3.4 Constitutive Equations and Newtonian Fluids

Constitutive equations are crucial in fluid dynamics, linking stress and deformation in fluids. They close the system of governing equations by characterizing a fluid's mechanical behavior, reflecting properties like viscosity and elasticity.

Newtonian fluids have a linear relationship between stress and strain rate, with constant viscosity. Non-Newtonian fluids show varying viscosity based on stress or strain rate. Understanding these differences is key for modeling complex fluid behaviors in engineering applications.

Constitutive Equations in Fluid Dynamics

Fundamental Concepts and Importance

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• Constitutive equations in fluid dynamics describe the relationship between stress and deformation in a fluid linking kinematic and dynamic aspects of fluid motion
• These equations characterize the mechanical behavior of fluids closing the system of governing equations in fluid dynamics
• Material-specific equations reflect unique properties of different fluids (viscosity, elasticity, plasticity)
• General form relates the stress tensor to the strain rate tensor and other relevant fluid properties
• Derived based on experimental observations and theoretical considerations often involving simplifying assumptions about fluid behavior
• Crucial for accurately modeling fluid behavior in various engineering and scientific applications (aerodynamics, hydraulics, oceanography)

Derivation and Applications

• Developed through rigorous mathematical analysis combined with empirical data
• Incorporate principles of continuum mechanics and thermodynamics
• Account for various fluid properties (compressibility, thermal conductivity, molecular structure)
• Applied in diverse fields (weather prediction, blood flow modeling, oil and gas extraction)
• Used in computational fluid dynamics (CFD) simulations to predict complex flow phenomena
• Help optimize design of fluid systems (engines, pipelines, chemical reactors)

Newtonian vs Non-Newtonian Fluids

Characteristics and Examples

• Newtonian fluids exhibit linear relationship between shear stress and strain rate with constant viscosity regardless of applied stress or strain rate
• Non-Newtonian fluids demonstrate non-linear relationship between shear stress and strain rate with varying viscosity based on applied stress or strain rate
• Newtonian fluid examples include water, air, and many common liquids (glycerin, mineral oil)
• Non-Newtonian fluid examples include blood, polymer solutions, and certain suspensions (ketchup, toothpaste)
• Non-Newtonian fluids classified into categories (shear-thinning, shear-thickening, viscoelastic, yield stress fluids)
• Shear-thinning fluids decrease in viscosity with increased shear rate (paint, whipped cream)
• Shear-thickening fluids increase in viscosity with increased shear rate (cornstarch in water)
• Viscoelastic fluids exhibit both viscous and elastic properties (silly putty)
• Yield stress fluids require minimum applied stress to initiate flow (mayonnaise)

Modeling and Applications

• Distinction between Newtonian and non-Newtonian fluids crucial in selecting appropriate constitutive equations
• Non-Newtonian fluids often require complex constitutive equations incorporating additional parameters and non-linear terms
• Power-law model used for many shear-thinning and shear-thickening fluids
• Bingham plastic model applied to yield stress fluids
• Maxwell model describes viscoelastic behavior
• Understanding fluid type impacts design of processing equipment (pumps, mixers, extruders)
• Non-Newtonian fluid behavior exploited in various applications (shock absorbers, bulletproof vests)

Constitutive Equation for Newtonian Fluids

Derivation and Components

• Constitutive equation for Newtonian fluids derived based on assumption of linear relationship between stress and strain rate
• Stress tensor for Newtonian fluid expressed as sum of pressure and viscous stress components
• Viscous stress tensor proportional to strain rate tensor with constant of proportionality being dynamic viscosity
• General form: $\tau_{ij} = 2\mu e_{ij}$, where $\tau_{ij}$ viscous stress tensor, $\mu$ dynamic viscosity, $e_{ij}$ strain rate tensor
• Derivation applies principle of material frame indifference and considers isotropy of fluid
• Resulting equation key component in Navier-Stokes equations governing motion of viscous fluids

Mathematical Formulation and Extensions

• Full tensor form of constitutive equation: $\sigma_{ij} = -p\delta_{ij} + \mu(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}) + \lambda(\nabla \cdot \mathbf{u})\delta_{ij}$
• $\sigma_{ij}$ total stress tensor, $p$ pressure, $\delta_{ij}$ Kronecker delta, $\lambda$ second coefficient of viscosity
• Incorporates bulk viscosity term for compressible fluids
• Simplified for incompressible fluids: $\nabla \cdot \mathbf{u} = 0$
• Extended to include temperature dependence of viscosity: $\mu = \mu(T)$
• Forms basis for more complex models (turbulence models, multiphase flow equations)

Viscous Stresses and Strain Rates

Problem-Solving Applications

• Constitutive equation for Newtonian fluids calculates viscous stresses given known strain rates or vice versa
• Simple shear flows simplify equation to $\tau = \mu(\frac{du}{dy})$, $\tau$ shear stress, $\frac{du}{dy}$ velocity gradient
• Analyzes fluid behavior in various geometries (flow between parallel plates, pipe flow, flow around objects)
• Combines with conservation laws (mass, momentum, energy) to solve complex fluid dynamics problems
• Determines velocity profiles, pressure drops, and drag forces in fluid systems
• Couette flow between parallel plates: $u(y) = U\frac{y}{h}$, $U$ plate velocity, $h$ gap width
• Poiseuille flow in circular pipe: $u(r) = \frac{\Delta P}{4\mu L}(R^2 - r^2)$, $\Delta P$ pressure difference, $L$ pipe length, $R$ pipe radius

Advanced Techniques and Numerical Methods

• Dimensional analysis and non-dimensionalization techniques identify important parameters and simplify problem-solving
• Reynolds number $Re = \frac{\rho UL}{\mu}$ relates inertial forces to viscous forces
• Numerical methods incorporate constitutive equation to simulate fluid behavior in complex geometries and flow conditions
• Finite difference method discretizes domain and approximates derivatives
• Finite element method divides domain into elements and solves weak form of equations
• Computational fluid dynamics (CFD) software packages utilize constitutive equations for advanced simulations
• Lattice Boltzmann methods model fluid as particles on a lattice incorporating constitutive behavior