Fluid Mechanics Unit 14 study guides

Key Concepts and Definitions

• Non-Newtonian fluids exhibit viscosity that varies with shear rate or shear stress, unlike Newtonian fluids which have constant viscosity
• Shear thinning (pseudoplastic) fluids decrease in viscosity as shear rate increases (paint, ketchup)
• Shear thickening (dilatant) fluids increase in viscosity as shear rate increases (cornstarch in water, silly putty)
  • Occurs due to increased particle interactions and jamming at high shear rates
• Yield stress fluids require a minimum stress to initiate flow (toothpaste, mayonnaise)
• Thixotropic fluids exhibit time-dependent decrease in viscosity under constant shear stress (yogurt, some gels)
• Rheopectic fluids exhibit time-dependent increase in viscosity under constant shear stress (gypsum pastes, some lubricants)
• Viscoelastic fluids exhibit both viscous and elastic properties, storing and dissipating energy during deformation (polymer solutions, melts)

Types of Non-Newtonian Fluids

• Pseudoplastic (shear thinning) fluids include polymer solutions, blood, paint, and shampoo
  • Viscosity decreases with increasing shear rate due to alignment of polymer chains or particles
• Dilatant (shear thickening) fluids include cornstarch in water, some dense suspensions, and certain colloids
• Bingham plastics are yield stress fluids that behave as Newtonian fluids above the yield stress (toothpaste, certain slurries)
• Herschel-Bulkley fluids are yield stress fluids that exhibit shear thinning or shear thickening behavior above the yield stress (drilling muds, food products)
• Thixotropic fluids include yogurt, some paints, and certain gels
  • Viscosity decreases over time at constant shear stress due to breakdown of internal structure
• Rheopectic fluids are less common and include some lubricants and gypsum pastes
• Viscoelastic fluids include polymer melts, dough, and some suspensions

Rheological Models

• Power-law (Ostwald-de Waele) model describes shear thinning and shear thickening behavior: $\tau = K \dot{\gamma}^n$
  • $\tau$ is shear stress, $K$ is consistency index, $\dot{\gamma}$ is shear rate, and $n$ is flow behavior index
  • $n < 1$ for shear thinning, $n > 1$ for shear thickening, and $n = 1$ for Newtonian fluids
• Bingham plastic model describes yield stress fluids: $\tau = \tau_y + \mu_p \dot{\gamma}$
  • $\tau_y$ is yield stress and $\mu_p$ is plastic viscosity
• Herschel-Bulkley model combines power-law and yield stress behavior: $\tau = \tau_y + K \dot{\gamma}^n$
• Casson model is another yield stress model: $\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\mu_c \dot{\gamma}}$
  • $\mu_c$ is Casson viscosity
• Carreau model captures shear thinning behavior at low and high shear rates: $\eta = \eta_\infty + (\eta_0 - \eta_\infty)[1 + (\lambda \dot{\gamma})^2]^{(n-1)/2}$
  • $\eta_0$ and $\eta_\infty$ are zero-shear and infinite-shear viscosities, $\lambda$ is relaxation time

Measuring Non-Newtonian Behavior

• Rheometers measure viscosity, yield stress, and viscoelastic properties under controlled shear conditions
  • Rotational rheometers apply shear using parallel plates, cone-and-plate, or concentric cylinders
  • Capillary rheometers measure pressure drop and flow rate through a small tube
• Oscillatory tests measure viscoelastic properties by applying sinusoidal shear strain and measuring stress response
  • Storage modulus $G'$ represents elastic behavior, loss modulus $G''$ represents viscous behavior
• Creep tests apply constant stress and measure strain over time to characterize viscoelastic behavior
• Yield stress can be measured using stress ramps, oscillatory amplitude sweeps, or creep tests
• Thixotropy and rheopexy are characterized by measuring viscosity over time at constant shear rate or stress
• Extensional rheology measures fluid behavior under extensional deformation (stretching, squeezing)
  • Techniques include filament stretching, capillary breakup, and squeeze flow

Applications in Industry

• Food processing involves shear thinning, yield stress, and viscoelastic fluids (sauces, dough, ice cream)
  • Rheological properties affect mixing, pumping, extrusion, and texture
• Polymer processing relies on understanding viscoelastic behavior during extrusion, injection molding, and fiber spinning
• Petroleum industry deals with yield stress and shear thinning drilling muds, fracturing fluids, and crude oil
• Cosmetics and personal care products (shampoo, lotion) are formulated for desired flow and texture properties
• Paints and coatings are designed to have shear thinning behavior for easy application and leveling
• Biomedical applications include blood flow, drug delivery systems, and tissue engineering scaffolds
• Additive manufacturing (3D printing) uses non-Newtonian fluids for improved print resolution and mechanical properties

Mathematical Descriptions

• Constitutive equations relate stress and strain rate tensors for non-Newtonian fluids
  • Generalized Newtonian fluid models (power-law, Bingham, Herschel-Bulkley) use scalar viscosity functions
  • Differential and integral viscoelastic models (Upper-Convected Maxwell, Oldroyd-B) incorporate elastic effects
• Navier-Stokes equations can be modified to include non-Newtonian constitutive equations
  • Momentum equation: $\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{g}$
  • Continuity equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$
• Dimensionless numbers characterize the relative importance of non-Newtonian effects
  • Deborah number $De = \frac{\lambda}{t_c}$ compares relaxation time to characteristic process time
  • Weissenberg number $Wi = \lambda \dot{\gamma}$ compares relaxation time to reciprocal shear rate
• Numerical methods (finite difference, finite element, finite volume) are used to solve complex flow problems
  • Challenges include handling yield stress, viscoelasticity, and free surface flows

Practical Examples and Case Studies

• Ketchup is a shear thinning fluid that exhibits yield stress, allowing it to remain on food without dripping
• Silly Putty is a viscoelastic fluid that bounces like a solid at high strain rates but flows like a liquid at low rates
• Blood is a shear thinning fluid due to red blood cell aggregation and deformation
  • Non-Newtonian behavior is important in modeling cardiovascular diseases and designing medical devices
• Hydraulic fracturing fluids are designed to have shear thinning and viscoelastic properties for efficient proppant transport
• 3D printing inks are formulated to have shear thinning behavior for improved print resolution and shape retention
• Chocolate is a yield stress fluid that requires careful tempering and processing to achieve desired texture and flow properties
• Debris flows and avalanches exhibit yield stress and shear thinning behavior, complicating hazard prediction and mitigation efforts

Challenges and Future Research

• Developing constitutive models that accurately capture complex non-Newtonian behavior across a wide range of flow conditions
• Improving numerical methods for simulating non-Newtonian flows with free surfaces, moving boundaries, and multiphase interactions
• Characterizing non-Newtonian behavior of complex fluids (nanocomposites, active matter, biological fluids) using advanced rheological techniques
• Designing microfluidic devices that exploit non-Newtonian effects for particle separation, mixing, and flow control
• Optimizing non-Newtonian fluid formulations for specific applications (3D printing, enhanced oil recovery, drug delivery)
• Investigating the role of non-Newtonian behavior in biological systems (mucus transport, cell mechanics, tissue engineering)
• Developing sustainable and environmentally friendly non-Newtonian fluids for industrial applications
• Advancing rheological characterization techniques for in-situ and real-time monitoring of non-Newtonian fluid properties