## Transition from laminar to turbulent ﬂow

**Introduction:** The transition from laminar Poiseuille ﬂow to turbulence in a circular tube is a familiar phenomenon that is generally understood to have a minimum lower critical Reynolds number between 1800 and 2300.1These values have been established on purely empirical grounds, and traditional linear stability does not adequately predict transition.2 Nonlinear theories,3, 4low-dimensional models,2and simulations3, 5, 6have been considered to explain and predict transition to turbulence in a circular tube, yet many questions remain open regarding the role of disturbances such as acoustic waves, vibrations, inlet agitation, and molecular motion. Such disturbances do not scale with the diameter of the tube, so one must allow for the possibility of their eﬀect when the diameter is reduced to the sub-100 ¹m level commonly dealt with in microﬂuidics.

In micro scale ﬂows of liquids the incompressible, viscous Navier-Stokes equations are expected to describe the ﬂuid motion down to scales of the order of 10 molecular spacing’s, 7 or until the tube diameter drops well below one micron. It is possible that there is a small eﬀect due to slip in the near vicinity of the wall. 8 For hydrophilic boundaries, closer investigations at the wall suggest that the no-slip boundary condition is valid, 9 but even if the wall material is hydrophobic, the slip length is less than 1 ¹m and the eﬀect of this conservative estimate of slip length on ﬂow resistance is likely to be within any experimental error for ﬂow diameters of the order of 300–400 microns. Other factors, such as such as weak non-Newtonian ﬂuid properties or micro-polar molecular structure, have negligible eﬀects on transition in macroscopic tubes, but might become important in the extremely high shear rates found in micro tubes at Reynolds numbers approaching transition. Like the ﬂuctuations described above, these eﬀects also fail to scale only with Reynolds number, and they therefore merit critical examination.

**The Darcy friction factor for ﬂow in a duct is deﬁned as:**

Where the hydraulic diameter D_{h} = 4A=P. A is the cross-sectional area, P is the wetted perimeter, ½ is the density, UB is the bulk velocity, x is the stream wise (axial) direction, and P is the mean pressure. If the ﬂuid obeys Newtonian rheology, the friction factor for steady, fully developed laminar should be given by

Where C_{1} is a constant that depends on the cross-sectional shape, R_{e}D_{h} = ½UBDh=¹ and ¹ is the dynamic viscosity

For a round cross-section, C_{1} = 8, and the numerator of Eq. 2 has the well-known value of 64. The ﬂow resistance may also be stated in terms of the Poiseuille number, whose deﬁnition is

**Transition and ﬂow resistance**: Before discussing the transition to turbulence, we ﬁrst show that the laminar ﬂow prior to

Transition obeyed the the relationships accepted for classical Poiseuille ﬂow in round tubes. The pressure drop versus ﬂow rate data from more than 1500 measurements are summarized the pressure drop is presented in a dimensionless

Form,

**Transition and the axial velocity:** Using the same ﬂow delivery and test sections described previously, micro-PIV experiments

Were also performed to quantitatively measure the axial, u, component of velocity within the micro tubes. In these experiments, a steady pressure was maintained inside the pressure vessel to within §0:4% to produce very nearly steady ﬂow and thereby to permit time averaging. In all cases, the measurements were obtained at a stream wise location, (x=D)

That was greater than 0:06R_{e}D, the entrance length needed to achieve fully developed ﬂow. Thus, the ﬂow was not expected to change in the stream wise direction across the length of the PIV image, unless there were spatial-time variations due to turbulence or spatial variations due to surface roughness eﬀects. The micro-PIV measurements were capable of detecting ﬂuctuations in t or x, thereby allowing us to assess each eﬀect.

**Summary and Conclusions:** The ﬂow of a liquid in micro channels should be represented well by continuum theory unless the channel dimensions approach the slip length at the wall, estimated to occur for channels and tubes whose dimensions lay below a few microns. Despite this expectation, signiﬁcant departures from continuum macro scale theory have been reported in the literature of microﬂuidics, and they have sometimes been attributed to unknown micro scale eﬀects that produce transition to turbulence at anomalously low Reynolds numbers. To resolve this controversy, experiments have been performed in round glass micro tubes with diameters ranging from 50 to 247 microns, using liquids with diﬀerent levels of polarity.