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4. Permeation flow. Poly(dimethylsiloxane) (PDMS) is an elastomer that is widely used in fabrication of microfluidic devices. This polymer is slightly permeable to water, and diffusion of water through a PDMS wall can lead to measurable velocities in thin chambers. Consider permeation-driven flow in the system shown in the figure, in which a channel of height h and length 2L has its ends connected reservoirs filled to equal heights. Assume that there is a constant velocity vw normal to the top of the channel, whereas the bottom surface (glass) is impermeable. (a) Assuming that the lubrication approximation is applicable, determine vz(x,z) in the channel. Find the cross-sectional average velocity, U(x). (b) For a channel of height h=2µm, width w=50µm, and length L=12.5mm, the average velocity was measured to be |U|=35µm/s at |x|=10mm. Assess the accuracy of the lubrication approximation for this channel, including end and edge effects.

          4. Permeation flow. Poly(dimethylsiloxane) (PDMS) is an elastomer that is widely used in fabrication of microfluidic devices. This polymer is slightly permeable to water, and diffusion of water through a PDMS wall can lead to measurable velocities in thin chambers. Consider permeation-driven flow in the system shown in the figure, in which a channel of height h and length 2L has its ends connected reservoirs filled to equal heights. Assume that there is a constant velocity vw normal to the top of the channel, whereas the bottom surface (glass) is impermeable.

(a) Assuming that the lubrication approximation is applicable, determine vz(x,z) in the channel. Find the cross-sectional average velocity, U(x).

(b) For a channel of height h=2µm, width w=50µm, and length L=12.5mm, the average velocity was measured to be |U|=35µm/s at |x|=10mm. Assess the accuracy of the lubrication approximation for this channel, including end and edge effects.

4. Permeation flow. Poly(dimethylsiloxane) (PDMS) is an elastomer that is widely used in fabrication of microfluidic devices. This polymer is slightly permeable to water, and diffusion of water through a PDMS wall can lead to measurable velocities in thin chambers. Consider permeation-driven flow in the system shown in the figure, in which a channel of height h and length 2L has its ends connected reservoirs filled to equal heights. Assume that there is a constant velocity vw normal to the top of the channel, whereas the bottom surface (glass) is impermeable.

(a) Assuming that the lubrication approximation is applicable, determine vz(x,z) in the channel. Find the cross-sectional average velocity, U(x).

(b) For a channel of height h=2µm, width w=50µm, and length L=12.5mm, the average velocity was measured to be |U|=35µm/s at |x|=10mm. Assess the accuracy of the lubrication approximation for this channel, including end and edge effects.

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