11.5 Viscoelastic flows
12 min read•august 20, 2024
Viscoelastic fluids combine viscous and elastic properties, resulting in unique flow behaviors. These fluids, like blood and , exhibit non-Newtonian characteristics such as and normal stress differences.
Understanding viscoelastic flows is crucial for many applications. This topic covers constitutive equations, dimensionless numbers, flow phenomena, and experimental techniques used to study and predict the complex behavior of these fascinating fluids.
Characteristics of viscoelastic fluids
- Viscoelastic fluids exhibit both viscous and elastic properties, resulting in unique flow behavior that differs from Newtonian fluids
- These fluids have a complex microstructure, often consisting of long-chain molecules or suspended particles, which contributes to their viscoelastic nature
Combination of viscous and elastic properties
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- Viscous properties cause the fluid to resist flow and dissipate energy due to internal friction
- Elastic properties enable the fluid to store energy and partially recover its original shape after deformation
- The interplay between viscous and elastic effects leads to time-dependent and non-linear flow behavior
Non-Newtonian behavior
- Viscoelastic fluids exhibit non-Newtonian behavior, meaning their viscosity is not constant and depends on the applied shear rate or stress
- Common non-Newtonian behaviors include shear-thinning (decreasing viscosity with increasing shear rate) and (increasing viscosity with increasing shear rate)
- The presence of , where the fluid behaves as a solid below a critical stress and flows above it, is another non-Newtonian characteristic
Examples in nature and industry
- Natural viscoelastic fluids include blood, saliva, and mucus, which play crucial roles in biological systems
- Polymer solutions and melts, such as plastics, rubbers, and adhesives, are widely used viscoelastic fluids in industrial applications
- Food products, like ketchup, mayonnaise, and yogurt, exhibit viscoelastic properties that influence their texture and processing
Constitutive equations for viscoelastic fluids
- Constitutive equations describe the relationship between stress and deformation in viscoelastic fluids, capturing their complex rheological behavior
- These equations are essential for mathematical modeling and simulation of viscoelastic flows, as they provide a link between the fluid's microstructure and its macroscopic flow properties
Linear vs nonlinear models
- Linear viscoelastic models, such as the Maxwell and Kelvin-Voigt models, are based on linear combinations of elastic and viscous elements
- These models are suitable for small deformations and can capture some basic viscoelastic phenomena
- However, they have limitations in describing more complex behaviors, such as shear-thinning or normal stress differences
- Nonlinear viscoelastic models, like the Oldroyd-B and Giesekus models, incorporate additional terms to account for nonlinear effects
- These models can better represent the behavior of real viscoelastic fluids under a wider range of flow conditions
- Nonlinear models are more computationally demanding but provide more accurate predictions
Maxwell model
- The is a simple linear viscoelastic model that consists of an elastic spring and a viscous damper connected in series
- It describes the fluid's response to deformation as a combination of instantaneous elastic deformation and time-dependent viscous flow
- The Maxwell model predicts , where the stress decays exponentially over time under constant strain
Oldroyd-B model
- The Oldroyd-B model is a nonlinear constitutive equation that extends the Maxwell model by including an additional term for the upper-convected time derivative of stress
- This model can capture shear-thinning behavior and predict non-zero normal stress differences in shear flows
- The Oldroyd-B model is widely used in simulations of polymer solutions and melts
Giesekus model
- The Giesekus model is another nonlinear constitutive equation that introduces a quadratic term in the stress tensor to account for anisotropic drag on polymer molecules
- This model can describe shear-thinning, normal stress differences, and extensional thickening in viscoelastic fluids
- The Giesekus model is particularly useful for modeling concentrated polymer solutions and melts with entangled microstructures
Dimensionless numbers in viscoelastic flows
- Dimensionless numbers are used to characterize the relative importance of different physical effects in viscoelastic flows
- These numbers help in understanding the flow behavior, comparing different flow scenarios, and designing experiments or simulations
Deborah number
- The (De) is the ratio of the fluid's relaxation time to the characteristic time scale of the flow
- It quantifies the importance of elastic effects relative to the flow time scale
- For De << 1, the fluid behaves mostly like a Newtonian fluid, as it has sufficient time to relax during the flow
- For De >> 1, the fluid exhibits significant viscoelastic effects, as the relaxation time is much longer than the flow time scale
- The Deborah number is important in determining the onset of viscoelastic instabilities and non-Newtonian flow phenomena
Weissenberg number
- The (Wi) is the product of the fluid's relaxation time and the characteristic shear rate of the flow
- It represents the ratio of elastic to viscous forces in the flow
- For Wi << 1, viscous effects dominate, and the fluid behaves mostly like a Newtonian fluid
- For Wi >> 1, elastic effects become significant, leading to viscoelastic phenomena such as rod climbing () and
- The Weissenberg number is crucial in understanding the flow behavior of viscoelastic fluids in shear-dominated flows
Elasticity number
- The (El) is the ratio of the Weissenberg number to the Reynolds number (Re)
- It compares the relative importance of elastic forces to inertial forces in the flow
- For El << 1, inertial effects dominate, and the flow behavior is primarily influenced by the Reynolds number
- For El >> 1, elastic effects are more significant than inertial effects, and the flow is governed by viscoelastic phenomena
- The elasticity number is useful in characterizing the flow behavior of viscoelastic fluids in mixed shear and extensional flows, such as in contractions and expansions
Shear and extensional flows
- Viscoelastic fluids exhibit distinct behavior in shear and extensional flows, which are fundamental types of deformation encountered in various applications
- Understanding the response of viscoelastic fluids to these deformations is crucial for predicting their flow behavior and optimizing processes
Shear-thinning and shear-thickening behavior
- Shear-thinning behavior, also known as pseudoplasticity, is characterized by a decrease in viscosity with increasing shear rate
- This behavior arises from the alignment and disentanglement of polymer molecules or the breakup of particle aggregates under shear
- Shear-thinning is common in polymer solutions, melts, and suspensions and is beneficial for improving flow and reducing energy consumption in processing applications
- Shear-thickening behavior, or dilatancy, is characterized by an increase in viscosity with increasing shear rate
- This behavior is less common and can occur in concentrated suspensions or solutions due to the formation of temporary particle or molecular networks under high shear
- Shear-thickening can be exploited in applications such as impact-resistant materials and vibration damping
Normal stress differences
- In shear flows, viscoelastic fluids exhibit normal stress differences, which are additional stresses perpendicular to the flow direction
- The first normal stress difference (N1) is the difference between the normal stresses in the flow and gradient directions, while the second normal stress difference (N2) is the difference between the normal stresses in the gradient and vorticity directions
- Normal stress differences arise from the anisotropic microstructure of viscoelastic fluids and are responsible for phenomena such as rod climbing (Weissenberg effect) and die swell
- Measuring normal stress differences is important for characterizing the viscoelastic properties of fluids and predicting their behavior in complex flows
Extensional viscosity
- Extensional viscosity describes the resistance of a fluid to extensional or elongational deformations, where the fluid is stretched along one or more axes
- In extensional flows, viscoelastic fluids often exhibit strain-hardening behavior, where the extensional viscosity increases with increasing strain rate
- This behavior is due to the stretching and alignment of polymer molecules or the formation of transient networks in the fluid
- Strain-hardening can lead to enhanced stability and reduced necking in processes such as fiber spinning and film blowing
- Measuring extensional viscosity is challenging but crucial for understanding the behavior of viscoelastic fluids in extensional-dominated flows, such as in contraction geometries and coating applications
Barus effect
- The Barus effect, also known as extrudate swell or die swell, refers to the increase in the cross-sectional area of a viscoelastic fluid as it exits a capillary or die
- This phenomenon occurs due to the recovery of elastic deformations accumulated within the fluid during flow through the constriction
- The extent of die swell depends on the viscoelastic properties of the fluid, the flow conditions, and the geometry of the die
- Predicting and controlling die swell is important in polymer processing operations, such as extrusion and injection molding, to ensure consistent product dimensions and quality
Instabilities and flow phenomena
- Viscoelastic fluids are prone to various instabilities and unique flow phenomena that arise from the interplay between elastic and viscous forces
- These instabilities can lead to complex flow patterns, interfacial distortions, and changes in flow resistance, which have implications for processing and performance
Elastic turbulence
- is a flow instability that occurs in viscoelastic fluids at low Reynolds numbers, where inertial effects are negligible
- It is characterized by chaotic, time-dependent flow patterns with increased mixing and energy dissipation
- Elastic turbulence arises from the nonlinear interaction between elastic stresses and the flow field, leading to the generation of elastic waves and secondary flows
- This phenomenon has potential applications in enhancing mixing, heat transfer, and mass transfer in microfluidic devices and polymer processing
Weissenberg effect
- The Weissenberg effect, also known as rod climbing, is a viscoelastic flow phenomenon where a fluid climbs up a rotating rod or shaft
- This behavior is caused by the development of normal stress differences in the fluid, which generate a radial force that pushes the fluid inwards and upwards along the rod
- The Weissenberg effect is a clear demonstration of the elastic properties of viscoelastic fluids and is often used as a qualitative test for viscoelasticity
- Understanding and controlling the Weissenberg effect is important in applications such as mixing, coating, and polymer processing
Die swell
- Die swell, or extrudate swell, refers to the increase in the cross-sectional area of a viscoelastic fluid as it exits a capillary or die
- This phenomenon is a result of the fluid's elastic recovery after being subjected to extensional and shear deformations within the die
- The extent of die swell depends on the viscoelastic properties of the fluid, the flow conditions, and the geometry of the die
- Predicting and controlling die swell is crucial in polymer processing operations, such as extrusion and injection molding, to ensure consistent product dimensions and quality
Sharkskin effect
- The is a surface instability that occurs in the extrusion of viscoelastic fluids, particularly polymer melts, at high shear rates
- It is characterized by a rough, matte surface with a series of ridges or grooves perpendicular to the flow direction, resembling sharkskin
- The sharkskin effect is believed to arise from the stick-slip motion of the fluid at the die wall, caused by the interplay between adhesive forces and elastic stresses
- This instability can affect the surface quality and mechanical properties of extruded products, and understanding its origins is important for process optimization and control
Numerical methods for viscoelastic flows
- Numerical simulation of viscoelastic flows is essential for understanding, predicting, and optimizing the behavior of these complex fluids in various applications
- However, the presence of elastic stresses and the coupling between the flow field and the fluid's microstructure pose significant challenges for numerical methods
Challenges in simulating viscoelastic flows
- The constitutive equations for viscoelastic fluids are often highly nonlinear and involve time derivatives, making them difficult to solve numerically
- The presence of elastic stresses can lead to numerical instabilities, such as the high Weissenberg number problem, where the solution becomes unstable or inaccurate at high Weissenberg numbers
- Viscoelastic flows often involve multiple time and length scales, requiring efficient and accurate multiscale modeling techniques
- The coupling between the flow field and the fluid's microstructure necessitates the development of specialized numerical methods that can handle this interaction
Finite element methods
- Finite element methods (FEM) are widely used for simulating viscoelastic flows due to their ability to handle complex geometries and their flexibility in terms of mesh refinement
- FEM discretizes the flow domain into a set of elements and approximates the solution variables (velocity, pressure, stress) using basis functions within each element
- Various formulations, such as the Galerkin and the streamline-upwind Petrov-Galerkin (SUPG) methods, have been developed to stabilize the numerical solution and capture sharp gradients in viscoelastic flows
- FEM has been successfully applied to simulate viscoelastic flows in a range of applications, including polymer processing, microfluidics, and biomedical engineering
Finite volume methods
- Finite volume methods (FVM) are another popular approach for simulating viscoelastic flows, particularly in the context of computational fluid dynamics (CFD)
- FVM discretizes the flow domain into a set of control volumes and solves the governing equations by enforcing conservation laws over each control volume
- The collocated grid arrangement, where all variables are stored at the same grid points, is commonly used in FVM for viscoelastic flows to avoid numerical instabilities
- FVM has been employed to simulate viscoelastic flows in various applications, such as polymer extrusion, mixing, and flow through porous media
Stabilization techniques
- Stabilization techniques are essential for overcoming numerical instabilities in the simulation of viscoelastic flows, particularly at high Weissenberg numbers
- The log-conformation representation is a popular stabilization approach that reformulates the constitutive equation in terms of the logarithm of the conformation tensor, which represents the microstructural state of the fluid
- This representation ensures the positive definiteness of the conformation tensor and improves the numerical stability at high Weissenberg numbers
- The log-conformation representation has been successfully combined with both FEM and FVM for simulating viscoelastic flows
- Other stabilization techniques include the discrete elastic viscous stress splitting (DEVSS) method, the matrix logarithm formulation, and the square root conformation representation
- These techniques aim to improve the robustness and accuracy of numerical methods for viscoelastic flows by addressing the challenges associated with the nonlinear and time-dependent nature of the governing equations
Experimental techniques for viscoelastic flows
- Experimental characterization of viscoelastic fluids is crucial for understanding their rheological properties, validating constitutive models, and providing insights into their flow behavior
- Various experimental techniques have been developed to measure the viscoelastic properties of fluids and visualize their flow patterns
Rheometry for measuring viscoelastic properties
- is the study of the flow and deformation of matter, and it is the primary experimental technique for characterizing the viscoelastic properties of fluids
- Shear rheometers, such as rotational and capillary rheometers, are used to measure the shear viscosity, normal stress differences, and dynamic viscoelastic properties of fluids
- Rotational rheometers apply a shear deformation to the fluid using a rotating geometry (e.g., cone-and-plate, parallel plates) and measure the resulting torque and normal forces
- Capillary rheometers force the fluid through a narrow capillary and measure the pressure drop and flow rate to determine the shear viscosity
- Extensional rheometers, such as the filament stretching rheometer and the capillary breakup extensional rheometer (CaBER), are used to measure the extensional viscosity of viscoelastic fluids
- These rheometers apply an extensional deformation to the fluid and measure the resulting stress or the evolution of the fluid filament during stretching or breakup
- Rheological measurements provide essential data for characterizing the viscoelastic behavior of fluids, developing constitutive models, and optimizing processing conditions
Flow visualization techniques
- Flow visualization techniques are used to observe and analyze the flow patterns, instabilities, and phenomena in viscoelastic fluids
- Particle tracking velocimetry (PTV) and particle image velocimetry (PIV) are optical techniques that use tracer particles to measure the velocity field in a fluid
- PTV tracks the motion of individual particles, while PIV measures the average velocity of particles within small interrogation windows
- These techniques provide quantitative information about the flow field and can reveal complex flow structures, such as recirculation zones and elastic turbulence
- Laser Doppler velocimetry (LDV) is another optical technique that measures the velocity of a fluid at a point by analyzing the Doppler shift of laser light scattered by tracer particles
- LDV offers high spatial and temporal resolution and is suitable for measuring velocity profiles in viscoelastic flows
- Flow-induced birefringence (FIB) is a technique that exploits the optical anisotropy of viscoelastic fluids to visualize the stress distribution and orientation of the microstructure
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Key Terms to Review (27)
Biological fluids: Biological fluids are liquids found within living organisms that serve crucial roles in physiological processes. These fluids include blood, lymph, saliva, and interstitial fluid, each playing a vital part in transporting nutrients, waste, and signaling molecules throughout the body. Understanding their properties is essential for comprehending how viscoelastic flows occur in various biological systems.
Biomechanics: Biomechanics is the study of the mechanical principles that govern movement and structure in living organisms. It combines the concepts of biology and mechanics to analyze how forces interact within biological systems, helping to understand movement, stability, and the functionality of various biological structures.
Creep behavior: Creep behavior refers to the tendency of materials, particularly viscoelastic materials, to deform gradually and permanently under constant stress over time. This phenomenon is crucial for understanding how materials respond to long-term loading, which can impact their mechanical properties and performance in various applications.
Deborah Number: The Deborah number is a dimensionless quantity that characterizes the time scale of relaxation of a viscoelastic material compared to the time scale of the flow or deformation it experiences. It helps determine whether a material behaves more like a viscous fluid or an elastic solid under given conditions, making it essential for understanding viscoelastic flows.
Die swell: Die swell refers to the phenomenon in which a polymer melt expands after passing through a die during extrusion, resulting in a larger cross-section than intended. This occurs because the flow of the material is influenced by its viscoelastic properties, which can cause it to behave differently compared to Newtonian fluids. Understanding die swell is crucial for accurately predicting the final shape and dimensions of extruded products.
Dynamic mechanical analysis: Dynamic mechanical analysis (DMA) is a technique used to study the viscoelastic properties of materials by applying oscillatory stress or strain and measuring the resulting response. This method provides insights into the material's stiffness, damping, and overall behavior under various temperature and frequency conditions, making it essential for understanding how materials perform in real-world applications.
Elastic turbulence: Elastic turbulence is a flow regime that occurs in viscoelastic fluids where elastic forces dominate over inertial forces, leading to chaotic behavior similar to that seen in traditional turbulence. This phenomenon typically arises in flows with high polymer concentrations, where the fluid's elastic properties influence its stability and flow patterns, causing the fluid to exhibit complex and unpredictable motion.
Elasticity number: The elasticity number is a dimensionless parameter that quantifies the relative importance of elastic effects compared to viscous effects in a fluid flow. It helps to determine how much a material deforms under stress and how it returns to its original shape after the stress is removed, which is particularly relevant in viscoelastic flows where both viscosity and elasticity play significant roles.
Jetting: Jetting refers to the phenomenon where a fluid is ejected at high velocity from an orifice, creating a jet-like flow that can interact with surrounding fluids. This process is significant in various applications, particularly in viscoelastic flows where the behavior of the fluid can be influenced by its elastic properties and viscosity, leading to complex flow patterns and effects such as stretching and retraction.
Kelvin-Voigt Model: The Kelvin-Voigt model is a mathematical representation of viscoelastic materials that combines both elastic and viscous behavior in response to deformation. It describes how materials exhibit both instantaneous elastic deformation and time-dependent viscous flow, making it particularly relevant for understanding the behavior of materials under stress over time.
Linear viscoelasticity: Linear viscoelasticity refers to the behavior of materials that exhibit both viscous and elastic characteristics when subjected to deformation. This means that these materials can both flow like a liquid and return to their original shape like a solid when the stress is removed, but only under small deformations where the relationship between stress and strain is linear. This concept is crucial in understanding how complex fluids behave under different flow conditions, particularly in viscoelastic flows.
Loss modulus: Loss modulus is a measure of the energy dissipated as heat during the deformation of a viscoelastic material when subjected to an oscillatory stress. It quantifies the viscous behavior of the material, distinguishing it from the elastic behavior measured by the storage modulus. Understanding loss modulus is crucial for analyzing the time-dependent deformation of materials in viscoelastic flows, where both elastic and viscous properties play a significant role.
Maxwell Model: The Maxwell Model is a mathematical representation used to describe the viscoelastic behavior of materials, combining both viscous and elastic properties. It captures how materials deform under stress and recover once the stress is removed, providing insight into how fluids behave when they exhibit both flow and elasticity, which is essential in understanding viscoelastic flows.
Molding processes: Molding processes are techniques used to shape materials, typically in liquid or semi-liquid form, into specific shapes using molds. These processes are essential in manufacturing, allowing for the efficient production of complex geometries and designs in various materials, including metals, plastics, and ceramics. Understanding these processes is crucial for manipulating viscoelastic materials, which can exhibit both liquid-like and solid-like behavior during deformation.
Nonlinear viscoelasticity: Nonlinear viscoelasticity refers to the behavior of materials that exhibit both viscous and elastic properties in a nonlinear manner when subjected to stress or strain. This means that the relationship between stress and strain is not proportional and can change depending on the amount of deformation and the rate at which it occurs. Understanding nonlinear viscoelasticity is crucial for analyzing complex flows of materials like polymers and biological fluids, where traditional linear models fail to accurately predict behavior under large deformations or varying rates of applied stress.
Polymer solutions: Polymer solutions are mixtures where long-chain molecules, known as polymers, are dissolved in a solvent, creating a homogeneous phase. These solutions exhibit unique properties due to the large size and flexibility of the polymer chains, significantly affecting their behavior in fluid flow and interactions with other materials.
Rheometry: Rheometry is the study of the flow and deformation of materials, particularly complex fluids like polymers and biological fluids. This field focuses on measuring the rheological properties of these materials to understand how they behave under different conditions, including stress and strain. It plays a crucial role in characterizing viscoelastic flows, where materials exhibit both viscous and elastic characteristics.
Sharkskin effect: The sharkskin effect refers to the unique surface texture observed in shark skin that reduces drag and enhances fluid flow. This phenomenon is significant in fluid dynamics, particularly when studying viscoelastic flows, as it demonstrates how surface textures can manipulate flow characteristics to minimize resistance and improve efficiency in various applications.
Shear-thickening: Shear-thickening refers to a behavior exhibited by some non-Newtonian fluids where their viscosity increases with an increase in shear rate. In simple terms, as you stir or apply force to these fluids, they become thicker and more resistant to flow, making them behave more like a solid. This phenomenon is crucial in understanding how certain materials, like cornstarch mixed with water or certain polymer solutions, respond under different conditions.
Shear-thinning: Shear-thinning is a property of certain fluids where their viscosity decreases with increasing shear rate. This behavior is commonly observed in materials like paints and biological fluids, making them easier to process and apply. In viscoelastic flows, shear-thinning affects the way materials respond to deformation, influencing both their flow behavior and stress response under different shear conditions.
Storage modulus: The storage modulus is a fundamental property of viscoelastic materials that measures their ability to store elastic energy when subjected to deformation. It reflects how much a material behaves like a solid under stress, indicating its stiffness and ability to recover its shape after deformation. In the context of viscoelastic flows, understanding the storage modulus is crucial as it influences the material's flow behavior, response to applied forces, and overall mechanical performance during processing and application.
Stress relaxation: Stress relaxation is the gradual decrease in stress in a material under constant strain over time. This phenomenon is particularly significant in viscoelastic materials, where the time-dependent behavior allows the material to adjust and redistribute internal stresses, leading to a reduction in the applied stress even as the strain remains constant.
Time-dependent viscosity: Time-dependent viscosity refers to the property of certain fluids where their viscosity changes over time under constant conditions of stress or strain. This behavior is particularly important in understanding how non-Newtonian fluids behave during processes such as flow and deformation, as their resistance to flow can vary significantly depending on the duration and nature of the applied force. In the context of viscoelastic materials, this concept highlights the interplay between viscosity and elasticity, where time plays a crucial role in how these materials respond to external stimuli.
Turbulence in viscoelastic flows: Turbulence in viscoelastic flows refers to the chaotic and irregular fluid motion that can occur in fluids that exhibit both viscous and elastic properties. This phenomenon is particularly significant because viscoelastic materials, like polymers, can behave differently under various flow conditions, leading to complex flow patterns that are not seen in purely viscous or elastic fluids. Understanding this turbulence is crucial for predicting flow behavior in applications involving polymer processing, biological fluids, and industrial mixing.
Weissenberg Effect: The Weissenberg Effect refers to the phenomenon where a viscoelastic fluid experiences a dramatic increase in its viscosity and exhibits elastic behavior when subjected to extensional flow. This behavior is particularly observed in non-Newtonian fluids, where the fluid's microstructure becomes aligned with the flow direction, leading to unique flow characteristics. The Weissenberg Effect highlights the complex interplay between viscosity and elasticity in viscoelastic flows, showcasing how these fluids respond differently compared to Newtonian fluids under similar conditions.
Weissenberg Number: The Weissenberg number is a dimensionless quantity that characterizes the relative importance of elastic and viscous effects in a fluid flow, particularly in viscoelastic materials. It helps in understanding how the material deforms under flow conditions, highlighting the balance between time-dependent strain and flow rate. A low Weissenberg number indicates a flow dominated by viscous forces, while a high number suggests that elastic forces play a more significant role.
Yield Stress: Yield stress is the minimum stress required to initiate flow in a material, marking the transition from a solid-like behavior to a fluid-like behavior in non-Newtonian fluids. This concept is crucial as it helps to understand how materials behave under different stress conditions, particularly in complex fluids that do not have a constant viscosity. Yield stress indicates that a certain amount of force must be applied before the material can deform and flow, which is especially significant in applications involving food, polymers, and slurries.