12.2 Non-Newtonian multiphase flows

Non-Newtonian fluids have complex flow behaviors that differ from simple liquids. Their viscosity changes with applied force, leading to unique properties like shear thinning or thickening. This impacts how they behave in multiphase systems.

Understanding non-Newtonian multiphase flows is crucial in many industries. From oil drilling to food processing, these flows affect product quality and process efficiency. Modeling them requires specialized equations and numerical methods to capture their distinct characteristics.

Non-Newtonian fluid properties

• Non-Newtonian fluids exhibit complex rheological behavior where the relationship between shear stress and shear rate is not linear
• The viscosity of non-Newtonian fluids is not constant and depends on the applied shear rate or shear stress
• Understanding the unique properties of non-Newtonian fluids is crucial for accurately modeling their behavior in multiphase flow systems

Shear rate vs shear stress

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• Non-Newtonian fluids display a non-linear relationship between shear rate and shear stress
• The shear stress-shear rate curve can be used to characterize the rheological behavior of non-Newtonian fluids
• The slope of the curve represents the apparent viscosity, which varies with the applied shear rate
• Examples of non-linear shear stress-shear rate relationships include shear thinning (pseudoplastic) and shear thickening (dilatant) behavior

Apparent viscosity

• Apparent viscosity is the ratio of shear stress to shear rate at a given point on the flow curve
• For non-Newtonian fluids, the apparent viscosity is not constant and depends on the applied shear rate or shear stress
• Shear thinning fluids exhibit a decrease in apparent viscosity with increasing shear rate (paints, ketchup)
• Shear thickening fluids display an increase in apparent viscosity with increasing shear rate (cornstarch suspensions, some colloidal dispersions)

Yield stress fluids

• Yield stress fluids require a minimum stress (yield stress) to be applied before they start to flow
• Below the yield stress, these fluids behave like solids, while above the yield stress, they flow like liquids
• Examples of yield stress fluids include toothpaste, mayonnaise, and drilling muds
• The presence of a yield stress can significantly impact the flow behavior and stability of multiphase systems

Shear thinning vs shear thickening

• Shear thinning (pseudoplastic) fluids exhibit a decrease in viscosity with increasing shear rate
• The viscosity of shear thinning fluids is highly dependent on the applied shear rate
• Examples of shear thinning fluids include polymer solutions, blood, and some suspensions
• Shear thickening (dilatant) fluids display an increase in viscosity with increasing shear rate
• The viscosity of shear thickening fluids increases rapidly beyond a critical shear rate
• Examples of shear thickening fluids include concentrated cornstarch suspensions and some colloidal dispersions

Rheological models

• Rheological models are mathematical expressions that describe the relationship between shear stress and shear rate for non-Newtonian fluids
• These models are essential for characterizing the flow behavior of non-Newtonian fluids and predicting their performance in multiphase flow systems
• The choice of rheological model depends on the specific fluid and the range of shear rates encountered in the application

Power law model

• The power law model, also known as the Ostwald-de Waele model, is a simple two-parameter model that describes the shear stress-shear rate relationship for non-Newtonian fluids
• The model is given by: $\tau = K \dot{\gamma}^n$, where $\tau$ is the shear stress, $K$ is the consistency index, $\dot{\gamma}$ is the shear rate, and $n$ is the flow behavior index
• For shear thinning fluids, $n < 1$, while for shear thickening fluids, $n > 1$
• The power law model is widely used due to its simplicity but has limitations in describing the behavior of fluids with yield stress or viscosity plateaus

Bingham plastic model

• The Bingham plastic model is a two-parameter model that accounts for the presence of a yield stress in non-Newtonian fluids
• The model is described by: $\tau = \tau_y + \mu_p \dot{\gamma}$, where $\tau_y$ is the yield stress and $\mu_p$ is the plastic viscosity
• Below the yield stress, the fluid behaves like a solid, while above the yield stress, it flows with a constant plastic viscosity
• The Bingham plastic model is suitable for describing the behavior of fluids such as toothpaste, mayonnaise, and some drilling muds

Herschel-Bulkley model

• The Herschel-Bulkley model is a three-parameter model that combines the features of the power law and Bingham plastic models
• The model is given by: $\tau = \tau_y + K \dot{\gamma}^n$, where $\tau_y$ is the yield stress, $K$ is the consistency index, and $n$ is the flow behavior index
• The Herschel-Bulkley model can describe the behavior of fluids that exhibit both a yield stress and shear thinning or shear thickening behavior
• Examples of fluids that can be modeled using the Herschel-Bulkley model include blood, yogurt, and some polymer solutions

Casson model

• The Casson model is a two-parameter model that is particularly useful for describing the flow behavior of fluids with a yield stress and a non-linear shear stress-shear rate relationship
• The model is given by: $\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\mu_c \dot{\gamma}}$, where $\tau_y$ is the yield stress and $\mu_c$ is the Casson viscosity
• The Casson model is often used to describe the flow behavior of blood, chocolate, and some food products
• The model can capture the shear thinning behavior and the presence of a yield stress in these fluids

Multiphase non-Newtonian flow regimes

• Multiphase non-Newtonian flow regimes describe the various patterns and distributions of phases that can occur when non-Newtonian fluids are present in a multiphase system
• The flow regime depends on factors such as the fluid properties, flow rates, and geometry of the system
• Understanding the different flow regimes is crucial for predicting the behavior and performance of non-Newtonian multiphase flows in various applications

Bubble flow

• Bubble flow occurs when the dispersed phase (gas) is present as discrete bubbles in a continuous non-Newtonian liquid phase
• The bubbles are typically small and spherical, and they are uniformly distributed throughout the liquid phase
• Bubble flow is often encountered in gas-liquid systems with low gas flow rates and can be observed in applications such as aeration and fermentation processes
• The presence of non-Newtonian fluid properties can affect the bubble size distribution, coalescence, and breakup behavior

Slug flow

• Slug flow is characterized by the presence of large, bullet-shaped gas bubbles (Taylor bubbles) that occupy a significant portion of the pipe cross-section
• The Taylor bubbles are separated by liquid slugs, which may contain smaller dispersed bubbles
• Slug flow occurs at higher gas flow rates compared to bubble flow and is commonly observed in vertical and inclined pipes
• Non-Newtonian fluid properties can influence the stability and length of the slugs, as well as the mixing and mass transfer between the phases

Churn flow

• Churn flow is a chaotic and highly turbulent flow regime that occurs at high gas and liquid flow rates
• The flow is characterized by the presence of irregularly shaped gas structures and intense mixing between the phases
• Churn flow is often a transitional regime between slug flow and annular flow
• The complex rheological properties of non-Newtonian fluids can significantly impact the onset and characteristics of churn flow

Annular flow

• Annular flow occurs when the gas phase flows in the center of the pipe as a continuous core, while the liquid phase flows as an annular film along the pipe wall
• The liquid film may contain entrained gas bubbles, and the interface between the gas core and the liquid film can be wavy or distorted
• Annular flow is commonly observed at high gas flow rates and is relevant in applications such as gas-lift operations and condensation processes
• The presence of non-Newtonian fluids can affect the stability and thickness of the liquid film, as well as the interfacial waves and droplet entrainment

Constitutive equations

• Constitutive equations are mathematical relationships that describe the behavior of non-Newtonian fluids in multiphase flow systems
• These equations relate the stress tensor to the rate of strain tensor and other relevant variables, such as temperature and concentration
• Constitutive equations are essential for closing the system of governing equations and enabling the numerical simulation of non-Newtonian multiphase flows

Momentum conservation

• The momentum conservation equation describes the balance of forces acting on a fluid element in a multiphase flow system
• For non-Newtonian fluids, the momentum conservation equation must account for the complex rheological behavior and the interactions between the phases
• The general form of the momentum conservation equation for a non-Newtonian fluid is: $\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{g}$, where $\rho$ is the fluid density, $\mathbf{u}$ is the velocity vector, $p$ is the pressure, $\boldsymbol{\tau}$ is the stress tensor, and $\mathbf{g}$ is the gravitational acceleration
• The stress tensor $\boldsymbol{\tau}$ is related to the rate of strain tensor through the constitutive equation, which depends on the specific rheological model used

Mass conservation

• The mass conservation equation, also known as the continuity equation, describes the balance of mass in a multiphase flow system
• For non-Newtonian fluids, the mass conservation equation must be satisfied for each phase present in the system
• The general form of the mass conservation equation for a phase $k$ is: $\frac{\partial (\alpha_k \rho_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \mathbf{u}_k) = 0$, where $\alpha_k$ is the volume fraction of phase $k$, $\rho_k$ is the density of phase $k$, and $\mathbf{u}_k$ is the velocity vector of phase $k$
• The mass conservation equation ensures that the total mass of each phase is conserved and accounts for the exchange of mass between phases due to processes such as phase change or chemical reactions

Stress-strain relationships

• Stress-strain relationships are constitutive equations that relate the stress tensor to the rate of strain tensor for non-Newtonian fluids
• These relationships are specific to the rheological model used to describe the fluid behavior
• For the power law model, the stress-strain relationship is given by: $\boldsymbol{\tau} = K |\dot{\boldsymbol{\gamma}}|^{n-1} \dot{\boldsymbol{\gamma}}$, where $\boldsymbol{\tau}$ is the stress tensor, $K$ is the consistency index, $\dot{\boldsymbol{\gamma}}$ is the rate of strain tensor, and $n$ is the flow behavior index
• For the Bingham plastic model, the stress-strain relationship is: $\boldsymbol{\tau} = (\tau_y / |\dot{\boldsymbol{\gamma}}| + \mu_p) \dot{\boldsymbol{\gamma}}$, where $\tau_y$ is the yield stress and $\mu_p$ is the plastic viscosity
• The stress-strain relationships for other rheological models, such as the Herschel-Bulkley and Casson models, can be similarly derived based on their respective constitutive equations

Numerical methods for non-Newtonian multiphase flows

• Numerical methods are essential for solving the complex system of governing equations that describe non-Newtonian multiphase flows
• These methods discretize the equations in space and time and provide approximate solutions for the velocity, pressure, and other relevant variables
• The choice of numerical method depends on factors such as the complexity of the geometry, the rheological model, and the desired accuracy and computational efficiency

Finite volume method

• The finite volume method (FVM) is a popular numerical method for solving the governing equations of non-Newtonian multiphase flows
• In FVM, the computational domain is divided into a set of control volumes, and the governing equations are integrated over each control volume
• The method ensures conservation of mass, momentum, and energy at the control volume level and can handle complex geometries and unstructured meshes
• FVM is widely used in commercial and open-source computational fluid dynamics (CFD) software packages for simulating non-Newtonian multiphase flows

Finite element method

• The finite element method (FEM) is another numerical method that can be used for solving the governing equations of non-Newtonian multiphase flows
• In FEM, the computational domain is discretized into a set of elements, and the governing equations are approximated using a weighted residual formulation
• FEM is particularly suitable for handling complex geometries and can provide high-order accuracy for the solution variables
• The method is often used in specialized CFD software packages and is well-suited for problems involving solid-fluid interactions and non-Newtonian fluid behavior

Lattice Boltzmann method

• The lattice Boltzmann method (LBM) is a mesoscopic numerical method that is based on the kinetic theory of gases
• In LBM, the fluid is represented by a set of discrete velocity distribution functions, which evolve according to a simplified version of the Boltzmann equation
• LBM is particularly attractive for simulating non-Newtonian multiphase flows due to its ability to handle complex rheological behavior and capture interfacial phenomena
• The method is computationally efficient and can be easily parallelized, making it suitable for large-scale simulations

Smoothed particle hydrodynamics

• Smoothed particle hydrodynamics (SPH) is a meshless Lagrangian numerical method that is based on the discretization of the fluid into a set of particles
• In SPH, the governing equations are approximated using a kernel function that determines the influence of neighboring particles on a given particle
• SPH is well-suited for simulating non-Newtonian multiphase flows with large deformations and free surface flows
• The method can handle complex rheological behavior and is particularly useful for problems involving fluid-structure interactions and multiphase mixing

Applications of non-Newtonian multiphase flows

• Non-Newtonian multiphase flows are encountered in a wide range of industrial and natural processes
• Understanding and predicting the behavior of these flows is crucial for optimizing process performance, ensuring product quality, and minimizing environmental impact
• Some key applications of non-Newtonian multiphase flows include:

Oil and gas industry

• Non-Newtonian multiphase flows are prevalent in the oil and gas industry, particularly in drilling and production operations
• Drilling muds, which are used to lubricate and cool the drill bit and transport rock cuttings to the surface, often exhibit non-Newtonian behavior due to the presence of polymers and other additives
• In multiphase flow pipelines, the presence of non-Newtonian fluids can affect the flow patterns, pressure drop, and phase distribution, which are critical for optimizing production and ensuring safe operations
• Understanding the rheological behavior of crude oils, which can be non-Newtonian, is essential for designing and operating pipelines, pumps, and separation equipment

Food processing

• Many food products, such as sauces, dressings, and dairy products, exhibit non-Newtonian behavior due to the presence of complex ingredients and microstructures
• In food processing operations, such as mixing, pumping, and extrusion, the non-Newtonian properties of the fluids can significantly impact the process performance and product quality
• Multiphase flows are common in food processing, such as in the production of emulsions (mayonnaise, salad dressings) and foams (whipped cream, ice cream)
• Modeling and simulating non-Newtonian multiphase flows in food processing can help optimize process conditions, improve product consistency, and reduce waste

Pharmaceutical manufacturing

• Non-Newtonian multiphase flows are encountered in various pharmaceutical manufacturing processes, such as mixing, granulation, and coating
• Many pharmaceutical formulations, such as suspensions, gels, and creams, exhibit non-Newtonian behavior due to the presence of active ingredients, excipients, and rheology modifiers
• In multiphase systems, such as emulsions and suspensions, the non-Newtonian properties of the continuous phase can affect the stability, homogeneity, and release kinetics of the drug
• Understanding and controlling non-Newtonian multiphase flows in pharmaceutical manufacturing is crucial for ensuring product quality, bioavailability, and patient safety

Polymer processing

• Polymer melts and solutions often exhibit non-Newtonian behavior, such as shear thinning and viscoelasticity, which can significantly impact their processing and final product properties
• In polymer processing operations, such as extrusion, injection molding, and fiber spinning, the non-Newtonian properties of the polymer can affect the flow behavior, heat transfer, and phase morphology
• Multiphase flows are common in polymer processing, such as in the production of polymer blends, composites, and foams