The analysis of developing flows at the entrance to circular and rectangular ducts has received considerable attention over the years. Whilst most researchers have concentrated their efforts on the no-slip (continuum) flow regime, several studies have considered the hydrodynamic entrance problem under rarefied conditions where the momentum transport starts to be affected by the discrete molecular composition of the gas. The present investigation examines the role of both the Reynolds and Knudsen numbers on the hydrodynamic development length at the entrance to circular and parallel-plate micro-channels. The circular pipe is important not only because of its fundamental geometrical properties but also because gas samples are commonly transported to MEMS devices in fine-bore micro-capillary tubing. The parallel-plate micro-channel is equally important as it forms the limiting flow condition for large aspect ratio rectangular ducts commonly encountered in silicon micro-machined devices. Knowledge of the expected hydrodynamic development length is particularly important when designing the layout of a microfluidic device or choosing a suitable location for the upstream boundary of a numerical model.
When a viscous fluid enters a duct, the uniform velocity distribution at the entrance is gradually redistributed into a parabolic velocity profile due to the retarding influence of the shear stresses along the side walls. Ultimately, the fluid will reach a location where the velocity profile no longer changes in the axial direction, and under such conditions the flow is said to be fully-developed. Theoretically, the required distance to reach the fully-developed solution is infinitely large. However, for practical engineering calculations, the hydrodynamic development length, L, can be arbitrarily defined as the axial distance required for the centreline velocity to reach 99% of the fully-developed value.
In the present study, we consider entrance development effects in the slip-flow regime (Knudsen number, Kn < 0.1). Within this regime, rarefaction effects are important but the flow can still be simulated using the Navier-Stokes equations provided slip-velocity boundary conditions are implemented along the walls of the flow domain.
The fully-developed slip-velocity profile in a circular micro-pipe can readily be obtained from the axial-direction Navier-Stokes equation as detailed by Barber & Emerson (2000). It can be shown that the theoretical velocity profile across a pipe of radius, R, is given by
where u is the mean velocity in the pipe, Kn is the Knudsen number based upon the diameter of the pipe and s is the tangential momentum accommodation coefficient (TMAC) which accounts for gas-surface interactions at the wall. In addition, the maximum velocity at the centreline of the pipe can be derived as:
Using the above equation in conjunction with a 99% velocity cut-off point allows the hydrodynamic development length to be defined as the axial location where the longitudinal velocity reaches a value of
Similarly, in the case of non-continuum slip-flow between a pair of parallel-plates separated by a distance, H, it can be shown that the fully-developed velocity distribution is given by
where Kn is the Knudsen number based upon the hydraulic diameter of the duct. In addition, the maximum velocity along the centreline ( y = H / 2 ) can be derived as
Using an analogous 99% velocity cut-off procedure to that employed for the circular pipe allows the hydrodynamic development length to be defined as the axial location where the longitudinal velocity reaches a value of
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 simulations assessed the hydrodynamic development length over a range of Reynolds and Knudsen numbers in the laminar slip-flow regime. In the present study, the Reynolds number was varied from Re=1 to Re=400 whilst the Knudsen number was varied from Kn=0 (continuum flow) to Kn=0.1 (a frequently adopted upper bound for the slip-flow regime). In the absence of additional information, the tangential momentum accommodation coefficient, s, was assumed to have a value of unity. The Reynolds and Knudsen numbers were defined using the hydraulic diameter of the cross-section as the characteristic length scale. Thus, in the case of the circular micro-pipe:
whilst for the parallel-plate micro-channel:
Preliminary validation of the hydrodynamic code was accomplished by comparing the analytical velocity profiles across the ducts against predictions from the downstream boundary of the numerical model. The tests involved a number of different grid resolutions, including meshes composed of 51x21, 101x41 and 201x81 nodes. In addition, numerical experimentation was used to decide upon a suitable length of duct. The computational domains were curtailed a finite distance downstream of the entrance, with the location chosen so as not to affect the computed entrance length. For Reynolds numbers up to 400, it was found that duct lengths of 40D for the circular pipe and 40H for the parallel-plate geometry were sufficient to achieve a reliable estimate of hydrodynamic development length. An exponential stretching of the meshes was implemented in the axial-direction to achieve a finer grid resolution in the critical boundary-layer formation zone at the entrance.
Non-dimensionalised entrance lengths for the circular pipe geometry were computed for three separate Knudsen numbers (Kn=0, 0.05 and 0.1) and a range of Reynolds numbers between 1 and 400. Figure 1 illustrates the entrance development length as a function of Reynolds number for the 201x81 mesh. Superimposed on the present results are the continuum development length equations presented by Atkinson et al. (1969) and Chen (1973). Atkinson et al. proposed that the non-dimensionalised hydrodynamic development length (L/D) could be related to the Reynolds number via a simple linear relationship:
whereas Chen (1973) proposed a more elaborate function of the form:
Figure 1 shows that the present continuum results (Kn=0) are in good agreement with the hydrodynamic entrance lengths predicted by Chen. It can also be seen that Atkinson et al.'s solution tends to over-predict the development length at all but the lowest Reynolds numbers. More importantly, the results demonstrate that rarefaction has only a marginal effect on the development length in circular pipes and therefore the development length equation proposed by Chen for continuum flows is equally valid in the slip-flow regime.
Figure 1: Non-dimensionalised development lengths for rarefied slip-flow in a circular pipe.
Figure 2 presents the corresponding hydrodynamic development lengths for the parallel-plate geometry. It should be noted that the non-dimensionalised development lengths and Reynolds number have been scaled according to the hydraulic diameter of the duct (Dh). Superimposed on the present results are the continuum development length equations presented by Atkinson et al. (1969) and Chen (1973). For the parallel-plate geometry, Atkinson et al. proposed that the non-dimensionalised hydrodynamic development length (L/Dh) could be related to the Reynolds number via the following linear relationship:
whereas Chen (1973) proposed:
Figure 2 indicates that the present continuum results (Kn=0) are in good agreement with the hydrodynamic entrance lengths predicted by Chen. It can also be seen that Atkinson et al.'s solution again tends to over-predict the development length at all but the lowest Reynolds numbers. More importantly, the present results show that the Knudsen number has a significant effect on the length of the development region. Inspection of Figure 2 reveals that at a Reynolds number of 400, the entrance length for a Knudsen number of 0.1 is approximately 25% longer than the corresponding continuum solution. It can therefore be concluded that the formulae proposed by Atkinson et al. and Chen for the parallel-plate geometry are no longer valid in the slip-flow regime and consequently a new development length equation accounting for both the Reynolds and Knudsen number must be evaluated.
Figure 2: Non-dimensionalised development lengths for rarefied slip-flow in a parallel-plate micro-channel.
A non-linear least-squares curve-fitting procedure employing the Levenberg-Marquardt method was used to determine the relationship between L/Dh, Re and Kn. The effect of the Knudsen number was taken into account by multiplying the coefficient in the second term of Chen's development length equation by a correction factor of the form:
where A and B are constants and Kn' is defined as
The form of the Knudsen number correction was chosen because the analytical expression for the fully-developed centreline velocity has a similar Knudsen number modification factor. Applying the Levenberg-Marquardt least-squares technique yields the following expression for the hydrodynamic development length:
Figure 3 illustrates the results of the least-squares fit. It can be seen that the proposed equation provides a good representation of the numerical development length data. Moreover, the linearity of the Reynolds number dependency in the second term of the above equation implies that the expression will provide a reliable estimate of hydrodynamic development length up to the transition to turbulence. The proposed entrance length equation should therefore be appropriate for the entire laminar slip-flow regime.
Figure 3: Least-squares fit of non-dimensionalised development lengths
for rarefied slip-flow in a parallel-plate micro-channel.
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For further information on 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
Dr. Robert Barber
Science and Technology Facilities Council
Daresbury Laboratory
Daresbury Science and Innovation Campus
Warrington WA4 4AD
Cheshire
United Kingdom