University:
Boston University
Course:
ENG ME 303 | Fluid Mechanics
Academic year:

2020
Views:

353

Pages:

35
Author:

Noupoortk3vc

Swelling of Elastic Materials Fluids Deforming Solids Fluids & Elasticity Flow through porous medium • Elastic deformation of medium Darcy’s Law • Biot’s Poroelasticity Fluids Deforming Solids • • • Surface Tension – Elastocapillarity Swelling & Growth Maxwell Stresses Swelling • Polymers Swelling a Sponge Pine Cones Tree-bound pine cones: Hydrated & closed, protecting seeds Fallen pine cones: Dried out & opened, releasing seeds Articular Cartilage Shape change caused by ion concentration. Residual strain at physiological conditions: 3-15% Tensile prestress in cartilage protective against frequent compresses forces. Lichens in the Rain Swelling & Growth Materials Science Swelling of a sponge. Mechanics Flow in Porous Media Flow in Porous Media Fluid Dynamics Navier-Stokes Equations (momentum conservation) Continuity Equation (mass conservation) Porous Media 2D Slice of Sandstone Fluid Behavior Inertial Forces Viscous Forces Trailing airplane vortices Coiling honey Reynolds Number: inertial/viscous Fluid Behavior Inertial Forces Viscous Forces Reynolds Number: inertial/viscous Steady inertial forcing due to the convective derivative: • 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 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 Reynolds Number: inertial/viscous Viscous Forces Typical Reynolds number: Low Reynolds number: viscous forces > inertial forces • Flows are linear. • Nonlinear terms in Navier-Stokes disappear • Linear, predictable Stokes flow Stokes Equations Stokes Equations (momentum & mass conservation) …but, in order to proceed, we need to prescribe BC’s on each grain… Porous Media Stokes Equations (momentum & mass conservation) …but, in order to proceed, we need to prescribe BC’s on each grain… Darcy-Brinkman-Stokes Equations Averaging over pressures and velocities. Void space: Viscous Drag Porous Media Darcy-Brinkman-Stokes Equations Averaging over pressures and velocities. Often negligible Darcy’s law (volumetric flux, isotropic medium) Porous Media Biot Poroelasticity Biot Poroelasticity Coupled problem: Pore pressure has time dependence, as does poroelastic stresses/strains. Poroelasticity: Cannot solve fluid flow problem independent of stress field. • Stress changes in fluid-saturated porous media typically produce significant changes in pore pressure. Increment in total work associated with strain increment and fluid content. Biot Poroelasticity Time dependence work is related to the fluid flux through Darcy’s law. Squeeze the soil – how much water comes out? Pressurize the water – how much water will go in the soil? Compression of the medium (e.g. soil) includes compression of pore fluid and particles plus the fluid expelled from an element by flow. Resistance of medium defined by bulk and shear moduli. Stress caused by (1.) hydrostatic pressure of water filling pores, and (2.) average stress in porous network. Stresses in the soil carried in part by the fluid and in part by solid. Biot Poroelasticity Swelling Spheres Polymers & Swelling Bad Solvent Good Solvent Swelling Free Energy Gibbs Free Energy of Dilution Equilibrium Swelling At constant pressure Determination via Osmotic Pressure Excess pressure required to keep mixed phase in equilibrium with the pure liquid. p: Vapor pressure of liquid in equilibrium with mixture. p0: Saturation pressure Free Energy Gibbs Free Energy of Dilution Enthalpy ~ Internal Energy Entropy of dilution (Boltzmann) Flory-Huggins Equation Free, long polymer chains Flory-Huggins Chi parameter: dimensionless, polymer/fluid interactions. Good solvents: Flory-Rehner Equation Crosslinked polymer networks • The entropy change caused by mixing of polymer and solvent. • The entropy change caused by reduction in numbers of possible chain conformations on swelling. • The heat of mixing of polymer and solvent, which may be positive, negative, or zero. Equilibrium swelling of a crosslinked network: Approximate equilibrium stretch: PDMS & Swelling Swelling a Disk λ is the initial swelling ratio. Swelling a Disk Swelling a Sphere Swelling Dynamics Linearized, similar to poroelasticity Incompressibility & Darcy’s law Volume change (e.g. sphere) Satisfied by diffusion relation:

Fluids Deforming Solids

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock