Abstract
We study the transient flow of a Newtonian fluid in rectangular microchannels taking into account boundary slip. An exact solution is derived by using the separation of variables in space and Fourier series expansion in time. It is found that, for different forms of driving pressure field, the effect of boundary slip on the flow behavior is qualitatively different. If the pressure gradient is constant, the flow rate is almost linearly proportional to the slip parameter when is large; if the pressure gradient is in a waveform, as the slip parameter increases, the amplitude of the flow rate increases until approaching a constant value when becomes sufficiently large.
1. Introduction
In recent years, many researchers worldwide focus on the study of behavior of materials at micro- and nanoscales [1, 2], leading to the development of many biological and engineering systems and devices. Most of these systems and devices involve fluid flow in microchannels, called microflows [1, 3–7]. Typical examples include drug delivery systems [8], fuel cell devices, energy conversion, and biological sensing devices [9]. As the functional characteristics of these systems depend on the behavior of fluid flow in the systems, the study of microflows is important and has attracted more and more attention from the engineering and science communities in order to derive a better understanding of the mechanism of microflows and consequently better design and control of the devices and systems [1, 6, 10].
The field equations governing the flow of Newtonian fluids are the continuity equation and the Navier-Stokes equations. These equations are subject to a set of boundary conditions. The no-slip boundary condition is used traditionally; namely, the tangential fluid velocity relative to the solid is zero on the fluid-solid interface [11]. However, recent molecular dynamic simulations and experiments in micrometer scale have shown that the fluid flow in microsystems is granular and slip may occur on the fluid-solid interface [10, 12–17]. Hence, for the study of microflow, it is important to take into account the boundary slip of fluids on the fluid-solid interfaces.
Over the last few decades, intensive research has been carried out to study various problems of fluid flow with the no-slip assumption or a slip boundary condition [1, 5, 17–33]. For flows under the no-slip assumption, exact solutions to many problems have been obtained and are available in the literature [11, 34–37]. But for slip flow very few exact solutions are available in the literature. Exact solutions for the fluid flow in circular microtubes and circular microannuals with boundary slip have been derived and discussed in the papers [17, 25, 28].
For microchannels with rectangular cross-section, a no-slip solution has been obtained [38–43]. For the slip case steady state solution has also been obtained [15, 35, 44–48]. However, so far no exact solution has been derived for the transient flow of fluids through rectangular channel under pulsatile pressure. As many microsystems and devices have microchannels of rectangular cross-section, it is important to derive exact analytical solutions for the behavior of transient flow through rectangular microchannels with slip boundary.
Based on the current development in the field, in this paper, we study the time-dependent flow of incompressible Newtonian fluids through a rectangular microchannel taking into account boundary slip. The rest of the paper is organized as follows. In Section 2, we give the underlying initial boundary value problem for the transient slip flow. In Section 3, we derive the exact solution for the velocity field. In Section 4, an analysis is conducted to study the effect of the slip length on the flow behavior. Finally, a conclusion is presented in Section 5.
2. Governing Boundary Value Problem
Consider the unsteady flow of an incompressible Newtonian fluid in a rectangular channel of cross-section dimension with the -axes being in the axial direction as shown in Figure 1. The field equations governing the flow include the Navier-Stokes equations and the continuity equation. As the flow is symmetric about the -plane and the -plane and is fully developed, there is no cross-sectional flow and thus the velocity components in the and directions vanish; that is, .
Hence, from the continuity equation and the Navier-Stokes equation the axial velocity, , is governed by the following equation: As a large class of functions may be expressed by Fourier series, we consider, in this work, the flow of a fluid driven by the pressure gradient that may be expressed in the form of Fourier series given by To define the problem completely, we supplement the field equation by the boundary condition. To take into account the boundary slip, the so-called Navier-slip boundary condition is used. On the fluid-solid interface and , the axial fluid velocity, relative to the solid surface, is assumed to be proportional to the shear stress on the interface. Let be the unit normal vector of the surface of the fluid, and let be the positive tangential direction. Also let the fluid velocity on the wall direction be , and let the velocity of the solid in the tangential direction of the surface be . Then, as shown in our previous work [17], the Navier-slip boundary condition can be expressed by where the negative sign indicates that the surface traction force which acted on the fluid by the solid is opposite to the tangential velocity of fluid relative to the solid. For our problems in the coordinate system, and . On the surface , and , and so and and consequently (5), on the surface , becomes On the surface , we have and , and hence and and consequently (5), on the surface , becomes Similarly, the boundary condition on the surface is We will remark here that, for , the slip boundary conditions (6)–(8) reduce to the no-slip condition; on the other extreme, where , (6)–(8) become the surface traction conditions for perfectly smooth surfaces; that is, .
3. Exact Solutions for Transient Velocity and Stress Fields
To solve the partial differential equation (3), complex numbers are used to express the Fourier series for the pressure gradient; namely, where and .
From the symmetry of the problem and the linearity of (3), we get , where is defined by Through some mathematical analysis, we get that, for , the boundary value problem (10) has solution of the form where is determined by with and being governed, respectively, by the following boundary value problems: Thus, the remaining work for finding and consequently and then is to solve the two BVPs: BVP1 and BVP2. We first solve BVP1 by the separation of variables. From the PDE and the homogeneous boundary conditions of BVP1, we obtain where are the roots of the nonlinear equation are respectively eigenvalues and eigenfunctions of BVP1 and The eigenvalues are the solution of the following nonlinear equation: which has infinite number of solutions as shown by the intersection of the graphs and where in Figure 2.
It can also be proved that the eigenfunctions are orthogonal and thus the coefficients can be determined from the nonhomogeneous boundary condition of BVP1 by Similarly, the solution of the BVP2 is where are the roots of the nonlinear equation Substituting (14) and (20) into (12) yields the solution For , proceeding as for finding , we obtain where and are as defined in (17) and (22) with ; that is, and ; and are as follows: Hence collecting all solutions of the subproblems, we have where and are determined, respectively, by (15) and (21), and are defined by (17) and (22), respectively, and are defined by (26) and (27), and and are defined by (19) and (23), respectively.
Now we determine the exact solutions of the flow rate and the stresses in the fluid. From the velocity solution (28), we obtain the flow rate as follows: where and denote, respectively, the flow rate corresponding to the constant component and the nth harmonic component of the pressure gradient and The stress tensor in the fluid can then be determined by the constitutive equation where is an identity matrix and the deformation rate depends on the velocity by As , we get From (28) and (32)–(34), we obtain and Thus, from the constitutive equation (32), we get where denotes the pressure gradient while is arbitrary and may be chosen to meet certain pressure condition.
4. Investigation of the Effect of Boundary Slip on the Flow Behavior
Based on the exact solutions obtained, we investigate the flow behaviour and the influence of the slip length in this section. As a general pressure field can be expressed by a Fourier series in the form of (4), from the principle of superposition, the exact solution of the problem is the superposition of the solutions corresponding to the constant pressure gradient plus the solutions corresponding to the sine or cosine waveform pressure gradients. In this work, without loss of generality, we consider flow problems under two different cases of driving pressure fields including the case with a constant pressure gradient and the case with a sine waveform pressure gradient. For simplicity, we introduce four dimensionless variables as follows:
Case 1 (). For this case, and for . The constant pressure gradient means that the pressure gradient does not depend on time. From (28), (30), (35), and (36), we get the following normalized velocity, flow rate, and shear stresses: As the pressure gradient does not depend on time, the normalized velocity and flow rate as well as stresses are influenced by the slip length only, which is implicitly contained in and .
Case 2 (). For this case, the pressure gradient is sinusoidal with amplitude , and , , for . From (27), we have Let where Then Using (19) and (23), through a lengthy derivation, we obtain where For convenience in discussion, transform (47) into the following form: where and denote the amplitude and phase angles of the normalized velocity defined, respectively, by The flow rate is Using real arithmetic, through lengthy calculation, we obtain the normalized flow rate where
In this study, we analyze the flow pattern through the rectangular microchannel having the same size of the cross-sectional area of for , Pa·s, , , and .
For Case 1, as the pressure gradient is constant, the normalized velocity and flow rate as well as shear stress vary with the slip length and the geometry of the cross-section, namely, the ratio of the cross-sectional area. The influences of the slip length and the ratio on the flow behavior are demonstrated by analyzing the solutions graphically. Figures 3 and 4 show, respectively, the three-dimensional and two-dimensional velocity profiles on a cross-section of the channel obtained from (39) for four different values of the slip length, . The results indicate that the axial velocity increases significantly when the slip length increases. Figure 5 shows the effects of the slip length and ratio on the flow rate . It is found that the flow rate is linear in slip length. For the same size of the cross-sectional area with the variation of the ratio , the flow rate increases significantly as the ratio increases.
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For Case 2, as the pressure gradient depends on time, the normalized velocity and flow rate as well as shear stress vary with time and the slip length. We investigate the axial velocity on a cross-section of the channel having the ratio of 1 and the slip length of 0.001 under the frequency with at various instants of time. Figures 6 and 7 show the transient velocity obtained from (43) for the slip length . For , the axial velocity has similar profile with , as is constant on the velocity equation (43) so that the curves coincide. The influences of the slip length, the frequency , and the ratio of the cross-section of the channel on the flow behavior are illustrated by analyzing the solutions graphically. Figure 8 shows the transient flow rate on a cross-section of the channel for four different values of the slip length, . Figure 9 presents the influence of on the amplitude of the flow rate for various frequencies with four different values: . The result indicates that the dependence of on is different for different . At high frequency (high ), the amplitude of the flow rate increases initially as increases but it then tends to a constant value once the slip length becomes sufficiently large. Figure 10 shows the influence of the ratio under the same cross-sectional area size on the flow rate for different slip lengths . It is noted that as the ratio increases the flow rate increases but tends to a constant value as becomes sufficiently large.
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5. Conclusions
In this paper, we derive an exact solution for the unsteady flow of an incompressible Newtonian fluid in a rectangular microchannel with a Navier-slip boundary. From the explicit analytical solutions of the velocity and flow rate, we investigate the effect of the slip length and the geometry of the cross-section on the flow of the fluid through the channel. The investigation shows the following.(1)For the flow through rectangular microchannels with constant pressure gradient, the axial velocity increases faster in the center of the cross-section than in other areas as the slip length increases, while for the flows, due to the waveform pressure gradient, the velocity changes significantly as the slip length increases.(2)For flow driven by a constant pressure gradient, the flow rate is linear with respect to the slip length and the ratio (), and, for different values of with constant (), the flow rate increases when increases and also when the slip length increases.(3)The amplitudes of flow rate initially increase significantly as slip length increases but tend to a constant value when becomes sufficiently large for various values. This profile is similar to the case shown in [39].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors wish to acknowledge the support of the Faculty of Science, Mahidol University. The second author gratefully acknowledges the support of the Australian Research Council through a discovery project grant. The authors are also grateful to both referees for their comments and suggestions which have led to significant improvement of this paper.