University:
Boston University
Course:
ENG ME 302 | Engineering Mechanics II
Academic year:

2023
Views:

433

Pages:

20
Author:

Marilyn Swanson

Flow: Cylindrical Tube Average velocity Recall the Reynolds number: For a fixed pressure gradient, and so a factor of two change in radius produces a factor of eight change in Reynolds number. The average velocity is proportional to the pressure gradient, writing the flow field in vector form gives Darcy’s law: - k is the permeability (dimensions length^2) - linear relation between pressure and velocity is due to lack of inertia effects - order-of-magnitude of permeability is typically the square of smallest dimension. Flow: Cylindrical Tube Hydrodynamic resistance recall: Electrical analogy: resistance in fluid channels depends on the fourth power of the radius rather than the second power in the electrical case. …………………small matters! Scaling on physical & dimensional arguments… • Can arrive at: dimensional analysis. and from • Fluid motion arises as a balance between the pressure drop driving motion and the frictional (viscous) resistance from the bounding walls. This balance applies independent of the cross-sectional channel shape. Flow: Rectangular Parabolic velocity distribution: Average velocity: Approximate flow rate (channel of width w) Hydrodynamic Resistance: Parabolic velocity distribution (known analytically in terms of a Fourier series): where: and, from the no-slip boundary conditions: Flow: Rectangular Parabolic velocity distribution: Average velocity: Approximate flow rate (channel of width w) Hydrodynamic Resistance: Corresponding flow rate where: Flow rate is nearly linear in aspect ratio h/w Flow: Rectangular Flow rate in a rectangular channel Approximate dimensionless flow rate Workout Problem While developing a microfluidic chip for your research, you decide a scale model would help you gain intuition about flow characteristics at the microscale. While most scale models are smaller that the original, your model will be 100 times larger (so that you can see it without a microscope). Honey Olive oil Water Molasses Viscosity 8750 cP 81 cP 1.025 cP @ 25°C 50,000 cP Density 1.42 g/mL 0.92 g/mL 1 g/mL 1.50 g/mL For the microfluidic chip, the flow conditions are: fluid velocity 1 mm/s, channel diameter 100 µm, and the liquid is water at T = 25°C. The Reynolds number for both systems is to be duplicated, and the microchannel is hemicylindrical (use the hydraulic diameter). For the scale model, which of the condiments listed above from your kitchen would you choose as the flow liquid, if the velocity in the model is: a.) 1 mm/s b.) 60 mm/s c.) 340 mm/s Recall: hydraulic diameter, D H , relates flow in noncircular channels to those in a circular channel: Workout Problem Hydraulic diameter of hemicylindrical channel: Reynold’s number of microfluidic chip: a. Fluid flowing at 1 mm/s in scaled up model Use Olive Oil. b. Fluid flowing at 60 mm/s in scaled up model Use Honey. c. Fluid flowing at 340 mm/s in scaled up model Use Molasses. Overview Confined Fluid Flow: Microfluidics and Capillarity Reynolds Number: Inertia vs. Viscous effects • Review of characteristic flows… Péclet Number: Transport phenomena in a continuum • Diffusion, separation, and mixing… Geometric confinement: Controlling and manipulating fluid flow • Microfluidic fabrication, valving, pumping… Capillary Number: Viscosity vs. Surface tension • Droplet formation, capillary rise, elasticity… Fluid Behavior Convection Diffusion Convection current as hot water mixes with cold water Diffusion of salt ions while curing pork Péclet Number: convection/diffusion Fluid Behavior Convection Péclet Number: convection/diffusion High Reynolds: eddies chaotically stretch and fold fluid elements. • Turbulent mixing & thermal convection move fluids. Low Reynolds: mixing occurs by diffusion only • Diffusion is remarkably slow – long mixing times. Diffusion T-Junction in which two fluids are injected and flow alongside each other How far down the channel must the fluids flow before the channel is homogenized? Fluid Behavior Convection Péclet Number: convection/diffusion How far down the channel must the fluids flow before the channel is homogenized? Time for particles/molecules to diffuse across channel: Diffusion During this time, the stripe moves down the channel: Number of channel widths required for complete mixing: Fluid Behavior Convection Péclet Number: convection/diffusion Relative importance: convection to diffusion: Diffusion Consider a small protein flowing with liquid: • Typical size: 5 nm • Diffusion constant: 40 µm2/s • e.g. Channel width: 100 µm • e.g. Flow velocity: 100 µm/s Sensing & Filtering Parallel Laminar Flows Pressure-driven laminar flow of two adjacent miscible streams. • • • Solution on left contains calcium. Solution on right contains calcium-dependent fluorophore, Fluo-3. In water, Fluo-3 and calcium form a fluorescent complex at a diffusion limited rate. Convective-diffusion equation: For high Péclet number flows, • Diffusive broadening down the channel depends on both y & x Away from boundaries, the interfacial region: Sensing & Filtering Parallel Laminar Flows Pressure-driven laminar flow of two adjacent miscible streams. • • • Solution on left contains calcium. Solution on right contains calcium-dependent fluorophore, Fluo-3. In water, Fluo-3 and calcium form a fluorescent complex at a diffusion limited rate. Away from boundaries, the interfacial region: Lévêque Problem: Near from boundaries, velocity varies linearly with distance into the channel. Diffusion across a linear flow field Diffusion is the only time scale in both x & y: Sensing & Filtering Parallel Laminar Flows Pressure-driven laminar flow of two adjacent miscible streams. • • • Edge of channel Solution on left contains calcium. Solution on right contains calcium-dependent fluorophore, Fluo-3. In water, Fluo-3 and calcium form a fluorescent complex at a diffusion limited rate. Away from boundaries, the interfacial region: Lévêque Problem: Near from boundaries, velocity varies linearly with distance into the channel. Diffusion across a linear flow field Center of channel Diffusion is the only time scale in both x & y: Sensing & Filtering H Filter Parallel Laminar Flows Filter particles by size without a membrane. • Pressure-driven laminar flow of two adjacent miscible streams. • One stream is a dilute solution of different sized particles. • Each particle has its own diffusivity and Péclet number. Péclet number determines the channel length required for each component to diffuse across the channel width. H filter works best when one Péclet number is large and the other is small. • e.g. small particles diffuse across channel, large ones do not. Sensing & Filtering H Filter - Variant Separation by Péclet number • Components to be separated spread across channel at different rates. • e.g. Separation of motile vs. nonmotile sperm. Motile sperm swim rapidly and randomly to fill the channel, as compared to nonmotile sperm. • Nonmotile sperm spread by diffusion alone. Motile sperm purity at inlet (blue) vs. motile sperm purity at outlet (purple) Sensing & Filtering Fabrication using multiple laminar streams …beating diffusion… Fabrication of a three electrode system • Multiple streams & large Péclet number. • Minimal mixing over large distances. Nonmixing, high Péclet flows can fabricate/ etch structures within microchannels. e.g. Parallel streams selectively etch a gold electrode • • A silver wire is formed by a precipitation reaction between the two streams. Result: three-electrode system. Mixing Rapid mixing means doing better than diffusion. Why mix? • Study chemical reaction kinetics… • Probe protein folding… Fluid stirring will stretch and fold inhomogenous fluid elements until mixing occurs. • Mixing: diffusive migration across streamlines. • Stirring motions reduce distance over which mixing must occur. General design principles for mixing: Dispersion of tracers occurs first by convective stretching with the fluid, followed by diffusive homogenization. Tracers: an object transported by and diffuses into the fluid, e.g. dyes, analyte molecules, proteins, cells, salt, or heat. Mixing Taylor Dispersion Thin stripe of tracers spans circular channel • Channel radius: w Pressure-driven (Poiseuille) flow stretches strip into a parabola with profile: Molecular diffusion across the channel smears the parabola into a plug of width: Each stripe is convectively stretched then diffusively smeared After N time steps of , the initial stripe grows into a Gaussian with width: