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

2024
Views:

372

Pages:

20
Author:

Nathan Carlson

Confined Fluid Flow: Microfluidics and Capillarity Why Small? Microfluidic chemostat …static chemical environment used as bioreactor… fresh medium is continuously added, culture liquid is continuously removed Device provided single-cell resolution to study microbial population growth. Why Small? Protein cystallization Droplets containing proteins are trapped in microchannel wells. Dyed droplets in the wells of the device. Proteins crystallized within droplets in a glass capillary. Device used to efficiently screen for optimal protein crystallization conditions Why Small? Bubbles and foams Bubble generation with tunable, monodisperse sizes Why Small? Capillary origami Using surface tension to deform thin structures Why Small? Cellular and developmental biology Study effect of temperature on the development of a fruitfly embryo. Immobilized embryo has water with cold (left) and warm (right) flow over it. Why Small? Inexpensive diagnostics Costs as low as $0.001 per device (€0.00089) Low-cost, simple paper-based microfluidics for diagnostics Why Small? Volume goes as L3 • Small decrease in size = large reduction of sample volume. How much sample volume do we need? Detection of biomolecules: Digoxin – heart stimulating drug Cortisol – stress hormone from adrenal gland Creatinine – level in blood is measure of kidney function Theophylline – drug used to treat respiratory diseases, e.g. asthma Microfluidics Published items in each year… Citations in each year… 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 Dynamics Navier-Stokes Equations (momentum conservation) Continuity Equation (mass conservation) Fluid Behavior Inertial Forces Viscous Forces Trailing airplane vortices Coiling honey Reynolds Number: inertial/viscous Fluid Behavior Inertial Forces Reynolds Number: inertial/viscous Viscous Forces Fluid element accelerating around curve. • During a turn time: • Loss of momentum density: • By exerting an inertial centrifugal force density: Fluid element in a channel of contracting length. • By mass conservation, velocity increases as: • Gain momentum at a rate: Fluid Behavior Inertial Forces Reynolds Number: inertial/viscous Viscous Forces Inertial Forces Viscous Forces • Viscous force densities result from gradients in viscous stress: Ratio of these two force densities is the Reynolds number: Fluid Behavior Inertial Forces Viscous Forces Reynolds Number: inertial/viscous Estimation of Reynolds numbers for common microfluidic devices. • Typical fluid – water • Viscosity: 1.025 cP @ 25°C • Density: 1 g/mL • Typical channel dimensions • Radius/height (smaller than width): 1 – 100 µm • Typical velocities • Average velocity: 1 µm/s – 1 cm/s Typical Reynolds number: Low Reynolds number: viscous forces > inertial forces • Flows are linear. • Nonlinear terms in Navier-Stokes disappear • Linear, predictable Stokes flow Fluid Behavior Inertial Forces Viscous Forces Reynolds Number: inertial/viscous Reynolds number is focused on steady inertial forcing due to the convective derivative: • • Almost always unimportant in microfluidic flows. Time dependence arises from U0 Linear unsteady term sets the inertial time scale to establish steady flows: Time scale estimated by balancing unsteady inertial force density with viscous force density Time required for a vorticity to diffuse a distance L0, with a diffusivity ν=µ/ρ, τi~10ms Fluids and Circuits Pressure-driven flow in a network of parallel Electrical circuit with resistances in parallel. channels. • First elementary rule of circuit design • Flow rates in three parallel channels. is Ohm’s law • Total flow rate: • Relates electrical potential to • Pressure drop and flow rate related by: current: Flow: Cylindrical Tube Continuity Equation (cylindrical coords): Flow only in z direction: Navier-Stokes equations: Assumptions: Reduces to the following: 1. Neglect gravity: 2. Steady state: 3. Axisymmetric: Pressure driven flow – linear change Flow: Cylindrical Tube Navier-Stokes equations - cylindrical pipe: …integrate once… Boundary conditions: 1. Velocity is at a maximum at the center of the piper: …integrate twice… 2. No-slip along the walls. Flow: Cylindrical Tube Parabolic velocity distribution: Flow rate (volume/time) The flow rate depends on the fourth power of the radius: important in small systems • e.g. For the same pressure gradient, reducing the radius by a factor of two causes a 16-fold reduction in flow rate. • e.g. Consider a blood vessel – a 10% decrease in radius produces more than a 40% decrease in the flow rate of blood (PSA: Eat more kale!)