Randall and Doyle (2005) discovered that diffusion of water through the poly(dimethylsiloxane) (PDMS) that formed the top of a microfluidic channel produced an observable flow. Their device is shown schematically in Fig. P8.3. A channel of thickness H and length 2L was connected at each end to reservoirs filled to equal heights. The channel width W greatly exceeded H, which justifies a two-dimensional model. The upper wall of the channel was PDMS and the lower wall was glass, which is impermeable to water. Diffusion of water through the PDMS and into the surrounding air created a velocity v_w normal to the upper wall. Although v_w was actually a function of x, you may assume here that it is constant.

(a) Relate the local mean velocity u(x) to v_w.

(b) Typical channel dimensions were H = 2 μm, W = 50 μm, and L = 12.5 mm, and it was found that u = 35 μm/s at |x| = 10 mm. Estimate the average value of v_w and explain why a lubrication approximation is justified.

(c) Derive expressions for v_x(x, y) and v_y(x, y).

Transcribed Image Text:

P=0 Vw 2L PDMS y Glass X TH Figure P8.3 Permeation-driven flow of water in a microchannel. P=0