Abstract
A novel approach based on the local entropy generation rate, also known as the second law analysis (SLA), is proposed to compute and visualize the flow resistance in mass transfer through a pipe/channel with a sudden contraction component (SCC) at low Reynolds number (Re) featuring velocity slip. The linear Navier velocity slip boundary condition is implemented using the explicit scheme. At small Reynolds number, i.e., Re ≤ 10.0, the flow resistance coefficient of the SCC, KSCC, is found to be a function of the dimensionless velocity slip length Lslip* and Re−1, and gradually increase to a constant value at contraction ratio Rarea ≥ 8, reaching a formula KSCC=(0.4454Lslip* 3−1.894Lslip* 2+2.917Lslip*+8.909)/Re. Over this range of Re, the equivalent length of the flow resistance is almost independent of Re, while out of this range, the equivalent length increases monotonically with Re. Moreover, the dimensionless drag force work around the SCC is negative and reaches a minimum at a critical Lslip*. The SLA reveals that the regions affected by the SCC mainly concentrate around the end section of the upstream pipe/channel rather than the initial partition of the downstream section reported in large Re turbulent flow, and this non-dimensional affected upstream length increases with Lslip*. The fluid physics are further examined using SLA to evaluate the energy loss over the entire domain, decomposed as the viscous dissipation inside the domain and the drag work on the wall boundary.
Highlights
The determination of the flow resistance in low Reynolds number (Re) mass transfer through micro and nanoscale pipes and channels with sudden contraction components (SCC) is a vital issue in the process of design and fabrication of efficient microfluidic devices
The main objective of the present study is to quantitatively examine the flow resistance coefficient and reveal the physics in flow through a microscale SCC with a linear Navier velocity slip boundary condition (SBC)
The resistance coefficients of the fully developed channel and pipe flow at low Re with linear Navier velocity slip boundary condition are solved with the NS equations and the second law analysis (SLA) approach which is extended to determine and visualize the flow resistance loss of velocity slip flow after considering the work contributed by wall drag force
Summary
The determination of the flow resistance in low Reynolds number (Re) mass transfer through micro and nanoscale pipes and channels with sudden contraction components (SCC) is a vital issue in the process of design and fabrication of efficient microfluidic devices (i.e., micro-electro-mechanical systems[1], heat exchangers[2] and cooling of electronic chips[3]). For fully developed laminar pipe flow without velocity slip, it is known that the Darcy friction factor (flow resistance coefficient is proportional to the friction factor in a straight pipe/channel) is 64/Re4–8 This value was reported in the well-known Moody diagram, stressed by Moody as independent on the wall relative roughness in laminar flow[9]. Gloss et al.[25] conducted experiments to test laminar flow in microchannels with length 130 μm, height from 20 μm to 400 μm and relative roughness ranging from 0 to 1 Their outcomes demonstrate that the classic macroscale no-slip theory underestimates the friction factor because an increased dissipation rate near the roughness element causes more pressure loss compared with smooth wall surfaces.
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