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Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function

  • Xue Gao , Guangpu Zhao EMAIL logo , Ying Zhang and Yue Zhang
Published/Copyright: December 31, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

Micro-devices fabrication has led to extensive scientific research on microfluidics and microelectromechanical systems. These devices are used for a wide range of technological applications, but research on microfluidic devices for nanofluids is relatively scarce. In response to this problem, the electrokinetic energy conversion (EKEC) efficiency of nanofluids is provided under the coupling effect of pressure gradient and magnetic field through porous microtubes using the Debye–Hückel linearization and the Green’s function method. The results show that the periodic excitation of the square waveform is more effective in increasing the EKEC efficiency. In addition, compared with previous studies, the average velocity is in good agreement with the cosine waveform at R = 0.2. It is worth noting that compared to cosine waves, the average velocity reaches 47% in triangular waves and 85% in square waves.

Keywords: electrokinetic energy conversion efficiency; streaming potential; Green’s function; nanofluids; porous microtubes

Nomenclature

B 0

magnetic field strength

Da

Darcy number

e

elementary charge

E 0

amplitude

E 1

characteristic electric field

E s

streaming potential

f

ionic friction coefficient

F

function of three time periods (cosine wave, square wave, and triangular wave)

Ha

Hartmann number

I 0

first class of zero order modified Bessel functions

I c

conduction current

I s

streaming current

κ

Debye–Hückel parameter

k

permeability of the porous media

K

electrokinetic width

k B

Boltzmann constant

n 0

ion density of the bulk liquid

n ±

number densities of the electrolyte cations and anions

P in

input powers

P out

output powers

Q in

volume flow rate of the input under pressure drive

R

radius

T

dimensionless time

T av

absolute temperature

u

axial velocity

u ±

combination of nanofluid advection velocity and electromigrative velocity

u e

Helmholtz–Smoluchowski electroosmotic velocity

z

valence of ions

φ

volume fraction of the nanoparticles

ψ

electrical potential

ψ 0

wall potential

ρ e

EDL charge density

ε

permittivity of the fluid

Ω′

amplitude of the pressure gradient

Ω

dimensionless frequency

ρ eff

effective density of the nanofluid

ρ s

density of a solid

ρ f

density of fluid

μ eff

effective viscosity of the nanofluid

μ f

viscosity of the base fluid

σ eff

effective electrical conductivity of the nanofluid

σ s

conductivity of the nanoparticles

σ f

conductivity of the base fluids

η

ratio of the viscosity of the base liquid to the effective viscosity

γ

ratio of the conductivity of the base fluid to the effective conductivity

ξ

efficiency of the electrokinetic energy conversion

1 Introduction

Recently, microfluidic technology [1,2,3] has important applications in the fields of microfluidic chip, medical diagnostics, fluid pumping technology, energy collection [4,5,6,7], etc. The high efficiency requirements corresponding to fluid flow and energy conversion in the resulting micro-device have received great attention [8,9,10]. Initially, similar to the macroscopic fluid drive mechanism, the liquid is driven by conventional pressure through a microfluidic device. However, due to the greatly reduced length scale of microfluidic devices, the fluid flow driven only by pressure will produce some disadvantages, such as power loss caused by friction and lack of precise flow control. Therefore, it is necessary to find solutions to optimize the drive mechanism for microfluidics. At present, the driving mechanism of micro-nano fluids includes pressure gradient, electric field, magnetic field or its appropriate combination [11,12,13], etc. Xie and Jian [14] studied the entropy generation analysis of a two-fluid dragging systems under the combined action of electric and magnetic fields. The results show that magnetic field can enhance local entropy production, while viscoelastic physical parameters can suppress local entropy production. Moreover, theoretical research can be used for the design of thermal flow control devices. Zheng and Jian [15] studied the rotational electroosmotic flow of two immiscible fluids in micro parallel channels under the action of an electric field. Chen et al. [16] analyzed the thermal transport of electromagnetohydrodynamic fluid in microtubules with electrokinetic effect and interfacial slip, which can be used to design exquisite and efficient magnetic fluid devices, especially within a specific range of thermal transport characteristics.

The chemical reaction generated by the contact of the electrolyte solution with the solid surface forms an electric double layer (EDL) near the wall of the microchannel, which is the main cause of electrokinetic flow. When a pressure gradient is applied upstream of the channel, the ions in EDL flow downstream to generate the streaming current. Subsequently, with the accumulation of downstream ions, the generation of potential difference between the upstream and downstream further results in an electric field opposite to the flow direction, which is called the streaming potential. At present, streaming potential is widely used in the analysis of electrokinetic energy conversion (EKEC) [17,18,19]. Gayen et al. [20] utilized the electrokinetic effects to use rigid baffles in two different concentration fluid micro mixers to improve the mixing efficiency. The results show that higher mixing quality can be achieved by considering the baffles in a specific orientation. Kumar et al. [21] studied the key parameters that affect mixing efficiency in a novel two-dimensional electroosmotic micromixer with nonaligned input and output microchannels. The results revealed that the mixing performance of SSAR-EM is strongly sensitive to the input fluid velocity, the phase difference applied to the micro-electrodes, the AC frequency. Jian [22] and Chen et al. [23] researched the variation in the streaming potential and EKEC efficiency of Newtonian fluids with pressure-dependent viscosity in parallel plates and circular tubes, respectively. The results show that the pressure-dependent viscosity effect can enhance the streaming potential and electromotive power output within a certain parameter range.

Porous medium is a kind of composite medium, which is composed of porous solid bone structure, and its main physical characteristics are extremely small pore size and large surface area. The research on porous media covers nature and industry, such as micro-nano-bubbles [24], oil exploitation [25], bioreactors [26], etc. At present, the microscopic research on porous media mainly focuses on the research of porous micro-nanotubes. At the microscopic level, the fluids in the pore space in porous media can be regarded as continuous terms. Biswas et al. [27] explored the mixed thermobioconvection of magnetically susceptible fluid containing copper nanoparticles and oxytactic bacteria in a novel W-shaped porous cavity. The results show that the magnitude of heat (Nu) and mass (Sh) transfer rate for the W-shaped cavity are high compared to conventional square and trapezoidal-shaped cavities. Mandal et al. [28] examined the magnetohydrodynamic (MHD) mixed bioconvection with oxytactic microorganisms suspended in copper-water nanofluid. The study can be helpful to understand the design and operation of many engineering and industrial systems and devices such as microbial fuel cell. Al-Farhany and Abdulsahib [29] studied the two-layer mixed convection of saturated porous media and nanofluids containing rotating cylinders. The results show that the local Nusselt number in the porous region is highest when rotating clockwise, and the local Nusselt number is highest in the nanofluid region when rotating counterclockwise.

Nanomaterial has attracted the attention of many scholars because of its wide range of applications, mainly due to unique heat transfer. Nanomaterials, including nanoparticles, nanotubes, nanofibers, and many other nanoscale structures, have been widely used. It is used to improve the performance of different applications, such as membrane filtration processes [30], optoelectronics [31], sewage treatment [32], etc. In terms of microfluidic technology, many scholars have devoted their attention to the research of nanofluids [33,34]. Zhao et al. [35] studied the heat transfer characteristics of heat development nanofluids (water-Al2O3) through porous microtubules under the action of applied magnetic fields. Turkyilmazoglu [36] analyzed the MHD flow and heat transfer characteristics of nanofluids by continuously stretching or contracting the permeable sheet under the conditions of velocity slip and temperature jump, and obtained closed analytical solutions for the flow and heat transfer parameters. Matin and Pop [37] mainly studied the forced convective heat and mass transfer of nanofluids in horizontal porous channels. The results show that an increase in the volume fraction of nanoparticles leads to an increase in the concentration of the electrolyte solution, which further leads to an increase in the temperature of the solution.

In contrast, the study of the electrokinetic phenomena of fluid flowing through micro-nano channels under unsteady conditions using the Green’s function method is relatively poor. The Green’s function method solves not only the stable field problem but also the non-stable field problem. Moghadam [38] used the Green’s function method to study the electroosmotic fully developed flow in a circular microchannel under an alternating electric field. The results show that when the diffusion time scale is much larger than the oscillation period (high frequency), the fluid in the double layer oscillates rapidly, while the bulk fluid almost remains stationary. Subsequently, Moghadam [39] again used the Green’s function method to study the effect of time-periodic electrokinetic-driven flow in a micro annular channel. The results show that the impact of specific excitation waveforms has proven to be more significant at lower frequencies, as the bulk fluid has more time to respond to transient changes in the applied unsteady field. Chen [40] combined Laplace transform for the time domain, Green’s function for the space domain and ε -algorithm acceleration method for fast convergence of the series solution to analyze hyperbolic heat conduction problems in cylindrical coordinates using a mixed Green’s function method. Kang et al. [41] studied the influence of the oscillation frequency of the sine alternating electric field on electroosmotic flow (EOF) by using the Green’s function. The results suggest that the geometry of the channel wall and the zeta potential have an effect on EOF. In addition, it was found that oscillation EOF depends on the frequency of the sine alternating electric field.

A conclusion that can be drawn from the literature analysis is that no one has examined the impact of porous medium at the pipelines side of one circular-shaped micromixer on EKEC. Subsequently, we numerically investigated the effect of the porous medium on EKEC and flow field by varying the Darcy number, Hartmann number, electrokinetic width, dimensionless frequency, and volume fraction of nanoparticles.

In this study, the streaming potential and EKEC efficiency of nanofluids through porous microtubes taking the combined consequences of pressure gradients and magnetic field into account is analyzed, where pressure gradients is the three time periodic functions [39,42] (cosine wave, square wave, and triangular wave). Meanwhile, the effects of dimensionless parameters (Darcy number, Hartmann number, electrokinetic width, dimensionless frequency and volume fraction of nanoparticles) on the velocity field, streaming potential field, and EKEC efficiency are considered. It is hoped that this study will provide a theoretical basis for the design and performance optimization of microfluidic devices in the future.

2 Theoretical derivation

2.1 Mathematical model

As shown in Figure 1, the flow problem of electrolyte solution in a cylindrical porous nano-channel with a radius of R driven by a pressure gradient in the axial direction of the channel is considered. The chemical reactions can take place between the electrolyte solution and solid surface to create EDL at the solid–liquid interface. At the same time, applied axial pressure gradient of the channel will result in potential difference, and thus generate streaming potential E s, contrary to the direction of flow. The magnetic field is applied in the perpendicular direction with the magnitude of B 0 . In addition, we know that nanoparticles can produce their own EDL, but we assume that these EDLs do not overlap.

Figure 1 Schematic theme of the problem geometry.
Figure 1

Schematic theme of the problem geometry.

2.2 EDL potential distribution

Consider the fully developed flow within the circular microchannel produced by the magnetic fields in the case of pressure gradients. In order to solve the velocity field, we should first obtain an analytical solution on electrical potential ψ and EDL charge density ρ e ( r ) . The electrical potential ψ and EDL charge density ρ e ( r ) satisfy the Poisson–Boltzmann equation, as follows:

(1) d 2 ψ d r 2 + 1 r d ψ d r = − ρ e ε ,

(2) ρ e ( r ) = z e ( n + − n − ) ,

where ε is the permittivity of the fluid, z is the valence of ions, e is the elementary charge, n + and n − are the number densities of the electrolyte cations and anions, respectively. The Boltzmann distribution is given by

(3) n ± = n 0 exp ∓ e z ψ ( r ) k B T av ,

where n 0 is the ion density of the bulk liquid, k B is the Boltzmann constant, and T av is the absolute temperature. Considering the small potential value of EDL, Eq. (3) can be approximated using Debye–Hückel linearization, namely,

(4) exp ∓ e z ψ ( r ) k B T av ≈ 1 ∓ e z ψ ( r ) k B T av .

Substituting Eqs (3) and (4) in Eq. (2), the net charge density is further obtained in the following form:

(5) ρ e ( r ) = − 2 e z n 0 sinh e z ψ ( r ) k B T av .

Next the following dimensionless group is introduced:

(6) r ¯ = r R , ψ ¯ = ψ ψ 0 , K = κ R ,

where R is the radius, ψ 0 = k B T av / e z is the wall potential, K stands for electrokinetic width, κ = ( 2 n 0 e 2 z 2 / ( ε k B T av ) ) 1 / 2 is the Debye–Hückel parameter, and 1 / κ represents the thickness of the EDL.

Substituting Eq. (6) in Eq. (1), the following form is obtained:

(7) d 2 ψ ¯ d r ¯ 2 + 1 r ¯ d ψ ¯ d r ¯ = K 2 sinh ( ψ ¯ ) .

The boundary conditions are

(8) ψ ¯ r ¯ ( r ¯ ) ∣ r ¯ = 0 = 0 , ψ ¯ ( r ¯ ) ∣ r ¯ = 0 = ψ 0 ,

where Eq. (7) is a Bessel function. Applying its general solution to solve Eqs (7) and (8), we finally obtain

(9) ψ ¯ ( r ¯ ) = ψ 0 I 0 ( K ) I 0 ( K r ¯ ) ,

where I 0 is the first class of zero-order modified Bessel functions. According to the relationship between electric potential and net charge density in Eq. (1), combined with Eq. (9), the final expression of net charge density is obtained as follows:

(10) ρ e ( r ) = − ε κ 2 ψ 0 I 0 ( K r ¯ ) I 0 ( K ) .

2.3 Velocity field

In circular porous microtube, applying a pressure gradient to promote the axial flow of fluid creates an electric field opposite to the flow direction, and the applied transverse magnetic field generates a magnetic force. In addition, porous media also generate resistance relative to nanofluids. Therefore, the corresponding Navier–Stokes equation for controlling flow can be obtained in the following form [35]:

(11) ρ eff ∂ u ∂ t = μ eff ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r − d P d x + ρ e E s − σ eff B 0 2 u − μ eff k u ,

where u is the axial velocity, − d P / d x = Ω ′ F ( ω t ) is the axial pressure gradient, Ω ′ is the amplitude of the pressure gradient, and F ( ω t ) is a function of three time periods (cosine wave, square wave, and triangular wave). E s is the streaming potential and is represented by E s = E 0 F ( ω t ) and E 0 is the amplitude, B 0 is the magnetic field strength, k is the permeability of the porous media, and ρ eff is the effective density of the nanofluid, which is given by

(12) ρ eff = φ ρ s + ( 1 − φ ) ρ f ,

where φ is the volume fraction of the nanoparticles, ρ s is the density of a solid, and ρ f is the density of fluid. μ eff is the effective viscosity of the nanofluid, given by Brinkman

(13) μ eff = μ f ( 1 − φ ) 2.5 ,

where μ f is the viscosity of the base fluid. In addition, σ eff is the effective electrical conductivity of the nanofluid, defined as

(14) σ eff = σ f 1 + 3 ( σ s / σ f − 1 ) φ ( σ s / σ f + 2 ) − ( σ s / σ f − 1 ) φ ,

where σ s is the conductivity of the nanoparticles and σ f is the conductivity of the base fluids. The following gives the slip free boundary conditions, symmetry conditions, and initial conditions of Eq. (11):

(15) u ∣ r = R = 0 , u r ∣ r = 0 = 0 , u ∣ t = 0 = 0 .

The following introduces dimensionless groups:

(16) u ¯ = u u e , u e = − ε ψ 0 E 1 μ f , η = μ f μ eff , γ = σ f σ eff , δ = η γ , E ¯ s = E s E 1 , Ha = B 0 R σ f μ f , Da = k R 2 , T = μ f ρ f R 2 t , Ω = ρ f R 2 μ f ω ,

where u e is the Helmholtz–Smoluchowski electroosmotic velocity and E 1 is the characteristic electric field. η is expressed as the ratio of the viscosity of the base liquid to the effective viscosity. γ is expressed as the ratio of the conductivity of the base fluid to the effective conductivity. Ha is the Hartmann number and Da is the Darcy number. T is the dimensionless time. Ω is the dimensionless frequency.

After substituting the dimensionless group of Eq. (16) into Eqs (11) and (15), the dimensionless equation and related boundary and initial conditions have the following form:

(17) ∂ u ¯ ∂ T − α 1 ∂ 2 u ¯ ∂ r ¯ 2 + 1 r ¯ ∂ u ¯ ∂ r ¯ − λ u ¯ = α 2 F ( Ω T ) + β K 2 I 0 ( K r ¯ ) I 0 ( K ) E ¯ s ︸ Q ( r ¯ , T ) ,

(18) u ¯ ∣ r ¯ = 1 = 0 , u ¯ r ¯ ∣ r ¯ = 0 = 0 , u ¯ ∣ T = 0 = 0 ,

where β = ρ f ρ eff , α 1 = β η , α 2 = β R 2 Ω ′ u e μ f , and λ = − β Ha 2 γ + β η Da .

2.4 Solution velocity of Green’s function method

Green’s function is commonly used to solve non-homogeneous differential equations with initial or boundary conditions, in order to obtain the numerical solution of the equation. Now, Green’s function can be used to solve Eq. (17) with boundary and initial conditions (18). The Green’s function is expressed as follows:

(19) ∂ g ∂ T − α 1 ∂ 2 g ∂ r ¯ 2 + 1 r ¯ ∂ g ∂ r ¯ − λ g = δ ( r ¯ − ℓ ) δ ( T − τ ) 2 π r ¯ 0 < r ¯ , ℓ < 1 , T , τ > 0 .

The boundary and initial conditions are as follows:

(20) g ( r ¯ , T ∣ ℓ , τ ) = g ( 1 , T ∣ ℓ , τ ) = 0 , g r ¯ ( r ¯ , T ∣ ℓ , τ ) = g r ¯ ( 0 , T ∣ ℓ , τ ) = 0 , g ( r ¯ , T ∣ ℓ , τ ) = g ( r ¯ , 0 ∣ ℓ , τ ) = 0 ,

where δ(x) is the Dirac delta function. First, applying the Laplace transform to Eq. (19), we obtain

(21) − α 1 ∂ 2 g ¯ ∂ r ¯ 2 + 1 r ¯ ∂ g ¯ ∂ r ¯ + ( s − λ ) g ¯ = δ ( r ¯ − ℓ ) e − s τ 2 π r ¯ .

Next perform the Hankel transform on Eq. (21), and it becomes the following form:

(22) g ¯ ( k i , s ∣ ℓ , τ ) = J 0 ( ℓ k i ) 2 π e − s τ s − ( λ − α 1 k i 2 ) .

After that, perform the inverse Laplace transform on Eq. (22), and apply the second shifting theorem to obtain

(23) g ˜ ( k i , T ∣ ℓ , τ ) = J 0 ( ℓ k i ) 2 π H ( T − τ ) e ( λ − α k i 2 ) ⋅ ( T − τ ) .

Finally, using inverse Hankel transform for Eq. (23), we obtain

(24) g ( r ¯ , T ∣ ℓ , τ ) = 1 π H ( T − τ ) ∑ i = 1 ∞ e ( λ − α k i 2 ) ⋅ ( T − τ ) J 0 ( ℓ k i ) J 0 ( r ¯ k i ) J 1 2 ( k i ) .

Subsequently, turn r̅, T to ℓ , τ in Eqs. (17) and (19)

(25) ∂ u ¯ ∂ τ − α 1 ∂ 2 u ¯ ∂ ℓ 2 + 1 ℓ ∂ u ¯ ∂ ℓ − λ u ¯ = Q ( ℓ , τ ) ,

(26) ∂ g ∂ τ − α 1 ∂ 2 g ∂ ℓ 2 + 1 ℓ ∂ g ∂ ℓ − λ g = δ ( r ¯ − ℓ ) δ ( T − τ ) 2 π ℓ .

Then, multiplying Eq. (25) by g and Eq. (26) by u̅, we obtain

(27) ∂ u ¯ ∂ τ g − α ∂ 2 u ¯ ∂ ℓ 2 + 1 ℓ ∂ u ¯ ∂ ℓ g − λ u ¯ g = Q ( ℓ , τ ) g ,

(28) ∂ g ∂ τ u ¯ − α ∂ 2 g ∂ ℓ 2 + 1 ℓ ∂ g ∂ ℓ u ¯ − λ g u ¯ = δ ( r ¯ − ℓ ) δ ( T − τ ) 2 π ℓ u ¯ .

Finally, subtracting and integrating over the radius r̅ and over the time T about Eqs (27) and (28), eventually we obtain

(29) u ¯ ( r ¯ , T ) = ∫ 0 T ∫ 0 1 Q ( ℓ , τ ) g ( r ¯ , T ∣ ℓ , τ ) d ℓ d τ ,

where Q ( ℓ , τ ) is defined in Eq. (17).

Next we introduce three time periodic functions to solve the analytical solution of velocity. If the time period function is the cosine waveform,

(30) F ( Ω T ) = cos ( Ω T ) .

The velocity distribution is

(31) u ¯ ( r ¯ , T ) = ∑ i = 1 ∞ α 2 J 0 ( r ¯ k i ) π J 1 2 ( k i ) ∫ 0 1 J 0 ( l k i ) d ℓ + β K 2 E ¯ 0 J 0 ( r ¯ k i ) π I 0 ( K ) J 1 2 ( k i ) ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) cos ( Ω T ) + Ω sin ( Ω T ) ) − ( α 1 k i 2 − λ ) ( α 1 k i 2 − λ ) 2 + Ω 2 ,

where E ¯ 0 = E 0 / E 1 .

If the time period function is the square waveform,

(32) F ( Ω T ) = 2 π ∑ m = 1 ∞ 1 − cos ( m π ) m sin ( m Ω T ) .

The velocity distribution is

(33) u ¯ ( r ¯ , T ) = ∑ i = 1 ∞ ∑ m = 1 ∞ 2 α 2 ( 1 − cos ( m π ) ) J 0 ( r ¯ k i ) π 2 m J 1 2 ( k i ) ∫ 0 1 J 0 ( ℓ k i ) d ℓ + 2 β K 2 E ¯ 0 ( 1 − cos ( m π ) ) J 0 ( r ¯ k i ) π 2 m I 0 ( K ) J 1 2 ( k i ) × ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 .

If the time period function is the triangular waveform,

(34) F ( Ω T ) = 8 π 2 ∑ m = 1 ∞ sin ( m π / 2 ) m 2 sin ( m Ω T ) .

The velocity distribution is

(35) u ¯ ( r ¯ , T ) = ∑ i = 1 ∞ ∑ m = 1 ∞ 8 α 2 sin ( m π / 2 ) J 0 ( r ¯ k i ) π 2 m 2 J 1 2 ( k i ) ∫ 0 1 J 0 ( ℓ k i ) d ℓ + 8 β K 2 E ¯ 0 sin ( m π / 2 ) J 0 ( r ¯ k i ) π 3 m 2 I 0 ( K ) J 1 2 ( k i ) × ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 .

2.5 Analytical solution of the streaming potential

According to the principle of streaming potential formation, when the electrolyte solution in the microchannel flows under a pressure gradient, a positive streaming current and a reverse conduction current will be generated, namely, I c + I s = 0 .

(36) I = 2 π e z ∫ 0 R ( n + u + − n − u − ) r d r = I c + I s = 0 ,

where I c is the conduction current, I s is the streaming current, u ± are the combination of nanofluid advection velocity ( u ) and electromigrative velocity (±ezE s/f), and hence they can be expressed as

(37) u ± = u ± e z E s f ,

where f is the ionic friction coefficient. Substituting Eqs (3), (4), and (37) in Eq. (36), we can obtain three time period streaming potentials. When the cosine wave is applied, the streaming potential is expressed as

(38) E s ¯ = B 1 + B 2 A ,

where

(39) B 1 = ∑ i = 1 ∞ α 2 π ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ ∫ 0 1 J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) cos ( Ω T ) + Ω sin ( Ω T ) ) − ( α 1 k i 2 − λ ) ( α 1 k i 2 − λ ) 2 + Ω 2 ,

(40) B 2 = ∑ i = 1 ∞ β K 2 E ¯ 0 π I 0 ( K ) ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) cos ( Ω T ) + Ω sin ( Ω T ) ) − ( α 1 k i 2 − λ ) ( α 1 k i 2 − λ ) 2 + Ω 2 ,

(41) A = e z E 1 2 f u e ψ 0 .

When the square wave is applied, the streaming potential is expressed as

(42) E ¯ s = B 3 + B 4 A ,

where

(43) B 3 = ∑ i = 1 ∞ ∑ m = 1 ∞ 2 α 2 π 2 1 − cos ( m π ) m ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ × ∫ 0 1 J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 ,

(44) B 4 = ∑ i = 1 ∞ ∑ m = 1 ∞ 2 β K 2 E ¯ 0 π 2 I 0 ( K ) 1 − cos ( m π ) m ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ × ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 .

When the triangular wave is applied, the streaming potential is expressed as

(45) E ¯ s = B 5 + B 6 A ,

where

(46) B 5 = ∑ i = 1 ∞ ∑ m = 1 ∞ 8 α 2 π 3 sin ( m π / 2 ) m 2 ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ × ∫ 0 1 J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 ,

(47) B 6 = ∑ i = 1 ∞ ∑ m = 1 ∞ 8 β K 2 E ¯ 0 π 3 I 0 ( K ) sin ( m π / 2 ) m 2 ∫ 0 1 J 0 ( r ¯ k i ) I 0 ( K r ¯ ) r ¯ J 1 2 ( k i ) I 0 ( K ) d r ¯ × ∫ 0 1 I 0 ( K ℓ ) J 0 ( ℓ k i ) d ℓ e ( λ − α 1 k i 2 ) T e ( α 1 k i 2 − λ ) T ( ( α 1 k i 2 − λ ) sin ( m Ω T ) − m Ω cos ( m Ω T ) ) + m Ω ( α 1 k i 2 − λ ) 2 + ( m Ω ) 2 .

2.6 Efficiency of EKEC

The mechanical energy of pressure-driven transmission and the chemical energy of EDL are converted into electric energy of flowing fluid. Its conversion efficiency ξ can be expressed as

(48) ξ = P out P in ,

where P in and P out , respectively, represent the input and output powers, and the expressions are as follows:

(49) P out = I s 2 E s 2 ,

(50) P in = − d P d x Q in ,

where Q in represents the volume flow rate of the input under pressure drive. Q in and I s can be identified as

(51) Q in = 2 π ∫ 0 R − 1 4 μ f d P d x ( R 2 − r 2 ) r d r ,

(52) I s = 2 π e z ∫ 0 R u ( n + − n − ) r d r .

Thus, we can get ξ from Eqs (48)–(52) as

(53) ξ = 4 e 2 z 2 n 0 E ¯ s 2 E 1 2 μ f Ω ′ 2 F ( Ω T ) 2 R 2 f .

3 Results and discussion

In the present work, the analytical expressions of velocity, streaming potential, and the EKEC efficiency for nanofluids in porous microtubes under the influences of axial pressure gradient and imposed magnetic field are debated. They change owing to some of the non-dimensional parameters defined above. In order to maintain consistency between the research results and practical applications (such as micromixer), the actual reference range of these dimensionless parameters should be afforded according to the relevant physical variables, as shown below: R = 100 μm , T av = 298 K , k B = 1.381 × 10 − 23 J K − 1 , ε = 7 × 10 − 10 C 2 N − 1 m − 2 , e = 1.6 × 10 − 19 C , z = 1, μ f = 8.91 × 10 − 3 kg m − 1 s − 1 , ρ s = 3 , 600 kg m − 3 , ρ f = 997.1 kg m − 3 , σ s = 10 − 12 s m − 1 , σ f = 0.05 s m − 1 , ψ 0 = − 0.025 V , and n 0 = 10 − 5 m − 3 . f is taken as 10 − 12 N s m − 1 , the electrokinetic width K ranges from 0 to 10 because of K = κ R . If the range of the imposed magnetic field B 0 is 40 mT to 0.44 T, the order of the Hartmann number Ha is evaluated as 3.6 × 10−6 from Eq. (16). The range of Da changes from 0.05 to 0.50 owing to Da = k R 2 , the range of φ changes from 0.00 to 0.05 and the range of Ω changes from 30 to 90 in the present work, which are in accordance with the physically acceptable values established in the studies.

In the present work, the average velocity ∣ u ¯ 0 ∣ of three waveforms in one period is defined as ∣ u ¯ 0 ∣ = ∫ 0 T ⁎ ∣ u ¯ ( r ¯ , T ) ∣ d T / T ⁎ , where T * = 2π/Ω for comparison with the study of Buren et al. [43]. If the Darcy number Da tends to infinity, then our research can be extended to be consistent with the research background of Buren et al. [43]. Figure 2 shows the change in average velocity with the radius r̅ when other parameters are constant. From the results of Figure 2, it can be seen that the average velocity of square waveform and triangular waveform are significantly higher than those studied by Buren et al. [43] under steady state conditions. This can be explained that square waveform and triangular waveform can be superimposed to provide a large velocity to fluid, thus improving the average velocity significantly. We can also see that the average velocity of cosine waveform is relatively small almost the same as that in the study by Buren et al. [43] starting from R = 0.2. This is because the cosine waveform is relatively single, and there is no multiple superposition of velocity, so the average velocity is smaller.

Figure 2 Comparisons of the average velocity ∣ u ¯ 0 ∣ | {\bar{u}}_{0}| between the present result and that of Buren et al. [43] when b 0 = 0 nm and ζ = −50 V, where Da = ∞, Ha = 0.2, K = 4, Ω = 7, and φ = 0.02.
Figure 2

Comparisons of the average velocity ∣ u ¯ 0 ∣ between the present result and that of Buren et al. [43] when b 0 = 0 nm and ζ = −50 V, where Da = ∞, Ha = 0.2, K = 4, Ω = 7, and φ = 0.02.

Figure 3 shows the dimensionless velocity distribution of different electrokinetic width with the cosine wave applied. Here the electrokinetic width is set to 3, 4, 5, and 6, respectively. From the four velocity profiles, it can be seen that the dimensionless velocity decreases with the increase in the electrokinetic width and changes periodically with time. The reason for this phenomenon is that the increase in the electrokinetic width means that the EDL becomes thinner, resulting in a decrease in the number of ions in the EDL. The number of ions driven by the pressure-driven flow decreases, which in turn leads to a decrease in velocity, because the overall velocity is a superposition of the pressure-driven flow and the reaction electroosmotic flow induced by the streaming potential.

Figure 3 Non-dimensional velocity distribution in microchannel with cosine waveform: (a) K = 3, (b) K = 4, (c) K = 5, and (d) K = 6 (Da = 0.5, Ha = 0.5, φ = 0.02, and Ω = 20).
Figure 3

Non-dimensional velocity distribution in microchannel with cosine waveform: (a) K = 3, (b) K = 4, (c) K = 5, and (d) K = 6 (Da = 0.5, Ha = 0.5, φ = 0.02, and Ω = 20).

The variation in dimensionless velocity distribution with dimensionless time when the dimensionless frequency is taken at different values under the periodic excitation of the square waveform is shown in Figure 4. Figure 4 shows the velocity distribution of dimensionless frequency of 30, 50, 70, and 90, respectively. It can be observed interestingly that the increase in the dimensionless frequency Ω leads to an overall decrease in dimensionless velocity and a shortening of the oscillation period. The argument for this discovery is based on the fact that larger dimensionless frequencies produce smaller time periods, which lead to shorter dimensionless velocity propagation times. Therefore, the oscillation accelerates, which indicates that the fluid cannot flow sufficiently in the microtubules resulting in a decrease in velocity. Obviously, dimensionless velocity is simple harmonic with time.

Figure 4 Non-dimensional velocity distribution in microchannel with square waveform: (a) Ω = 30, (b) Ω = 50, (c) Ω = 70, and (d) Ω = 90 (Da = 0.5, Ha = 1, K = 5, and φ = 0.02).
Figure 4

Non-dimensional velocity distribution in microchannel with square waveform: (a) Ω = 30, (b) Ω = 50, (c) Ω = 70, and (d) Ω = 90 (Da = 0.5, Ha = 1, K = 5, and φ = 0.02).

Figure 5 shows the dimensionless streaming potential distribution under different Da , Ha , K , and φ in the case of square waves.

Figure 5 Dimensionless streaming potential changes with the dimensionless time at square waveform. (a) When Darcy number takes different values, where Ha = 1, K = 3, φ = 0.02, and Ω = 3; (b) when Hartmann number takes different values, where Da = 0.5, K = 3, φ = 0.01, and Ω = 4; (c) when the electrokinetic width takes different values, where Da = 0.3, Ha = 2, φ = 0.02, and Ω = 4; (d) when the nanoparticle volume fraction takes different values, where Da = 0.5, Ha = 2, K = 3, and Ω = 5.
Figure 5

Dimensionless streaming potential changes with the dimensionless time at square waveform. (a) When Darcy number takes different values, where Ha = 1, K = 3, φ = 0.02, and Ω = 3; (b) when Hartmann number takes different values, where Da = 0.5, K = 3, φ = 0.01, and Ω = 4; (c) when the electrokinetic width takes different values, where Da = 0.3, Ha = 2, φ = 0.02, and Ω = 4; (d) when the nanoparticle volume fraction takes different values, where Da = 0.5, Ha = 2, K = 3, and Ω = 5.

Figure 5(a) displays the variation in the streaming potential with dimensionless time under different Darcy numbers. It can be clearly seen that the increase in the Darcy number leads to an overall increase in the streaming potential. The reason is that the increase in the Darcy number will lead to small obstacles and fluid friction in porous media. That is to say, as Da increases, obstacles in the porous medium decrease and the fluid friction decreases. Therefore, the pressure driven fluid velocity accelerates, resulting in the accumulation of more ions downstream of the channel, which in turn leads to a larger streaming potential. In addition, it is evident that for a fixed Darcy number, the streaming potential increases over time. Overall, the Darcy number affects the streaming potential distribution within the microchannel.

In Figure 5(b), we provide the variation in dimensionless streaming potential with dimensionless time under different Hartmann numbers. It is obvious from Figure 5(b) that the larger the dimensionless Hartmann number within the specified time range, the smaller the dimensionless streaming potential overall. In addition, we can find that the magnetic field possesses a retardant effect on the whole flow field. At the same time, it is observed that for a specific Ha value, the dimensionless streaming potential increases with the dimensionless time. The reason for this phenomenon is that the change in Hartmann number will trigger the increase in magnetic field force in the microchannel, resulting in greater resistance in the channel. In this way, the ability to drive the flow of ions in the EDL by pressure is weakened, which leads to a decrease in the streaming potential.

Figure 5(c) shows the variation in dimensionless streaming potential with dimensionless time under different electrokinetic widths. It is interesting to observe that the dimensionless streaming potential decreases with the increase in the electrokinetic width. In addition, for a fixed value of K , the dimensionless streaming potential increases with the increase in dimensionless time. The reason is that the increase in the electrokinetic width causes the thickness of the EDL to decrease, that is, the number of ions in the EDL decreases. Therefore, the number of ions in the EDL is reduced by pressure, which in turn leads to the smaller streaming potential. Figure 5(c) also shows the electrodynamic effects near the wall of the microfluidic system.

Figure 5(d) presents the change in dimensionless streaming potential with dimensionless time under different values of nanoparticle volume fractions. It is obvious from Figure 5(d) that with the increase in the volume fraction of nanoparticles, the dimensionless streaming potential shows an overall downward trend. The reason is that the increase in the volume fraction of nanoparticles can improve the effective viscosity of nanoparticles. Therefore, the increase in viscosity leads to a slowdown driven by pressure and a decrease in the number of ions downstream, which ultimately leads to an overall decrease in the streaming potential.

As shown in Figure 6(a), the change in dimensionless streaming potential under the periodic excitation of the triangular waveform with dimensionless time when different values are taken in Darcy numbers is plotted. It can be seen from Figure 6(a) that the dimensionless streaming potential increases with the increase in Darcy number, and for the fixed Darcy number, the dimensionless streaming potential first increases and then decreases with dimensionless time. The reason for this phenomenon is that the increase in the Darcy number will lead to an increase in the permeability of the porous medium, so more ions will be accumulated downstream of the channel, triggering the larger streaming potential.

Figure 6 Dimensionless streaming potential changes with the dimensionless time at triangular waveform. (a) When Darcy number takes different values, where Ha = 2, K = 3, φ = 0.02, and Ω = 5; (b) when Hartmann number takes different values, where Da = 0.5, K = 3, φ = 0.02, and Ω = 5; (c) when electrokinetic width takes different values, where Da = 0.4, Ha = 2, φ = 0.02, and Ω = 5; and (d) when dimensionless frequency takes different values, where Da = 0.5, Ha = 0.5, K = 3, φ = 0.02.
Figure 6

Dimensionless streaming potential changes with the dimensionless time at triangular waveform. (a) When Darcy number takes different values, where Ha = 2, K = 3, φ = 0.02, and Ω = 5; (b) when Hartmann number takes different values, where Da = 0.5, K = 3, φ = 0.02, and Ω = 5; (c) when electrokinetic width takes different values, where Da = 0.4, Ha = 2, φ = 0.02, and Ω = 5; and (d) when dimensionless frequency takes different values, where Da = 0.5, Ha = 0.5, K = 3, φ = 0.02.

As shown in Figure 6(b), the change in dimensionless streaming potential under the periodic excitation of the triangular waveform with dimensionless time when the Hartmann number is taken at different values is obtained. It can be seen from the figure that with the increase in Hartmann number, the dimensionless streaming potential decreases as a whole, and it shows an increase-decrease trend with time. The reason can be explained that the magnetic field has an obstructive effect on the entire flow field, and a larger Hartmann number means that the magnetic field force is larger, so the decrease in the velocity of driving fluid flow leads to a decrease in the number of ions downstream, which further leads to a decrease in the streaming potential.

As shown in Figure 6(c), the variation in dimensionless streaming potential under the periodic excitation of the triangular waveform with dimensionless time when the electrokinetic width is taken at different values is obtained. It can be clearly seen from the figure that the dimensionless streaming potential decreases with the increase in electrokinetic width, and shows a growth-decline trend with time. This can be explained by the fact that the increase in the electrokinetic width leads to the thinning of the EDL, which further leads to the decrease in the ion transport capacity in the EDL, and finally obtains the smaller streaming potential.

As shown in Figure 6(d), the change in dimensionless streaming potential with dimensionless time under the periodic excitation of the triangular waveform is obtained when the dimensionless frequency is taken at different values. It is obvious from the figure that with the increase in dimensionless frequency Ω , the dimensionless streaming potential shows an overall downward trend. This may be because the larger the dimensionless frequency, the smaller the oscillation period, in other words the time required for nanofluids to flow in microtubules is shortened. Therefore, the ions in the microtubules become unstable and the resistance to the downstream of the channel increases, which eventually leads to a decrease in dimensionless streaming potential. In addition, with the increase in time, dimensionless streaming potential takes a simple harmonic form.

Figure 7(a) describes the change in EKEC efficiency under the periodic excitation of the square wave with dimensionless time when the Darcy number takes different values. Here Darcy numbers are set to 0.05, 0.10, 0.20, and 0.50, respectively. From Figure 7(a), it can be interestingly observed that the larger the Darcy number, the higher the efficiency of EKEC in the channel. On the one hand, this may be because the permeability of porous media is large when the Darcy number increases, so the increase in the velocity of fluid flow in the channel leads to an increase in the number of ions downstream of the channel, and eventually a large streaming potential. On the other hand, according to the change in the streaming potential of Figure 5(a) with the Darcy number, the increase in the streaming potential obtained by the combined Eq. (53) further leads to the increase in the EKEC efficiency.

Figure 7 Change in EKEC efficiency with dimensionless time. (a) Square waveform when Ha = 1, K = 3, φ = 0.02, and Ω = 0.2; (b) square waveform when Da = 0.1, Ha = 2, φ = 0.05, and Ω = 0.4; (c) triangular waveform when Da = 0.3, K = 3, φ = 0.02, and Ω = 0.3; and (d) triangular waveform when Da = 0.5, Ha = 1, K = 3, and Ω = 0.2.
Figure 7

Change in EKEC efficiency with dimensionless time. (a) Square waveform when Ha = 1, K = 3, φ = 0.02, and Ω = 0.2; (b) square waveform when Da = 0.1, Ha = 2, φ = 0.05, and Ω = 0.4; (c) triangular waveform when Da = 0.3, K = 3, φ = 0.02, and Ω = 0.3; and (d) triangular waveform when Da = 0.5, Ha = 1, K = 3, and Ω = 0.2.

Figure 7(b) represents the change in EKEC efficiency with dimensionless time when the electrokinetic width is taken at different values under the periodic excitation of the square wave. It can be seen from Figure 7(b) that with the increase in electrokinetic width, the EKEC efficiency shows a downward trend, and with the passage of time, the EKEC efficiency shows exponential rapid growth. The reason is that the increase in the electrokinetic width will lead to the decrease in the streaming potential, and the decrease in the EKEC efficiency according to Eq. (53).

Figure 7(c) sketches the change in EKEC efficiency under periodic excitation of the triangle wave with dimensionless time when the Hartmann number is taken at different values. Here the Hartmann numbers are set to 0.5, 1.5, 2.0, and 3.0, respectively. The results show that the increase in Hartmann number reduces the EKEC efficiency overall. This may be due to the fact that a larger Hartmann number triggers an enhanced electrical resistance in nanofluids. Therefore, the nanofluidic velocity in microtubules slows down, resulting in incomplete energy conversion. In other words, the larger the Hartmann number, the greater the magnetic field force in the flow field, which ultimately leads to a decrease in the efficiency of EKEC. Similarly, in the process of dimensionless time, the EKEC efficiency has an increasing tendency for a certain Hartmann number.

Figure 7(d) pictures the change in EKEC efficiency under periodic excitation of the triangular wave with dimensionless time when the volume fraction of nanoparticles is taken at different values. It can be clearly seen from the figure that with the increase in the volume fraction of nanoparticles, the EKEC efficiency generally shows a downward trend. On the other hand, when the volume fraction of nanoparticles is fixed, the EKEC efficiency increases over dimensionless time. The evidence for this finding is based on the fact that the emergence of nanoparticles will increase the effective viscosity of nanofluids, thereby reducing the flow rate, which in turn leads to a decrease in the streaming potential, which further triggers a decrease in the efficiency of EKEC. Based on the above analysis, the streaming potential can be used as an indicator to measure the rise and fall of EKEC efficiency.

Figure 8 shows a comparison of the EKEC efficiency under the three time period functions with the same Darcy number. It can be seen from Figure 8 that under the same Darcy number, the EKEC efficiency under the applied square wave is higher than that of the triangle wave, followed by the cosine wave. The occurrence of this phenomenon can be explained as follows. First, at low dimensionless frequencies, specific waveforms have a greater influence on the flow field. Second, square waveforms produce higher local velocities than triangular waveforms. Finally, due to the square wave, the superposition of the triangle wave leads to an increase in the streaming potential, which further leads to the increase in the EKEC efficiency.

Figure 8 Comparison of EKEC efficiency under three time-period functions for the same Darcy number, with Ha = 1, K = 3, φ = 0.02, and Ω = 0.3.
Figure 8

Comparison of EKEC efficiency under three time-period functions for the same Darcy number, with Ha = 1, K = 3, φ = 0.02, and Ω = 0.3.

4 Conclusion

In this chapter, a theoretical study of the streaming potential and EKEC efficiency of porous microtubules are obtained considering the combined influence of pressure gradient and magnetic field, respectively. Among them, the velocity analytical solution under the time period excitation is obtained by using the Green’s function method. Next the variation in streaming potential and EKEC efficiency with time under three time period excitations with different dimensionless parameters, such as Darcy number, Hartmann number, electrokinetic width, dimensionless frequency, and nanoparticle volume fraction, is discussed. The results show that the EKEC efficiency increases with the increase in Darcy number, and decreases with the increase in Hartmann number, electrokinetic width, and nanoparticle volume fraction. The recommended parametric conditions for the best characteristics are as follows: Da = 0.1, Ha = 2, K = 6, Ω = 0.4, and φ = 0.05. As shown in Figure 2, average velocity ∣ u ¯ 0 ∣ has high similarity under the condition of Da = ∞ starting from R = 0.2 compared with the results obtained by Buren et al. [43] in the case of cosine waveform. Also, the average velocity of fluid is the highest in the case of square wave. It is worth noting that compared to cosine waves, the average velocity reaches 47% in the triangular waves and 85% in the square waves. Here microfluidic devices are usually used to analyze biological fluids, polymer solutions, and colloidal suspensions, so we hope that our research on nanofluid can provide some reference for engineering, medicine, etc.

  1. Funding information: The work was supported by the National Natural Science Foundation of China (Grant No. 11802147), the Foundation of Inner Mongolia Autonomous Region University Scientific Research Project (Grant No. NJZZ23076), the Basic Research Foundation for Universities of Inner Mongolia (Grant No. JY20230031), the Natural Science Foundation of Inner Mongolia (Grant No. 2023MS01012).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-11-12
Revised: 2023-12-13
Accepted: 2023-12-18
Published Online: 2023-12-31

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
  5. Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
  6. Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
  7. Analysis of the working mechanism and detection sensitivity of a flash detector
  8. Flat and bent branes with inner structure in two-field mimetic gravity
  9. Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
  10. Weighted survival functional entropy and its properties
  11. Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
  12. Study on the impulse mechanism of optical films formed by laser plasma shock waves
  13. Analysis of sweeping jet and film composite cooling using the decoupled model
  14. Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
  15. Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
  16. Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
  17. Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
  18. Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
  19. Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
  20. Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
  21. Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
  22. Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
  23. Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
  24. Controlling the physical field using the shape function technique
  25. Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
  26. Complex dynamics of a sub-quadratic Lorenz-like system
  27. Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
  28. Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
  29. Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
  30. A modified power-law model: Properties, estimation, and applications
  31. Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
  32. Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
  33. Predictability of machine learning framework in cross-section data
  34. Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
  35. Isomorphic shut form valuation for quantum field theory and biological population models
  36. Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
  37. Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
  38. Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
  39. A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
  40. Tangential electrostatic field at metal surfaces
  41. Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
  42. Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
  43. Influence of cationic surfactants on the growth of gypsum crystals
  44. Study on instability mechanism of KCl/PHPA drilling waste fluid
  45. Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
  46. A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
  47. Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
  48. Conserved vectors and solutions of the two-dimensional potential KP equation
  49. The reciprocal linear effect, a new optical effect of the Sagnac type
  50. Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
  51. The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
  52. Research on absolute ranging technology of resampling phase comparison method based on FMCW
  53. Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
  54. Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
  55. Fractional-order modeling: Analysis of foam drainage and Fisher's equations
  56. Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
  57. Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
  58. Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
  59. On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
  60. 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
  61. Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
  62. Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
  63. Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
  64. Does the Mott problem extend to Geiger counters?
  65. Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
  66. Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
  67. Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
  68. Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
  69. Study of fractional variable order COVID-19 environmental transformation model
  70. Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
  71. Influence of the regularization scheme in the QCD phase diagram in the PNJL model
  72. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
  73. Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
  74. Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
  75. High-sensitivity on-chip temperature sensor based on cascaded microring resonators
  76. Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
  77. Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
  78. Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
  79. Numerical simulation and analysis of Airy's-type equation
  80. A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
  81. Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
  82. ΛCDM and the principle of equivalence
  83. Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
  84. HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
  85. Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
  86. Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
  87. Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
  88. A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
  89. Application of IoT network for marine wildlife surveillance
  90. Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
  91. Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
  92. Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
  93. Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
  94. Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
  95. Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
  96. Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
  97. Micropolar flow and heat transfer within a permeable channel using the successive linearization method
  98. Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
  99. Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
  100. Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
  101. Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
  102. Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
  103. Review Article
  104. Effects of stretching on phase transformation of PVDF and its copolymers: A review
  105. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
  106. Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
  107. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
  108. Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
  109. Special Issue on Advanced Energy Materials - Part II
  110. Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
  111. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
  112. Adomian decomposition method for solution of fourteenth order boundary value problems
  113. New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
  114. On traveling wave solutions to Manakov model with variable coefficients
  115. Rational approximation for solving Fredholm integro-differential equations by new algorithm
  116. Special Issue on Predicting pattern alterations in nature - Part I
  117. Modeling the monkeypox infection using the Mittag–Leffler kernel
  118. Spectral analysis of variable-order multi-terms fractional differential equations
  119. Special Issue on Nanomaterial utilization and structural optimization - Part I
  120. Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations
https://doi.org/10.1515/phys-2023-0173
Keywords for this article
electrokinetic energy conversion efficiency; streaming potential; Green’s function; nanofluids; porous microtubes
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
  5. Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
  6. Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
  7. Analysis of the working mechanism and detection sensitivity of a flash detector
  8. Flat and bent branes with inner structure in two-field mimetic gravity
  9. Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
  10. Weighted survival functional entropy and its properties
  11. Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
  12. Study on the impulse mechanism of optical films formed by laser plasma shock waves
  13. Analysis of sweeping jet and film composite cooling using the decoupled model
  14. Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
  15. Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
  16. Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
  17. Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
  18. Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
  19. Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
  20. Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
  21. Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
  22. Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
  23. Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
  24. Controlling the physical field using the shape function technique
  25. Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
  26. Complex dynamics of a sub-quadratic Lorenz-like system
  27. Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
  28. Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
  29. Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
  30. A modified power-law model: Properties, estimation, and applications
  31. Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
  32. Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
  33. Predictability of machine learning framework in cross-section data
  34. Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
  35. Isomorphic shut form valuation for quantum field theory and biological population models
  36. Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
  37. Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
  38. Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
  39. A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
  40. Tangential electrostatic field at metal surfaces
  41. Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
  42. Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
  43. Influence of cationic surfactants on the growth of gypsum crystals
  44. Study on instability mechanism of KCl/PHPA drilling waste fluid
  45. Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
  46. A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
  47. Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
  48. Conserved vectors and solutions of the two-dimensional potential KP equation
  49. The reciprocal linear effect, a new optical effect of the Sagnac type
  50. Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
  51. The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
  52. Research on absolute ranging technology of resampling phase comparison method based on FMCW
  53. Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
  54. Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
  55. Fractional-order modeling: Analysis of foam drainage and Fisher's equations
  56. Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
  57. Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
  58. Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
  59. On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
  60. 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
  61. Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
  62. Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
  63. Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
  64. Does the Mott problem extend to Geiger counters?
  65. Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
  66. Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
  67. Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
  68. Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
  69. Study of fractional variable order COVID-19 environmental transformation model
  70. Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
  71. Influence of the regularization scheme in the QCD phase diagram in the PNJL model
  72. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
  73. Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
  74. Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
  75. High-sensitivity on-chip temperature sensor based on cascaded microring resonators
  76. Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
  77. Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
  78. Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
  79. Numerical simulation and analysis of Airy's-type equation
  80. A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
  81. Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
  82. ΛCDM and the principle of equivalence
  83. Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
  84. HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
  85. Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
  86. Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
  87. Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
  88. A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
  89. Application of IoT network for marine wildlife surveillance
  90. Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
  91. Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
  92. Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
  93. Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
  94. Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
  95. Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
  96. Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
  97. Micropolar flow and heat transfer within a permeable channel using the successive linearization method
  98. Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
  99. Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
  100. Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
  101. Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
  102. Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
  103. Review Article
  104. Effects of stretching on phase transformation of PVDF and its copolymers: A review
  105. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
  106. Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
  107. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
  108. Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
  109. Special Issue on Advanced Energy Materials - Part II
  110. Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
  111. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
  112. Adomian decomposition method for solution of fourteenth order boundary value problems
  113. New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
  114. On traveling wave solutions to Manakov model with variable coefficients
  115. Rational approximation for solving Fredholm integro-differential equations by new algorithm
  116. Special Issue on Predicting pattern alterations in nature - Part I
  117. Modeling the monkeypox infection using the Mittag–Leffler kernel
  118. Spectral analysis of variable-order multi-terms fractional differential equations
  119. Special Issue on Nanomaterial utilization and structural optimization - Part I
  120. Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations
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