💨Fluid Dynamics Unit 11 – Multiphase & Non-Newtonian Fluid Flows

Key Concepts

• Multiphase flows involve the simultaneous presence of two or more phases (gas, liquid, or solid) in a fluid system
• Non-Newtonian fluids exhibit viscosity that depends on the applied shear stress or shear rate (shear-thinning or shear-thickening behavior)
• Interfacial phenomena play a crucial role in multiphase flows, including surface tension, wettability, and interfacial instabilities
• Coupling between phases occurs through mass, momentum, and energy transfer across phase boundaries
• Rheology is the study of flow and deformation behavior of non-Newtonian fluids
  • Describes the relationship between stress and strain in complex fluids
• Constitutive equations are used to model the stress-strain relationship in non-Newtonian fluids (power-law, Bingham plastic, Herschel-Bulkley models)
• Dimensionless numbers characterize the relative importance of different forces in multiphase and non-Newtonian flows (Reynolds number, Capillary number, Deborah number)

Types of Multiphase Flows

• Gas-liquid flows: Bubbles or droplets of one phase dispersed in another phase (bubbly flow, slug flow, annular flow)
• Liquid-liquid flows: Immiscible liquids forming droplets or layers (oil-water flow, emulsions)
• Gas-solid flows: Solid particles suspended in a gas phase (pneumatic conveying, fluidized beds)
• Liquid-solid flows: Solid particles transported by a liquid phase (slurry flow, sediment transport)
• Three-phase flows: Simultaneous presence of gas, liquid, and solid phases (gas-liquid-solid reactors, oil-water-gas pipelines)
• Separated flows: Distinct phases flow separately with a clear interface between them (stratified flow, annular flow)
• Dispersed flows: One phase is dispersed as small elements (droplets, bubbles, or particles) in another continuous phase (bubbly flow, mist flow, particulate flow)

Non-Newtonian Fluid Behavior

• Shear-thinning (pseudoplastic) fluids: Viscosity decreases with increasing shear rate (polymer solutions, blood, paint)
  • Apparent viscosity is a function of shear rate
• Shear-thickening (dilatant) fluids: Viscosity increases with increasing shear rate (suspensions, cornstarch in water)
• Bingham plastic fluids: Exhibit a yield stress and a constant viscosity above the yield stress (toothpaste, mayonnaise, drilling mud)
• Herschel-Bulkley fluids: Combine yield stress and power-law behavior (food products, cosmetics, some slurries)
• Thixotropic fluids: Viscosity decreases with time under constant shear stress and recovers when stress is removed (yogurt, some paints)
• Viscoelastic fluids: Exhibit both viscous and elastic properties (polymer melts, dough, some gels)
  • Elastic effects become significant at high Deborah numbers

Governing Equations

• Conservation of mass (continuity equation) for each phase: $\frac{\partial \rho_k}{\partial t} + \nabla \cdot (\rho_k \mathbf{u}_k) = 0$
  • $\rho_k$ is the density and $\mathbf{u}_k$ is the velocity of phase $k$
• Conservation of momentum (Navier-Stokes equations) for each phase: $\rho_k (\frac{\partial \mathbf{u}_k}{\partial t} + \mathbf{u}_k \cdot \nabla \mathbf{u}_k) = -\nabla p_k + \nabla \cdot \boldsymbol{\tau}_k + \rho_k \mathbf{g} + \mathbf{F}_{k,interfacial}$
  • $p_k$ is the pressure, $\boldsymbol{\tau}_k$ is the stress tensor, $\mathbf{g}$ is the gravitational acceleration, and $\mathbf{F}_{k,interfacial}$ represents interfacial forces
• Constitutive equations relate stress to strain rate for non-Newtonian fluids (power-law model: $\boldsymbol{\tau} = K |\dot{\boldsymbol{\gamma}}|^{n-1} \dot{\boldsymbol{\gamma}}$)
• Interfacial conditions: Jump conditions for mass, momentum, and energy across phase boundaries
• Closure models: Additional equations for interfacial forces, turbulence, and phase interactions

Flow Patterns and Regimes

• Flow patterns describe the spatial distribution of phases in multiphase flows (bubbly, slug, churn, annular, stratified, dispersed)
• Flow regime maps: Graphical representation of flow patterns based on dimensionless numbers (gas and liquid superficial velocities)
• Transition criteria: Conditions for the transition between different flow patterns (void fraction, slip velocity, stability criteria)
• Influence of fluid properties, pipe geometry, and inclination on flow patterns
• Intermittent flows: Alternating presence of different flow patterns (slug flow, plug flow)
• Stratified flows: Gravity-driven separation of phases with a distinct interface (smooth or wavy stratified flow)
• Dispersed flows: One phase is dispersed as small elements in another continuous phase (bubbly flow, mist flow, particulate flow)

Measurement Techniques

• Visualization methods: High-speed imaging, particle image velocimetry (PIV), laser-induced fluorescence (LIF)
  • Provide qualitative and quantitative information on flow patterns and phase distributions
• Intrusive methods: Conductivity probes, optical probes, hot-wire anemometry
  • Measure local phase fractions, velocities, and interfacial properties
• Non-intrusive methods: X-ray and gamma-ray tomography, electrical capacitance tomography (ECT), ultrasonic techniques
  • Enable non-invasive measurements of phase distributions and velocities
• Pressure drop measurements: Differential pressure transducers, manometers
  • Determine the pressure gradient and infer flow characteristics
• Rheological measurements: Rheometers (rotational, capillary, oscillatory) to characterize non-Newtonian fluid behavior
• Sampling techniques: Isokinetic sampling, phase separation methods to obtain representative samples of multiphase flows
• Data analysis: Signal processing, statistical methods, and machine learning techniques to extract meaningful information from measurement data

Industrial Applications

• Oil and gas industry: Multiphase flow in pipelines, well bores, and processing equipment (separators, desalters)
• Chemical processing: Reactors, mixing vessels, and separation units involving multiphase flows and non-Newtonian fluids
• Food processing: Handling and processing of food products with complex rheological properties (extrusion, mixing, pumping)
• Pharmaceutical industry: Manufacturing and processing of drug formulations, emulsions, and suspensions
• Environmental engineering: Wastewater treatment, sedimentation processes, and contaminant transport in multiphase systems
• Power generation: Boiling and condensation in heat exchangers, steam generators, and cooling systems
• Polymer processing: Extrusion, injection molding, and fiber spinning of polymeric materials with non-Newtonian behavior
• Biotechnology: Bioreactors, fermentation processes, and cell culture systems involving multiphase flows and complex fluids

Modeling and Simulation

• Eulerian-Eulerian approach: Treats each phase as an interpenetrating continuum with its own set of conservation equations
  • Requires closure models for interfacial forces and phase interactions
• Eulerian-Lagrangian approach: Continuous phase is treated as a continuum, while dispersed phase is tracked as individual particles or droplets
  • Suitable for dilute dispersed flows with low volume fractions
• Volume of Fluid (VOF) method: Captures the interface between immiscible fluids using a volume fraction function
  • Enables simulation of interfacial phenomena and topological changes
• Level-Set method: Represents the interface as a zero level-set of a higher-dimensional function
  • Provides accurate interface tracking and handles complex interface geometries
• Discrete Element Method (DEM): Models the motion and interactions of individual particles in a Lagrangian framework
  • Coupled with fluid flow equations to simulate particle-laden flows
• Constitutive modeling: Incorporates non-Newtonian fluid behavior into computational fluid dynamics (CFD) simulations
  • Requires appropriate constitutive equations and numerical techniques
• Turbulence modeling: Addresses the challenges of turbulent multiphase flows using approaches like Reynolds-Averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES)