17-03-2014, 05:35 PM
Abstract— In this paper, the two-dimensional steady slip flow
in microchannels is investigated. Research on micro flow,
especially on micro slip flow, is very important for designing
and optimizing the micro electromechanical system (MEMS).
The Navier–Stokes equations for 2-dimensional steady slip
flow in microchannels are reduced to a nonlinear three-order
differential equation by using similarity solution. Variational
iteration method (VIM) is used to solve this nonlinear equation
analytically. Comparison of the result obtained by the present
method with numerical solution reveals that the accuracy and
fast convergence of the new method. It is predicted that the
VIM can have wide application in engineering problems.
Keywords- Slip flow; Navier–Stokes equations; Variational
iteration method; Microchannel (key words)
I. INTRODUCTION
Nonlinear phenomena play a crucial role in applied
mathematics and physics. As we know most of engineering
problems are nonlinear, and it is difficult to solve them
analytically. Various powerful mathematical methods have
been proposed for obtaining exact and approximate analytic
solutions [1-10].
The variational iteration method (VIM) was first
proposed by He [11,12] and systematically illustrated in
1999 [13] and used to give approximate solutions of the
problem of seepage flow in porous media with fractional
derivatives. The VIM is useful to obtain exact and
approximate solutions of linear and nonlinear differential
equations. In this method, general Lagrange multipliers are
introduced to construct correction functional for the
problems. The multipliers can be identified optimally via the
variational theory. It has been used to solve effectively,
easily and accurately a large class of nonlinear problems
with approximation [14,15]. It was shown by many authors
[16–21] that this method is more powerful than existing
techniques such as the Adomian method [22].
In recent years, progress in micro-fabrication and
assembly techniques has led to the development of
extremely small-scale machines commonly referred to as
MEMS (Micro-Electro-Mechanical Systems) which are
generally defined as electro-mechanical deviceshaving a
characteristic length scale between 1mm and 1μm [23].
Microchannels, however, are the basic structures in these
systems. Based on Kn number, Beskok [24] classified the
gas flow in microchannel into four flow regimes: continuum
flow regime (Kn < 0.001), slip flow regime (0.001 < Kn <
0.1), transition flow regime (0.1 < Kn <10) and free
molecular flow regime (Kn > 10). The flow in most
application of these systems, such as Micro Gyroscope,
Accelerometer, Flow Sensors, Micro Nozzles, Micro Valves,
is in slip flow regime, which is characterized by slip flow at
wall. Traditionally, the no-slip condition at wall is enforced
in the momentum equation and an analogous notemperature-
jump condition is applied in the energy
equation. Strictly speaking, no-slip/no-jump boundary
conditions are valid only if the fluid flow adjacent to the
surface is in thermodynamic equilibrium. This requires an
infinitely high frequency of collisions between the fluid and
the solid surface [25]. In practice, the no-slip/no-jump
condition leads to fairly accurate predictions as long as Kn <
0.001. Beyond that, gas flow in devices shows significant
slip since characteristic length is on the order of the mean
free path of the gas molecules. It means that the collision
frequency is simply not high enough to ensure equilibrium
and a certain degree of tangentialvelocity slip and
temperature jump must be appeared. Slip at wall is the most
important feature in micro-scale that differs from
conventional internal flow. So, slip flow characteristics are
very important for designing and optimizing the micro
systems [26]. Many investigators have studied on these
problems experimentally and numerically [27-31]. Zhang et
al. [26] obtain an analytic solution for slip flow in
microchannels using homotopy analysis method. However,
there are few analytic solutions on these problems because
they are strongly non-linear problems which are very hard to
be solved in an analytical way.
in microchannels is investigated. Research on micro flow,
especially on micro slip flow, is very important for designing
and optimizing the micro electromechanical system (MEMS).
The Navier–Stokes equations for 2-dimensional steady slip
flow in microchannels are reduced to a nonlinear three-order
differential equation by using similarity solution. Variational
iteration method (VIM) is used to solve this nonlinear equation
analytically. Comparison of the result obtained by the present
method with numerical solution reveals that the accuracy and
fast convergence of the new method. It is predicted that the
VIM can have wide application in engineering problems.
Keywords- Slip flow; Navier–Stokes equations; Variational
iteration method; Microchannel (key words)
I. INTRODUCTION
Nonlinear phenomena play a crucial role in applied
mathematics and physics. As we know most of engineering
problems are nonlinear, and it is difficult to solve them
analytically. Various powerful mathematical methods have
been proposed for obtaining exact and approximate analytic
solutions [1-10].
The variational iteration method (VIM) was first
proposed by He [11,12] and systematically illustrated in
1999 [13] and used to give approximate solutions of the
problem of seepage flow in porous media with fractional
derivatives. The VIM is useful to obtain exact and
approximate solutions of linear and nonlinear differential
equations. In this method, general Lagrange multipliers are
introduced to construct correction functional for the
problems. The multipliers can be identified optimally via the
variational theory. It has been used to solve effectively,
easily and accurately a large class of nonlinear problems
with approximation [14,15]. It was shown by many authors
[16–21] that this method is more powerful than existing
techniques such as the Adomian method [22].
In recent years, progress in micro-fabrication and
assembly techniques has led to the development of
extremely small-scale machines commonly referred to as
MEMS (Micro-Electro-Mechanical Systems) which are
generally defined as electro-mechanical deviceshaving a
characteristic length scale between 1mm and 1μm [23].
Microchannels, however, are the basic structures in these
systems. Based on Kn number, Beskok [24] classified the
gas flow in microchannel into four flow regimes: continuum
flow regime (Kn < 0.001), slip flow regime (0.001 < Kn <
0.1), transition flow regime (0.1 < Kn <10) and free
molecular flow regime (Kn > 10). The flow in most
application of these systems, such as Micro Gyroscope,
Accelerometer, Flow Sensors, Micro Nozzles, Micro Valves,
is in slip flow regime, which is characterized by slip flow at
wall. Traditionally, the no-slip condition at wall is enforced
in the momentum equation and an analogous notemperature-
jump condition is applied in the energy
equation. Strictly speaking, no-slip/no-jump boundary
conditions are valid only if the fluid flow adjacent to the
surface is in thermodynamic equilibrium. This requires an
infinitely high frequency of collisions between the fluid and
the solid surface [25]. In practice, the no-slip/no-jump
condition leads to fairly accurate predictions as long as Kn <
0.001. Beyond that, gas flow in devices shows significant
slip since characteristic length is on the order of the mean
free path of the gas molecules. It means that the collision
frequency is simply not high enough to ensure equilibrium
and a certain degree of tangentialvelocity slip and
temperature jump must be appeared. Slip at wall is the most
important feature in micro-scale that differs from
conventional internal flow. So, slip flow characteristics are
very important for designing and optimizing the micro
systems [26]. Many investigators have studied on these
problems experimentally and numerically [27-31]. Zhang et
al. [26] obtain an analytic solution for slip flow in
microchannels using homotopy analysis method. However,
there are few analytic solutions on these problems because
they are strongly non-linear problems which are very hard to
be solved in an analytical way.