a ¯ = 4; (d) a ¯ = 5.
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3D velocity distributions corresponding to half-sinusoidal pulse wave for different pulse widths a ¯ ( K = 20 )⁠. (a) a ¯ = 2⁠; (b) a ¯ = 3⁠; (c) a ¯ = 4⁠; (d) a ¯ = 5⁠.

Close modal

Interestingly, the evolution (including the transient phase) of velocity over time related to the two pulse waves is vividly presented in Figs. 3 and 4. As a pulsed electric field is applied to the fluid, its response does not immediately reach a stable state (where “stable state” is a relative concept). The transient phase represents the transitional process that the fluid undergoes under the initial influence of the electric field, involving the complex and subtle interactions between the fluid's internal inertial effects and viscous effects. A deep understanding of this transient behavior is crucial for revealing how the fluid gradually transitions from its initial state to a stable state. Specifically, in practical applications, fully considering transient characteristics can provide a more comprehensive description of flow phenomena, thereby offering concrete support for optimizing engineering designs. The presentation of this evolutionary process not only enriches our overall understanding of fluid dynamics but also provides valuable references for practical applications in related fields, promoting an effective integration of theory and practice.

The influence of electrokinetic width K on the velocities associated with the triangular and half-sinusoidal pulse waves at various times t ¯ is described in Figs. 5 and 6, respectively. From these plots, it can be seen that for a given time, the electrokinetic width K has a substantial impact on the near-wall velocity of any pulse wave, while exerting minimal effect on the near-center velocity. This phenomenon is attributed to the concentration of electroosmotic forces within the electric double layer (EDL) region adjacent to the wall. Increasing the electrokinetic width K leads to higher near-wall velocities. This is due to the fact that a larger electroosmotic width indicates a thinner EDL characteristic thickness [refer to Eq. (12)]. Consequently, the charge density on the solid surface grows, resulting in a more pronounced electric field effect that enhances the pulse EOF velocity. Meanwhile, we can observe from these two figures that owing to the variations of the pulse electric field force with time, the near-wall velocity surpasses the near-center velocity during the first half-cycle ( t ¯ = 0.5 , 1 . 5 )⁠, whereas the situation is reversed during the second half-cycle ( t ¯ = 2.5 , 3 . 5 )⁠. This phenomenon reveals that the fluid exhibits markedly different flow characteristics at various stages of the pulse, highlighting the intricate and profound impact of the pulse electric field on fluid behavior. Furthermore, by comparing Figs. 5 and 6, it is clear that the velocity related to the half-sinusoidal pulse wave exceeds that of the triangular pulse wave. This is because the continuous and smooth pulse electric field force generated by the half-sinusoidal pulse wave, paired with its higher frequency, facilitates easier movement of ions, leading to an increment in the pulse EOF velocity.

FIG. 5.
Variations of velocity corresponding to triangular pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 ). (a) t ¯ = 0.5; (b) t ¯ = 1.5; (c) t ¯ = 2.5; (d) t ¯ = 3.5.
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Variations of velocity corresponding to triangular pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 )⁠. (a) t ¯ = 0.5⁠; (b) t ¯ = 1.5⁠; (c) t ¯ = 2.5⁠; (d) t ¯ = 3.5⁠.

FIG. 5.
Variations of velocity corresponding to triangular pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 ). (a) t ¯ = 0.5; (b) t ¯ = 1.5; (c) t ¯ = 2.5; (d) t ¯ = 3.5.
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Variations of velocity corresponding to triangular pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 )⁠. (a) t ¯ = 0.5⁠; (b) t ¯ = 1.5⁠; (c) t ¯ = 2.5⁠; (d) t ¯ = 3.5⁠.

Close modal
FIG. 6.
Variations of velocity corresponding to half-sinusoidal pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 ). (a) t ¯ = 0.5; (b) t ¯ = 1.5; (c) t ¯ = 2.5; (d) t ¯ = 3.5.
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Variations of velocity corresponding to half-sinusoidal pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 )⁠. (a) t ¯ = 0.5⁠; (b) t ¯ = 1.5⁠; (c) t ¯ = 2.5⁠; (d) t ¯ = 3.5⁠.

FIG. 6.
Variations of velocity corresponding to half-sinusoidal pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 ). (a) t ¯ = 0.5; (b) t ¯ = 1.5; (c) t ¯ = 2.5; (d) t ¯ = 3.5.
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Variations of velocity corresponding to half-sinusoidal pulse wave with channel location y ¯ for different electrokinetic widths K and times t ¯ ( a ¯ = 2 )⁠. (a) t ¯ = 0.5⁠; (b) t ¯ = 1.5⁠; (c) t ¯ = 2.5⁠; (d) t ¯ = 3.5⁠.

Close modal

Figure 7 demonstrates the volumetric flow rate linked to the triangular and half-sinusoidal pulse waves as a function of time t ¯ for varying pulse widths a ¯⁠. Examining Fig. 7 holistically, we can find that the volumetric flow rates associated with the two pulse waves show periodic variations over time, with the period and amplitude increasing as the pulse width a ¯ expands. Also, it can be noted from Fig. 7 that the shapes of these volumetric flow rate profiles bear resemblance to the waveforms of the corresponding pulse electric field forces. Clearly, these findings are consistent with the results displayed in the velocity distribution, primarily due to the integral relationship between velocity and volumetric flow rate,28 as outlined in Eq. (26).

FIG. 7.
Variations of volumetric flow rate corresponding to various pulse waves with time t ¯ for different pulse widths a ¯ ( K = 20 ). (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.
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Variations of volumetric flow rate corresponding to various pulse waves with time t ¯ for different pulse widths a ¯ ( K = 20 )⁠. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.

FIG. 7.
Variations of volumetric flow rate corresponding to various pulse waves with time t ¯ for different pulse widths a ¯ ( K = 20 ). (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.
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Variations of volumetric flow rate corresponding to various pulse waves with time t ¯ for different pulse widths a ¯ ( K = 20 )⁠. (a) Triangular pulse wave; (b) half-sinusoidal pulse wave.

Close modal

Figures 8 and 9 present the volumetric flow rate related to the triangular and half-sinusoidal pulse waves as a function of electrokinetic width K during two half-cycles, respectively. It can be appreciated from Figs. 8 and 9 that within each half-cycle, the volumetric flow rates associated with these two pulse waves experience an initial sharp increase, succeeded by a relatively stable trend with increasing the electrokinetic width K. This phenomenon occurs because initially raising the electrokinetic width significantly amplifies the electric field effect, thereby propelling a boost in volumetric flow rate. Nevertheless, as the electrokinetic width K continues to increase, the restrictions imposed by the electric field effect intensify, resulting in a decelerated growth of the volumetric flow rate, ultimately converging towards a constant value. Moreover, these images indicate that the volumetric flow rate corresponding to the two pulse waves rises during the first half-cycle ( t ¯ = 0.5 , 1 , 1 . 5 , 2 ) and declines during the second half-cycle ( t ¯ = 2.5 , 3 , 3 . 5 , 4 ) over time. The reason behind this fact lies in the fluctuation of the driving force: a surge in the driving force during the initial half-cycle leads to an escalation of the volumetric flow rate, whereas a drop in the driving force in the subsequent half-cycle results in a reduction in the volumetric flow rate. Finally, upon comparison between Figs. 8 and 9, it is apparent that the volumetric flow rate linked to the half-sinusoidal pulse wave is notably higher than that of the triangular pulse wave.

FIG. 8.
Variations of volumetric flow rate corresponding to triangular pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 ). (a) First half-cycle; (b) second half-cycle.
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Variations of volumetric flow rate corresponding to triangular pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 )⁠. (a) First half-cycle; (b) second half-cycle.

FIG. 8.
Variations of volumetric flow rate corresponding to triangular pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 ). (a) First half-cycle; (b) second half-cycle.
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Variations of volumetric flow rate corresponding to triangular pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 )⁠. (a) First half-cycle; (b) second half-cycle.

Close modal
FIG. 9.
Variations of volumetric flow rate corresponding to half-sinusoidal pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 ). (a) First half-cycle; (b) second half-cycle.
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Variations of volumetric flow rate corresponding to half-sinusoidal pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 )⁠. (a) First half-cycle; (b) second half-cycle.

FIG. 9.
Variations of volumetric flow rate corresponding to half-sinusoidal pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 ). (a) First half-cycle; (b) second half-cycle.
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Variations of volumetric flow rate corresponding to half-sinusoidal pulse wave with electrokinetic width K during different half-cycles ( a ¯ = 2 )⁠. (a) First half-cycle; (b) second half-cycle.

Close modal

Figure 10 depicts the volumetric flow rates associated with the triangular and half-sinusoidal pulse waves across various electrokinetic widths K and pulse widths a ¯⁠. As expected, a thorough analysis of Fig. 10 reveals that irrespective of the electrokinetic width and pulse width, the volumetric flow rate in response to the half-sinusoidal pulse wave surpasses that of the triangular pulse wave. Additionally, it can be found from the figure that independent of the electrokinetic width, an enlarged in pulse width prolongs the period of volumetric flow rate variation related to these two pulse waves over time, accompanied by a corresponding amplification in amplitude. It is worth emphasizing that although these findings align with the previous ones, they provide a more exhaustive and higher-dimensional analysis by combining electrokinetic width K and pulse width a ¯ to thoroughly assess the impact of relevant parameters on the volumetric flow rate. A particularly intriguing observation from Fig. 10 is that regardless of the pulse width, the volumetric flow rates linked to both pulse waves grow with higher values of electrokinetic width. This phenomenon is especially noticeable at the alternating intervals of the two half-cycles within a complete cycle (namely, t ¯ = n T + a ¯ = ( 2 n + 1 ) a ¯ , n = 0 , 1 , 2 , …⁠), which can be attributed to the peak driving force at that juncture in time.

FIG. 10.
Comparison of volumetric flow rates corresponding to triangular and half-sinusoidal pulse waves for different electrokinetic widths K and pulse widths a ¯. (a) a ¯ = 2; (b) a ¯ = 5.
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Comparison of volumetric flow rates corresponding to triangular and half-sinusoidal pulse waves for different electrokinetic widths K and pulse widths a ¯⁠. (a) a ¯ = 2⁠; (b) a ¯ = 5⁠.

FIG. 10.
Comparison of volumetric flow rates corresponding to triangular and half-sinusoidal pulse waves for different electrokinetic widths K and pulse widths a ¯. (a) a ¯ = 2; (b) a ¯ = 5.
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Comparison of volumetric flow rates corresponding to triangular and half-sinusoidal pulse waves for different electrokinetic widths K and pulse widths a ¯⁠. (a) a ¯ = 2⁠; (b) a ¯ = 5⁠.

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The analysis above indicates that the flow velocity and volumetric flow rate of the fluid are highly influenced by the applied pulse electric field, particularly with respect to their profiles closely correlating with the pulse waveform. This correlation aligns with previous studies28–30 on rectangular pulse EOF, further reinforcing the reliability and universality of this discovery. The importance of this work lies in breaking from the previous single-mode rectangular pulse waveform and expanding the diversity of pulse waveforms to encompass various shapes such as triangular and half-sinusoidal. This provides a more comprehensive and diversified perspective for understanding the flow behavior of pulse EOF. Not only does it aid researchers in selecting the optimal pulse shape for specific fluid mechanics studies or experimental designs, but it also opens up new possibilities and directions for exploration and innovation in related fields in the future.

Finally, it should be noted that this study focuses commonly on used Newtonian fluids and relatively simple pulse EOF, as research on triangular and half-sinusoidal pulse EOFs is still in its preliminary stages. Future work will emphasize the exploration of non-Newtonian fluids like Burgers, Oldroyd-B, and Jeffrey fluids, alongside the investigation of other complex flows such as pulse electromagnetic flow and pulse electromagnetic EOF flow. These endeavors aim to uncover new insights and phenomena, thereby broadening our knowledge base and fortifying the ongoing research momentum in this area.

In the present work, we have delved into the pulse electroosmotic flow (EOF) of Newtonian fluids in a parallel plate microchannel driven by triangular and half-sinusoidal pulse electric fields. The analytical solutions for the velocity and volumetric flow rate of the pulse EOF associated with these two pulse waves are achieved by employing the Laplace transform method and the residual theorem, respectively. Moreover, the impacts of various parameters on these solutions are addressed. The primary findings can be consolidated as follows:

  • The velocities and volumetric flow rates linked to the triangular and half-sinusoidal pulse waves demonstrate periodic variations over time. The period and amplitude of their profiles increase as the pulse width a ¯ widens, with the shape of the profiles resembling the waveform of each pulse electric field force.

  • The electrokinetic width K significantly affects the near-wall velocity of any pulse wave, while having minimal impact on the near-center velocity. Expanding the electrokinetic width K leads to an augmented in near-wall velocity. During the first half-cycle ( t ¯ = 0.5 , 1 . 5 )⁠, the near-wall velocity is higher than the near-center one, while the scenario reverses during the second half-cycle ( t ¯ = 2.5 , 3 . 5 )⁠.

  • The volumetric flow rate corresponding to the triangular and half-sinusoidal pulse waves tends to rise sharply, eventually reaching a constant value as the electrokinetic width K grows. These volumetric flow rates ascend during the initial half-cycle ( t ¯ = 0.5 , 1 , 1 . 5 , 2 )⁠, only to descend during the subsequent half-cycle ( t ¯ = 2.5 , 3 , 3 . 5 , 4 ) as time elapses.

  • Regardless of the electrokinetic width K and pulse width a ¯⁠, the volumetric flow rate related to the half-sinusoidal pulse wave exceeds that of the triangular pulse wave. A larger electrokinetic width K results in an elevated volumetric flow rate, which is particularly discernible at t ¯ = ( 2 n + 1 ) a ¯ , n = 0 , 1 , 2 , … .⁠.

These revelations significantly deepen our understanding of pulse EOF dynamics while also offering essential guidance for the design of microfluidic systems. In such systems, the ability to exert precise control over fluid behavior like flow velocity and volumetric flow rate is paramount for a variety of applications, including lab-on-a-chip technologies and chemical analysis.

This work was partially funded by the National Natural Science Foundation of China (Grant No.11962021), the Natural Science Foundation of Inner Mongolia (Grant No. 2021MS05020), and the Basic Science Research Fund in the Universities Directly under the Inner Mongolia Autonomous Region (Grant Nos. JY20220138, JY20220331, and ZTY2024015).

The authors have no conflicts to disclose.

Dongsheng Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Haibin Li: Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Jiaofei Liu: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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