The development of precision fabrication techniques for constructing Micro-Electro-Mechanical-Systems (MEMS) has emerged as one of the most exciting and revolutionary new areas of technology. Miniaturisation of conventional fluidic systems offers a wide range of benefits to the chemical and bio-chemical industries by enabling faster mixing times, improved heat transfer rates, increased chemical yields and faster throughputs of chemical assays. In addition, the small length scales employed in microfluidic systems offer the prospect of developing pressure and chemical sensors having extremely high frequency responses.
In the present investigation, numerical simulations are used to study the effects of rarefaction on axisymmetric flow past a confined microsphere within a circular pipe, as illustrated schematically in Figure 1. The study of rarefied gases by measuring the viscous damping on a rotating sphere dates back to Maxwell and nowadays forms the basis of conventional spinning-rotor vacuum gauges. Essentially the rate of damping of an electro-magnetically suspended rotating sphere can be used to measure a number of important properties of a rarefied gas including pressure, viscosity or molecular weight. Similar principles have been envisaged in the MEMS community for measuring flow rates, pressures or viscosities in microchannels. Alternatively, a microsphere located axisymmetrically within a circular cone could be used as a microvalve to control flow rates. Adjusting the position of the sphere along the axis of the cone (via a magnetic field) would allow the blockage ratio to be altered thereby controlling the pressure difference across the device.
The investigation considers the case of low Reynolds number, isothermal, rarefied gas flow past a confined microsphere within a circular pipe and focuses on the estimation of the hydrodynamic drag forces on a stationary (non-rotating) sphere. Knudsen numbers covering the continuum and slip-flow regimes (0 < Kn < 10-1) are studied whilst the Reynolds number is varied between 10-2 < Re < 1. In addition, blockage effects have been investigated by varying the ratio between the diameter of the pipe, H, and the diameter of the sphere, D.
Figure 1: Problem formulation.
The governing hydrodynamic equations for a continuous (infinitely divisible) fluid can be written in tensor notation as follows:
continuity:
momentum:
where u is the velocity, p is the pressure, r is the density and tik is the second-order stress tensor. For a Newtonian, isotropic fluid, the stress tensor is given by
where m and l are the first and second coefficients of viscosity and dik is the unit second-order tensor. Implementing Stokes' continuum hypothesis allows the first and second coefficients of viscosity to be related via
although the validity of the above equation has occasionally been questioned for fluids other than dilute monatomic gases.
To account for non-continuum effects in the slip-flow regime (Kn < 10-1), the Navier-Stokes equations are solved in conjunction with the slip-velocity boundary condition originally proposed by Basset (1888):
where ut is the tangential slip-velocity at the wall, tt is the tangential shear stress on the wall and b is the slip coefficient. Schaaf & Chambre (1961) have shown that the slip coefficient can be related to the mean free path of the molecules as follows:
where µ is the fluid viscosity and s is the tangential momentum accommodation coefficient (TMAC).
The governing hydrodynamic equations were solved using THOR-2D – a two-dimensional finite-volume Navier-Stokes solver developed by the Computational Engineering Group at CLRC Daresbury Laboratory (Gu & Emerson, 2000). Since the conditions investigated in the study had relatively low Mach numbers compressibility effects were ignored and the flow was assumed to be incompressible and isothermal.
The numerical model was validated in the continuum regime by comparing the computed drag on the sphere against the numerical results presented by Liu et al. (1998) and the analytical solution developed by Haberman & Sayre (1958). Unconfined creeping flow past a sphere was first analysed by Stokes who demonstrated that in the absence of inertial forces, the total drag due to the flow of an unbounded incompressible Newtonian fluid could be written as
where U denotes the uniform velocity distribution infinitely far from the sphere and a is the radius of the sphere. Non-dimensionalising the drag by the dynamic pressure and the cross-sectional area of the sphere allows the drag coefficient, CD, to be written as
The drag coefficient for a confined sphere can be normalised using the procedure adopted by Liu et al. (1998) to give
where f(H/D) denotes a function of the blockage ratio and Re is the Reynolds number based upon the mean velocity in the pipe.
Figure 2: Variation of normalised total drag coefficient on a confined sphere as a function of blockage ratio in the continuum flow regime (Kn = 0).
Figure 2 presents the normalised drag predictions for a range of blockage ratios from H/D=2 to H/D=40. The results show that the sphere experiences significant blockage effects for H/D<10 with a very large increase in the drag coefficient being observed for H/D<5. The numerical predictions were compared against the analytical drag formula presented by Haberman & Sayre (1958) who used an algebraic stream function approach. Normalising the equation presented by Haberman and Sayre yields:
The excellent agreement between the numerical predictions and the analytical solution over the entire range of blockage ratios indicates that the computational scheme provides an accurate representation of the flow past a confined sphere in the continuum regime.
The second part of the study investigated the drag experienced by the sphere in the slip-flow regime. Simulations were conducted using Reynolds numbers in the range, 10-2 < Re < 1 whilst the Knudsen number (based upon the diameter of the sphere) was varied up to Kn = 10-1 (the upper limit of the slip-flow regime). In an incompressible Newtonian fluid, it can be shown that the normal stress component must vanish along any rigid no-slip impermeable boundary. In contrast, the tangential slip-velocity boundary condition in rarefied gas flows generates a non-zero normal stress component which produces an important additional drag force on the sphere.
Normalised drag results for the least confined blockage ratio (H/D = 40) provide a useful validation of the hydrodynamic code since the results should approach the asymptotic limit of low Reynolds number slip flow past an unconfined sphere. Barber & Emerson (2000) have previously described an extension of Stokes’ analytical solution for creeping flow past a sphere which accounts for non-continuum effects. The analysis follows the methodology originally proposed by Basset (1888) and provides expressions for the total drag and the individual drag components experienced by an unconfined sphere in the slip-flow regime. The total drag can be shown to be given by
whilst the individual drag components are given by
Figure 3: Variation of normalised drag coefficients on a confined sphere in the slip-flow regime as a function of Knudsen number (H/D = 40).
Figure 3 illustrates the variation in normalised drag components on the sphere as a function of Knudsen number for a blockage ratio of H/D = 40. It should be noted that in normalising the analytical drag equations, the unconfined drag components presented earlier have been multiplied by a factor of 2 to account for the parabolic velocity profile in the pipe. Small discrepancies can be seen in the normal stress drag predictions which in turn affect the total drag. This can be confirmed by noting that the numerical model fails to predict a zero normal stress drag component in the continuum regime (Kn = 0). Previous numerical studies by Beskok & Karniadakis (1994) on rarefied gas flows past circular cylinders have confirmed the difficulty in obtaining accurate estimates of the normal stress distribution. Nevertheless, the general agreement between the predictions and the unconfined analytical solution provides an important validation test.
Figure 4: Variation of normalised drag coefficients on a confined sphere in the slip-flow regime as a function of Knudsen number.
Figure 4 illustrates the variation in normalised total drag coefficient with Knudsen number for three separate blockage ratios. In the slip-flow regime, the total drag on the sphere decreases as the Knudsen number is increased. More importantly, the results indicate that the drag amplification effect caused by the blockage ratio becomes less significant as rarefaction starts to influence the flow. At the upper limit of the slip-flow regime (Kn = 10-1), blockage amplification effects are reduced by almost 50% for a pipe-sphere geometry of H/D = 2. This may have important consequences for the design of microfluidic components which operate over a wide range of Knudsen numbers. It should be noted that the tangential momentum accommodation coefficient, s, is assumed to have a value of unity in the present simulations. However, a sub-unity accommodation coefficient would further reduce the blockage amplification effect. Finally, Figure 5 illustrates the effect of the Knudsen number on the magnitude of the flow velocities around the sphere. As the Knudsen number is increased, the velocity distribution becomes more uniform explaining the reduction in the blockage amplification effect.
Figure 5: Distributions of flow speed around the microsphere (H/D = 2).
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R. W. Barber & D. R. Emerson
Analytical
solution of low Reynolds number slip flow past a sphere
Daresbury Laboratory Technical Report DL-TR-00-001, December 2000. (file size 304KB)
A. Beskok & G. E. Karniadakis
Simulation of heat and momentum transfer in complex microgeometries
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R. W. Barber & D. R. Emerson
Analytical
solution of low Reynolds number slip flow past a sphere
Daresbury Laboratory Technical Report DL-TR-00-001, December 2000. (file size 304KB)
R. W. Barber & D. R. Emerson
Numerical
simulation of low Reynolds number slip flow past a confined microsphere
Daresbury Laboratory Technical Report DL-TR-01-001, May 2001. (file size 893KB)
Robert W. Barber & David R. Emerson
Numerical
simulation of low Reynolds number slip flow past a confined microsphere
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20-25 July 2002. (file size 80.5KB)
Robert W. Barber
Numerical
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For more information about this work please contact:
Professor David Emerson
Science and Technology Facilities Council
Daresbury Laboratory
Daresbury Science and Innovation Campus
Warrington WA4 4AD
Cheshire
United Kingdom
For further information on the rarefied gas work please contact:
Dr. Robert Barber
Science and Technology Facilities Council
Daresbury Laboratory
Daresbury Science and Innovation Campus
Warrington WA4 4AD
Cheshire
United Kingdom