Study of a Newtonian Fluid through Circular Channels with Slip Boundary Taking into Account Electrokinetic Effect

Abstract

We study the slip flow of fluids driven by the combined effect of electrical force and pressure gradient. The underlying boundary value problem is solved through the use of Fourier series expansion in time and Bessel function in space. The exact solutions and numerical investigations show that the slip length and electrical field parameters have significant effects on the velocity profile. By varying these system parameters, one can achieve smooth velocity profiles or wave form profiles with different wave amplitude and frequency. This opens the way for optimizing the flow by choosing the slip length, the electrical field, and electrolyte solutions.

1. Introduction

Over the past few decades, advances in nanoscience and nanotechnology have led to the development of many microelectromechanical systems and devices such as heat exchanger [1], micropump [2], lab-on-a-chip diagnostic devises [3], drug delivery systems [4], energy conversion, and biological sensing devices [5]. Most of these systems and devices involve fluid flow in microtubes and microchannels. To control the microfluidics in microchannels so as to achieve optional system performance, it is essential to study the fundamental mechanics of microflows and derive better models and understanding of the flow mechanism and flow behaviour.

Flow of microfluidics may be driven by pressure gradient or by electrical forces. For electrical-driven flow, the solid surface of the microchannel is electrically charged to generate a region in the fluid with a distribution of electrical charges near the channel surface. This region is called the electrical double layer (EDL). The EDL, on one hand, has the effect of retarding liquid flow driven by external pressure gradient to form a streaming potential, whereas, on the other hand, it can induce fluid flow by applying an external electric field. In this work, we will study the flow of microfluidics in microchannels driven by the combination of pressure gradient and externally applied electric field.

The equations governing the flow of microfluidics in microchannels include the Navier-Stokes equations, the incompressible continuity equation, and boundary conditions. Traditionally, the no-slip boundary condition is used [6, 7]. However, recent molecular dynamic simulations and experiments in micrometer scale show that the flow of fluids in micrometer scale is granular and slip occurs between the fluid and the solid surface [8–12]. Therefore, the no-slip condition does not work for fluid flow in microchannels. In this work, the Navier slip boundary condition will be used; namely, the tangential fluid velocity relative to the solid surface is proportional to the shear stress on the solid-fluid interface. The validity of the Navier slip boundary condition is supported by many experimental results [13–15].

For many fluid flow problems, under the nonslip assumption, exact and numerical solutions have been obtained and can be found in the literature [6, 16–18]. Steady state solutions under slip conditions have also been established for flows of Newtonian fluids through pipes, channels, and annulus [19, 20]. More recently, various analytical solutions for pressure-driven time-dependent slip flows of Newtonian fluids through microtubes and microannulus were derived [16, 18]. Various attempts have also been made to study the electrically driven fluid flows. Rice and Whitehead [21] investigated the steady-state liquid flow due to an electric field in circular capillaries. Levine et al. [22] analysed the electrokinetic steady flow in a narrow parallel-plate microchannel. Yang et al. [23] studied the flow in rectangular microchannels and Mala et al. [24] in parallel-plate microchannels.

Motivated by the previous work, this work aims to generalize the result in [25] for the pressure-driven slip flow to the case with the combined effect of pressure-driving forces and electrically driving forces. The rest of the paper is organized as follows. In Section 2, we present the mathematical model for the problem, consisting of the governing field equations and boundary conditions. In Section 3, we derive the exact solutions for the velocity field through the use of Bessel functions in space and Fourier series expansion in time. Based on the velocity solutions, we then establish the analytical solutions for the stress tensor and flow rate. In Section 5, we investigate the influence of the electric field and the surface slippage on the flow behaviour. Then a conclusion is given in Section 6.

2. Mathematical Model and Formulation

In this paper, we consider the flow of microfludics through a circular microchannel driven by both pressure gradient and external electric field. The cylindrical polar coordinate (𝑟, 𝜃, 𝑧), with the 𝑧-axis being in the axial direction, is used in the formulation. The governing field equations for the problem include the Navier-Stokes equations and the continuity equation. Let 𝑣=(𝑣_𝑟,𝑣_𝜃,𝑢) be the velocity vector with 𝑣_𝑟,𝑣_𝜃, and 𝑢 being, respectively, the components of velocity in the radial direction, the transverse direction, and axial direction. Then the continuity equation and the Navier-Stokes equation in the 𝑧-direction are 𝜕²𝑣_𝑟𝜕𝑟²+𝑣_𝑟𝑟+𝜕𝑣_𝜃+𝑟𝜕𝜃𝜕𝑢𝜕𝑧=0,𝜕𝑢𝜕𝑡+𝑣_𝑟𝜕𝑢+𝑣𝜕𝑟𝑟^𝜃𝜕𝑢𝜕𝜃+𝑢𝜕𝑢𝜕𝜃=𝑔_𝑧−1𝜌_𝑓𝜕𝑝+𝜇𝜕𝑧𝜌_𝑓𝜕²𝑢𝜕𝑟²+𝜕𝑢+1𝑟𝜕𝑟𝑟²𝜕²𝑢𝜕𝜃²+𝜕²𝑢𝜕𝑧²+𝜌_𝑒𝜌_𝑓𝐸^′_𝑧, (1) where 𝑝 is the pressure, 𝜇 is the dynamic viscosity, 𝜌_𝑓 and 𝜌_𝑒, are respectively, the fluid density and the electric charge density, and 𝐸^′_𝑧 is the externally applied electric field.

Assuming that the flow is axially symmetric and the radial and transverse velocity components are negligible, then (1) admits solution of the form 𝑣_𝑟=0,𝑣_𝜃=0,𝑢=𝑢(𝑟,𝑡) (2) with 𝑢(𝑟,𝑡) determined by 𝜕𝑢−𝜇𝜕𝑡𝜌_𝑓𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢1𝜕𝑟=−𝜌_𝑓𝜕𝑝−𝜌𝜕𝑧𝜌^𝑒_𝑓𝐸^′_𝑧. (3) Based on [25], the velocity boundary condition for the problem can be written as 𝜕𝑢()()𝜕𝑟0,𝑡=0,𝑢𝑅,𝑡=−𝑙𝜕𝑢().𝜕𝑟𝑅,𝑡 (4) In this paper, we study the flow driven by both pressure gradient 𝜕𝑝/𝜕𝑧 and an externally applied electrical field 𝐸^′_𝑧. Without loss of generality, we express the pressure gradient by Fourier series; namely, 𝜕𝑝=𝜕𝑧𝑞(𝑡)=𝑎₀+^∞_𝑛=1𝑎_𝑛cos(𝑛𝜔𝑡)+𝑏_𝑛sin(𝑛𝜔𝑡), (5) which can also be expressed by 𝜕𝑝𝜕𝑧=Re^∞_𝑛=0𝑐_𝑛𝑒^{𝑖𝑛𝜔𝑡}, (6) where Re() is to take the real part of the complex quantity. To determine the free charge density 𝜌_𝑒, let 𝜓(𝑟) be the electric potential associated with double layer at the equilibrium state. Then based on [26], we have 𝜌_𝑒=−𝜅²𝑑𝜓,₂𝜓𝑑𝑟²+1𝑟𝑑𝜓𝑑𝑟=𝜅²𝜓(𝑅)𝜓,=𝜓_𝑠,𝜓(0)isfinite, (7)(8)(9) where 𝜓_𝑠 is the surface potential at the wall 𝑟=𝑅. Now substituting (7) into (3), we have 𝜕𝑢−𝜇𝜕𝑡𝜌_𝑓𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢1𝜕𝑟=−𝜌_𝑓𝜕𝑝+1𝜕𝑧𝜌_𝑓𝜅²𝜓𝐸^′_𝑧. (10) Hence the problem is to solve the boundary value problem (8)-(9) for 𝜓 and then the partial differential equation (10) subject to boundary conditions (4) for 𝑢, which will be done in Section 3.

3. Solution for the Transient Velocity Field

Let 𝜅𝑟=𝑟; (8) becomes 𝑟²𝑑²𝜓𝑑𝑟²+𝑟𝑑𝜓𝑑𝑟−𝑟²𝜓=0 (11) which gives solution 𝜓=𝐴𝐽₀(𝜅𝑟)+𝐵𝑌₀(𝜅𝑟), (12) where 𝐴 and 𝐵 denote integration constants and 𝐽₀ and 𝑌₀ are the zero-order Bessel functions of the first and second kinds, respectively. From (12) and boundary condition (9), we obtain 𝜓𝜓=^𝑠𝐽₀()𝐽𝜅𝑅⁰(𝜅𝑟) (13) and (10) becomes 𝜕𝑢−𝜇𝜕𝑡𝜌_𝑓𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢1𝜕𝑟=−𝜌_𝑓Re^∞_𝑛=1𝑐_𝑛𝑒^{𝑖𝑛𝜔𝑡}+1𝜌_𝑓𝜅²𝜓_𝑠𝐽₀()𝐽𝜅𝑅⁰(𝜅𝑟)𝐸^′_𝑧 (14) which admits solution of the form 𝑢(𝑟,𝑡)=𝑢_𝑝(𝑟,𝑡)+𝑢_𝑒(𝑟,𝑡), (15) where 𝑢_𝑝 is the solution corresponding to the first term on the right hand side and 𝑢_𝑒 is the solution contributed from the second term of the right hand side. From the superposition principle, based on the work of [25], we have 𝑢_𝑝()𝑎𝑟,𝑡=𝐶+⁰𝑟4𝜇²+^∞_𝑛=1Re𝑑_𝑛𝐽₀𝛽_𝑛𝑟+𝑐_𝑛𝑖𝜌_𝑓𝑒𝑛𝜔^{𝑖𝑛𝜔𝑡}, (16) where 𝛽²_𝑛=𝑛𝛽² with 𝛽²=(𝜌_𝑓𝜔/𝜇)𝑖. Now, we proceed to find the solution contributed from the second term on the right hand side of (15). Let 𝑢_𝑒(𝑟)=𝐶₀(𝑟)𝑡+𝐶₁(𝑟) and substitute it into (14); then we have −𝜇𝜌_𝑓𝐶₀^′′(𝑟)+1𝑟𝐶^′₀(𝑟)𝑡+𝐶₀(𝑟)−𝜇𝜌_𝑓𝐶₁^′′(𝑟)+1𝑟𝐶^′₁(𝑟=1)𝜌_𝑓𝜅²𝜓_𝑠𝐽₀()𝐸𝜅𝑅^′𝑧𝐽₀(𝜅𝑟). (17) For the above equation to hold for any instant of time 𝑡, we require 𝐶₀^′′(𝑟)+1𝑟𝐶^′₀(𝑟)𝐶=0,₀(𝑟)−𝜇𝜌_𝑓𝐶₁^′′(𝑟)+1𝑟𝐶^′₁(𝑟=1)𝜌_𝑓𝜅²𝜓_𝑠𝐽₀()𝐸𝜅𝑅^′𝑧𝐽₀(𝜅𝑟). (18) Thus, 𝐶₀=𝐷₁ln𝑟+𝐷₂. (19) As 𝐶₀ must be bounded at 𝑟=0 but ln𝑟 has singularity at 𝑟=0, we require that 𝐷₁=0. From (18), we have 𝐶₁=1𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰()+𝜌𝜅𝑟^𝑓𝜇𝐷₂4𝑟², (20) and then by substituting 𝐶₀ and 𝐶₁ into 𝑢_𝑒(𝑟,𝑡), we have 𝑢_𝑒(𝑟,𝑡)=𝐷₂1𝑡+𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰()+𝜌𝜅𝑟^𝑓𝜇𝐷₂4𝑟², (21) Hence, by substituting (16) and (21) into (15), we obtain 𝑢()𝑎𝑟,𝑡=𝐶+⁰𝑟4𝜇²+^∞_𝑛=1Re𝑑_𝑛𝐽₀𝛽_𝑛𝑟+𝑐_𝑛𝑖𝜌_𝑓𝑒𝑛𝜔^{𝑖𝑛𝜔𝑡}+𝐷₂1𝑡+𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰()+𝜌𝜅𝑟^𝑓𝜇𝐷₂4𝑟². (22)

Substituting (22) into boundary condition (4) yields 𝑎𝐶+⁰𝑅4𝜇²𝑎+𝑙⁰12𝜇𝑅+𝜇𝜓_𝑠𝐸^′_𝑧−𝑙𝜅𝜓_𝑠𝐸^′_𝑧𝜇𝐽₀()𝐽𝜅𝑅¹(𝜅𝑅)+𝐷₂𝜌𝑡+^𝑓𝐷₂𝑅²𝜌4𝜇+𝑙^𝑓𝐷₂𝑅+2𝜇_∞𝑛=1Re𝑑_𝑛𝐽₀+𝑐𝛽𝑅^𝑛𝑖𝜌_𝑓𝑛𝜔−𝑙𝛽_𝑛𝑑_𝑛𝐽₁𝛽_𝑛𝑅𝑒^{𝑖𝑛𝜔𝑡}=0. (23) For the above equation to hold for any instant of time 𝑡, we require 𝐷₂=0 and 𝑎𝐶+⁰𝑅4𝜇²𝑎+𝑙⁰12𝜇𝑅+𝜇𝜓_𝑠𝐸^′_𝑧𝜅−𝑙𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹()𝑑𝜅𝑅=0,_𝑛𝐽₀+𝑐𝛽𝑅^𝑛𝑖𝜌_𝑓𝑛𝜔−𝑙𝛽_𝑛𝑑_𝑛𝐽₁𝛽_𝑛𝑅=0, (24) which give 𝑎𝐶=−⁰𝑅²4𝜇1+2𝑙𝑅−1𝜇𝜓_𝑠𝐸^′_𝑧𝜅+𝑙𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹(),𝑑𝜅𝑅_𝑛=−𝑐_𝑛𝑖𝜌_𝑓𝑛𝜔𝐽₀𝛽𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅. (25) Hence, by substituting (25) into (22), we obtain 𝑎𝑢=−⁰𝑅²4𝜇1+2𝑙𝑅−𝑟²𝑅²−1𝜇𝜓_𝑠𝐸^′_𝑧𝜅+𝑙𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹()−𝜅𝑅_∞𝑛=1𝑐Re^𝑛𝑖𝜌_𝑓𝐽𝑛𝜔⁰𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅−1𝑒^{𝑖𝑛𝜔𝑡}+1𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰(𝜅𝑟). (26) We should remark that if 𝑙=0, solution (26) reduces to the no-slip solution 𝑎𝑢=−⁰𝑅²𝑟4𝜇1−²𝑅²−1𝜇𝜓_𝑠𝐸^′_𝑧−^∞_𝑛=1𝑐Re^𝑛𝑖𝜌_𝑓𝐽𝑛𝜔⁰𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑅−1𝑒^{𝑖𝑛𝜔𝑡}+1𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰(𝜅𝑟). (27)

4. Solutions for the Flow Rate and Stresses

From the velocity solution (26), we get the flow rate as follows: 𝑄(𝑡)=^𝑅₀𝑎2𝜋𝑟𝑢𝑑𝑟=−⁰𝜋𝑅³𝑅2𝜇𝑙+4+𝑙𝜋𝑅²𝜅𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹(𝜅𝑅)+^𝑅₀1𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅⁰(𝜅𝑟)𝑟𝑑𝑟−2𝜋𝜌_𝑓𝜔×Re^∞_𝑛=1𝑐_𝑛𝑖𝑒^{𝑖𝑛𝜔𝑡}𝑛^𝑅₀𝐽₀𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑅−𝑙𝐽₁𝛽_𝑛𝑅−1𝑟𝑑𝑟−𝜋𝑅²𝜇𝜓_𝑠𝐸^′_𝑧. (28) From the identity 𝑑𝑥𝑑𝑥^𝑛𝐽_𝑛(𝑥)=𝑥^𝑛𝐽_𝑛−1(𝑥), (29) we have 𝑑𝑑𝑟𝑟𝐽₁𝛽_𝑛𝑟=𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑟, (30) and hence ^𝑅₀𝑟𝐽₀𝛽_𝑛𝑟1𝑑𝑟=𝛽_𝑛𝑟𝐽₁𝛽_𝑛𝑟^𝑅₀=1𝛽_𝑛𝑅𝐽₁𝛽_𝑛𝑅. (31) Therefore from (28) and the above formula, we get 𝑄(𝑡)𝑎=−⁰𝜋𝑅³𝑅2𝜇𝑙+4+𝑙𝜋𝑅²𝜅𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹()+𝑅𝜅𝑅𝜓𝜇𝜅^𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹(𝜅𝑅)−2𝜋𝜌_𝑓𝜔×Re^∞_𝑛=1𝑐_𝑛𝑖𝑒^{𝑖𝑛𝜔𝑡}𝑛𝑅𝐽₁𝛽_𝑛𝑅𝛽_𝑛𝐽₀𝛽_𝑛𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅−𝑅²2−𝜋𝑅²𝜇𝜓_𝑠𝐸^′_𝑧. (32) The total amount of fluid, flowing through the tube during the period [0,𝑇], can then be determined as follows: 𝑄_𝑇=^𝑇₀𝑄(𝑡)𝑎𝑑𝑡=−⁰𝜋𝑅³𝑇𝑅2𝜇𝑙+4+𝑙𝜋𝑅²𝜅𝜇𝜓_𝑠𝐸^′_𝑧𝑇𝐽₀()𝐽𝜅𝑅¹()+𝜅𝑅𝜋𝑅𝜓𝜇𝜅^𝑠𝐸^′_𝑧𝑇𝐽₀()𝐽𝜅𝑅¹(𝜅𝑅)−2𝜋𝜌_𝑓𝜔²×Re^∞_𝑛=1𝑐_𝑛𝑒^{𝑖𝑛𝜔𝑇}−1𝑛²×𝑅𝐽₁𝛽_𝑛𝑅𝛽_𝑛𝐽₀𝛽_𝑛𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅−𝑅²2−𝜋𝑅²𝜇𝜓_𝑠𝐸^′_𝑧𝑇. (33)

Now, we determine the stresses in the fluid. From 𝑣=𝑒_𝑟0+𝑒_𝜃0+𝑒_𝑧𝑢(𝑟), we get ∇𝑣=00𝜕𝑢𝜕𝑟000𝜕𝑢𝜕𝑟00. (34) From the above formulae and (26), we obtain 𝑑_𝑟𝜃=𝑑_𝜃𝑧=𝑑_𝑟𝑟=𝑑_𝜃𝜃=𝑑_𝑧𝑧,𝑑_𝑟𝑧=𝑎₀𝑟+2𝜇^∞_𝑛=1𝑐Re^𝑛𝑖𝜌_𝑓𝛽𝑛𝜔^𝑛𝐽₁𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅𝑒^{𝑖𝑛𝜔𝑡}−𝜅𝜇𝜓_𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹(𝜅𝑟). (35) Thus from the constitutive equations for Newtonian fluid, we get 𝜎_𝑟𝑟=𝜎_𝜃𝜃=𝜎_𝑧𝑧=−𝑝=𝑞(𝑡)𝑥+𝑝₀(𝑡),𝜎_𝑟𝜃=𝜎_𝜃𝑧𝜎=0,_𝑟𝑧=𝑎₀𝑟+2𝜇^∞_𝑛=1𝑐Re^𝑛𝑖𝜌_𝑓𝛽𝑛𝜔^𝑛𝐽₁𝛽_𝑛𝑟𝐽₀𝛽_𝑛𝑅−𝑙𝛽_𝑛𝐽₁𝛽_𝑛𝑅𝑒^{𝑖𝑛𝜔𝑡}𝜓−2𝜇𝜅^𝑠𝐸^′_𝑧𝐽₀()𝐽𝜅𝑅¹(𝜅𝑟), (36) where 𝑝₀(𝑡) is an arbitrary constant which may be chosen to meet certain pressure conditions.

5. Numerical Investigation

In this section, we use the solutions obtained in previous sections to study the influence of the combined effect of electrical field and pressure gradient on the velocity profile and flow rate. As the pressure gradient can be expressed as a Fourier series, without loss of generality, we consider here only two cases of pressure gradient, including the constant pressure gradient and a cosine wave form pressure gradient.

Case 1. Consider 𝑑𝑝𝑑𝑧=𝑎₀. (37)
In this case, we have 𝑐_𝑛=0 for all 𝑛≥1, and thus, from (26) and (32), we get the normalized velocity and flow rate as follows: 𝑢^∗=−4𝜇𝑎₀𝑅²𝑢=1+2𝑙𝑅−𝑟²𝑅²−4𝜇𝑎₀𝑅²𝑙𝜅𝜇𝐽₁(𝜅𝑅)𝐽₀()+1𝜅𝑅𝜇𝐽₀(𝜅𝑟)𝐽₀()−1𝜅𝑅𝜇𝜓_𝑠𝐸^′_𝑧,𝑄^∗(𝑡)=−2𝜇𝑎₀𝜋𝑅³𝑄(𝑡)=𝑅𝑙+4−2𝜇𝑎₀𝜋𝑅²×𝑙𝜋𝜅𝑅𝜇𝐽₁(𝜅𝑅)𝐽₀()+1𝜅𝑅𝐽𝜇𝜅¹(𝜅𝑅)𝐽₀()−𝜅𝑅𝜋𝑅𝜇𝜓_𝑠𝐸^′_𝑧. (38) We should address here that the velocity and flow rate are linearly propositional to the slip length 𝑙.

Case 2. Consider 𝑑𝑝𝑑𝑧=𝑎₁cos(𝜔𝑡). (39)
In this case, we have 𝑎₀=0,𝑐₁=𝑎₁∈𝑅,𝑐_𝑛=0 for all 𝑛≥2. For simplification, we normalize the variables as follows: 𝛽^∗=𝛽𝑅,𝑟^∗=𝑟𝑅∈[0,1],𝑙^∗=𝑙𝑅,𝑡^∗=𝜔𝑡2𝜋,𝑢^∗𝜌=−^𝑓𝜔𝑎₁𝑢,𝐾=𝜅𝑅. (40) Then from (26), we get 𝑢^∗𝐽=Re⁰𝛽^∗𝑟^∗𝐽₀𝛽^∗−𝑙^∗𝛽^∗𝐽₁𝛽^∗−1𝑖𝑒^{2𝜋𝑖𝑡∗}−𝜌_𝑓𝜔𝑎₁1𝜇𝐽₀𝐾𝑟^∗𝐽₀(𝐾)+𝜅𝑅𝑙^∗𝜇𝐽₁(𝐾)𝐽₀(𝐾)−1𝜇𝜓_𝑠𝐸^′_𝑧. (41)

Let 𝑀=𝑀𝑟^∗𝐽=Re⁰𝛽^∗𝑟^∗𝐽₀𝛽^∗−𝑙^∗𝛽^∗𝐽₁𝛽^∗,𝑁=𝑁𝑟^∗𝐽=Im⁰𝛽^∗𝑟^∗𝐽₀𝛽^∗−𝑙^∗𝛽^∗𝐽₁𝛽^∗,𝐴₁𝜌=−^𝑓𝜔𝜇𝐽₀(𝐾),𝐴₂=𝜌_𝑓𝜔𝐾𝜇𝐽₀(𝐾)𝐽₁(𝐾)𝐴𝑅,₃=𝜌_𝑓𝜔𝜇, (42) then 𝑢^∗=𝐴𝑟^∗cos2𝜋𝑡^∗+𝐴−𝜃₁𝐽₀𝐾𝑟^∗+𝐴₂𝑙^∗+𝐴₃𝜓_𝑠𝐸^′_𝑧𝑎₁, (43) where 𝐴=𝑁²+(1−𝑀)²,𝜃=arctan𝑀−1𝑁. (44) As the 𝑀 and 𝑁 defined above are in terms of the complex parameter 𝛽^∗ and the Bessel functions with complex arguments, the relationship between the velocity and the slip length 𝑙 needs to be investigated. We first proceed to derive more explicit formulae relating 𝑢^∗ and 𝑙 in the real domain. As 𝛽²=−(𝜌𝜔/𝜇)𝑖=−(𝜌𝜔/𝜇)𝑒^−𝜋/2, we get 𝛽=𝜌𝜔()=2𝜇1−𝑖𝛽𝑅(),11−𝑖𝛽=𝑅2𝛽(1+𝑖),𝛽^∗=𝛽(1−𝑖), (45) where 𝛽=𝑅𝜌𝜔/2𝜇. For ‖𝑦‖≪1, then we have the following asymptotic formulae for approximating Bessel functions [12]: 𝐽_𝑛𝑥+𝑦𝑖≈𝐽_𝑛(𝑥)−𝑖𝑦2𝐽_𝑛+1(𝑥)−𝐽_𝑛−1(𝑥,𝐽_𝑛)𝑦≃1𝑦𝑛!2². (46) Thus, for 𝛽≪1, we have 𝐽₀𝛽^∗=𝐽₀𝛽−𝛽𝑖≈𝐽₀𝛽+𝑖𝛽𝐽₁𝛽≈1+𝛽²2𝐽𝑖,₁𝛽^∗=𝐽₁𝛽−𝛽𝑖≈𝛽2−𝛽21−𝛽²8𝐽𝑖,₀𝛽^∗𝑟^∗≈1+𝛽²𝑟^∗22𝐽𝑖,₁𝛽^∗𝑟^∗=𝛽𝑟^∗2−𝛽𝑟^∗21−𝛽𝑟^∗28𝑖. (47)

Substituting the above into (48) yields 𝑀≈1−𝛽⁴𝑙^∗1−8𝑟^∗2/161+𝛽⁴𝑙^∗(7/8)+𝑙^∗,𝑁≈−𝛽²1−𝑟^∗2+2𝑙^∗21+𝛽⁴𝑙^∗(7/8)+𝑙^∗. (48) Hence we have 𝐴=𝐴𝑟^∗=𝛽²21+𝛽⁴𝑙^∗(7/8)+𝑙^∗×1−𝑟^∗2+2𝑙^∗2+𝛽⁴𝑙^∗26415−8𝑟²+16𝑙^∗2,𝑢^∗=𝐴𝑟^∗cos2𝜋𝑡^∗+𝐴−𝜃₁𝐽₀𝐾𝑟^∗+𝐴₂𝑙^∗+𝐴₃𝜓_𝑠𝐸^′_𝑧𝑎₁. (49)

Thus, it can be concluded that the velocity consists of a time-dependent part in terms of a trigonometric function and a time-independent part in terms of a Bessel function of the first kind of order zero.

To demonstrate the velocity profile of the mixed pressure-driving and electrical-driving flow and the influence of the slip parameter and electric field on the flow, we carry out numerical investigation under various conditions. The typical values chosen for the model parameters 𝑅,𝜌_𝑓,𝜔,𝜇 are 𝑅=20 mm, 𝜌_𝑓=1000 km/m³, 𝜔=1/2𝜋,𝜇=0.86 Ns/m². Based on [27–29], the 𝐾 value varies from one electrolyte solution to another; thus in this work we use a wide range of values to demonstrate the influence of 𝐾 on the flow. The slip length 𝑙 is related to the smoothness of the surface of the tubes and is controllable within certain range; hence we also use a wide range of 𝑙 values in the investigation to demonstrate the influence of 𝑙 on the flow profile and flow rate. The 𝜓_𝑠𝐸^′_𝑧 is set to 0.05 V in all experiments except for the experiment in which the influence of the intensity of electric field relative to the pressure-driving force is to be investigated.

As the velocity profile consists of two parts due, respectively, to pressure gradient and electrical field, we first show the profile of these two velocity components in Figures 1 and 2 under different electric fields 𝜓_𝑠𝐸^′_𝑧/𝑎₁ and 𝐾 values. From the figure, it is clear that it is possible to control the velocity profile in an optimal manner based on the need of application for the mixed driving flow by finding proper combination of the pressure field and electrical field. Figures 3 and 4 show typical velocity profile for mixed driving flows.

Figure 1 

Typical velocity profile of the velocity component due to pressure gradient under different 𝑙∗ value.

Figure 2 

Typical profiles of the velocity component due to electric field for 𝑙∗=0.5 under different 𝜓𝑠𝐸′𝑧/𝑎1 and 𝐾 values.

Figure 3 

Typical velocity profile for various different values of slip length 𝑙∗ at a typical time 𝑡∗=𝜃/2𝜋, obtained with 𝜓𝑠𝐸′𝑧/𝑎1=10−4, 𝐾=200.

Figure 4 

Typical velocity profile for the mixed pressure and electric driven flow at various instants of time 𝑡∗, obtained with slip length 𝑙∗=1, 𝜓𝑠𝐸′𝑧/𝑎1=10−6, and 𝐾=1500.

To demonstrate the influence of the slip length on the velocity, we vary the value of 𝑙^∗ from 0 to 100 while maintaining other parameter values unchanged. Figure 5 shows the effect of slip length on the velocity on the wall at a typical time step 𝑡^∗=𝜃/2𝜋, obtained with 𝜓_𝑠𝐸^′_𝑧/𝑎₁=10⁻⁶, 𝐾=3000, and 𝐾=2000. It is clear that the relationship between the slip parameter and the velocity is nonlinear and there is an optimal value for which the slip velocity takes the maximum value.

Figure 5 

The influence of slip length on the slip velocity on the wall at a typical time 𝑡∗=𝜃/2𝜋, obtained with 𝜓𝑠𝐸′𝑧/𝑎1=10−3, 𝑙∗=1.

As the 𝐾 value changes across different electrolyte solutions, we investigate the influence of 𝐾 on the velocity profile. Figure 6 shows the velocity profiles for different 𝐾 values for 𝜓_𝑠𝐸^′_𝑧/𝑎₁=10⁻³ and under the same pressure gradient and slip length 𝑙^∗=1. It is clear from the result that the 𝐾 value influences the velocity profile significantly. For lower 𝐾, the velocity profile is smooth, while under high 𝐾 values, the velocity shows wave form profile and the frequency of the waves increases as 𝐾 increases. It is also noted that the amplitude of the waves increases as the 𝜓_𝑠𝐸^′_𝑧/𝑎₁ value increases. The magnitude of the velocity is also affected significantly by the 𝜓_𝑠𝐸^′_𝑧/𝑎₁ value as shown in Figure 7. As 𝜓_𝑠𝐸^′_𝑧/𝑎₁ increases, the magnitude of the velocity increases.

Figure 6 

Influence of 𝐾 value on the velocity profile at a typical time 𝑡∗=𝜃/2𝜋, obtained with 𝜓𝑠𝐸′𝑧/𝑎1=10−3,𝑙∗=1.

Figure 7 

Influence of magnitude of electric field on the velocity profile at a typical time 𝑡∗=𝜃/2𝜋, 𝑙∗=0.5, 𝐾=200.

6. Conclusions

In this paper, an exact solution for the transient flow of an incompressible Newtonian fluid in microtubes is derived taking into account the electrokinetic effect and the boundary slip. We have shown that both boundary slip and electrokinetic field have significant effect on the flow. The results are summarized as follows.

(i) Both the pressure-gradient driving force and electrical-driving force influence the velocity profile significantly, and it is possible to construct an optimal control problem to achieve the desired velocity profile based on the need of the application.

(ii) For the case of constant pressure gradient, the relationship between the slip parameter and the velocity is linear. However, for the pressure gradient in wave form, the influence of the slip parameter on the velocity magnitude is nonlinear and there exists a critical slip length where the magnitude and consequently the transient flow rate attain maximum values, which indicates that there exists an optimal slip length in terms of the magnitude of velocity and flow rate.

(iii) The 𝐾 value of the electrolyte solution has significant influence on the velocity profile on the cross-section of the tube. At low 𝐾 value, no wave form profile exists. However, as 𝐾 increases, the cross-section velocity exhibits wave form profile, and the frequency of the wave increases as 𝐾 increases.

(iv) The magnitude of the intensity of electric field relative to the pressure gradient, measured by 𝜓_𝑠𝐸^′_𝑧/𝑎₁, also has very significant influence on the velocity profile. As 𝜓_𝑠𝐸^′_𝑧/𝑎₁ increases, the velocity component increases and the amplitude of the wave in the waved form velocity profile also increases.

Acknowledgments

Qian Sun, Yonghong Wu, and Lishan Liu were supported financially by the National Natural Science Foundation of China (11071141, 11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.