In continuum mechanics, a
Contents 1 Definition 1.1
2 Examples 3 See also 4 References Definition[edit] An element of a flowing liquid or gas will suffer forces from the surrounding fluid, including viscous stress forces that cause it to gradually deform over time. These forces can be mathematically approximated to first order by a viscous stress tensor, which is usually denoted by τ displaystyle tau . The deformation of that fluid element, relative to some previous state, can be approximated to first order by a strain tensor that changes with time. The time derivative of that tensor is the strain rate tensor, that expresses how the element's deformation is changing with time; and is also the gradient of the velocity vector field v displaystyle v at that point, often denoted ∇ v displaystyle nabla v . The tensors τ displaystyle tau and ∇ v displaystyle nabla v can be expressed by 3×3 matrices, relative to any chosen coordinate system. The fluid is said to be Newtonian if these matrices are related by the equation τ = μ ( ∇ v ) displaystyle mathbf tau =mathbf mu (nabla v) where μ displaystyle mu is a fixed 3×3×3×3 fourth order tensor, that does not depend on
the velocity or stress state of the fluid.
τ = μ d u d y displaystyle tau =mu frac du dy where τ displaystyle tau is the shear stress ("drag") in the fluid, μ displaystyle mu is a scalar constant of proportionality, the shear viscosity of the fluid d u d y displaystyle frac du dy is the derivative of the velocity component that is parallel to the direction of shear, relative to displacement in the perpendicular direction. If the fluid is incompressible and viscosity is constant across the fluid, this equation can be written in terms of an arbitrary coordinate system as τ i j = μ ( ∂ v i ∂ x j + ∂ v j ∂ x i ) displaystyle tau _ ij =mu left( frac partial v_ i partial x_ j + frac partial v_ j partial x_ i right) where x j displaystyle x_ j is the j displaystyle j th spatial coordinate v i displaystyle v_ i is the fluid's velocity in the direction of axis i displaystyle i τ i j displaystyle tau _ ij is the j displaystyle j th component of the stress acting on the faces of the fluid element perpendicular to axis i displaystyle i . One also defines a total stress tensor σ displaystyle mathbf sigma ) that combines the shear stress with conventional (thermodynamic) pressure p displaystyle p . The stress-shear equation then becomes σ i j = − p δ i j + μ ( ∂ v i ∂ x j + ∂ v j ∂ x i ) displaystyle mathbf sigma _ ij =-pdelta _ ij +mu left( frac partial v_ i partial x_ j + frac partial v_ j partial x_ i right) For anisotropic fluids[edit] More generally, in a non-isotropic Newtonian fluid, the coefficient μ displaystyle mu that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nine-element viscosity tensor μ i j displaystyle mu _ ij . There is general formula for friction force in a liquid: The vector differential of friction force is equal the viscosity tensor increased on vector product differential of the area vector of adjoining a liquid layers and rotor of velocity: d F = μ i j d S × r o t u displaystyle d mathbf F = mu _ ij ,mathbf dS times mathrm rot ,mathbf u where μ i j displaystyle mu _ ij – viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity.[5] Examples[edit] Water, air, alcohol, glycerol, and thin motor oil are all examples of Newtonian fluids over the range of shear stresses and shear rates encountered in everyday life. Single-phase fluids made up of small molecules are generally (although not exclusively) Newtonian. See also[edit] Continuum mechanics Laws Conservations Energy Mass Momentum Inequalities Clausius–Duhem (entropy) Solid mechanics Stress Deformation Compatibility Finite strain Infinitesimal strain Elasticity (linear) Plasticity Bending Hooke's law Material failure theory Fracture mechanics Contact mechanics (frictional)
Fluids Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure Liquids Surface tension Capillary action Gases Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law Plasma Rheology Viscoelasticity Rheometry Rheometer Smart fluids Magnetorheological Electrorheological Ferrofluids Scientists Bernoulli Boyle Cauchy Charles Euler Gay-Lussac Hooke Pascal Newton Navier Stokes v t e Non-newtonian fluid References[edit] ^ Panton, Ronald L. (2013).
v t e Branches of physics Divisions Applied Experimental Theoretical Energy Motion Thermodynamics Mechanics Classical Ballistics Lagrangian Hamiltonian Continuum Celestial Statistical Solid Fluid Quantum Waves Fields Gravitation Electromagnetism Optics Geometrical Physical Nonlinear Quantum Quantum field theory Relativity Special General By speciality Accelerator Acoustics Astrophysics Nuclear Stellar Heliophysics Solar Space Astroparticle Atomic–molecular–optical (AMO) Communication Computational Condensed matter Mesoscopic Solid-state Soft Digital Engineering Material Mathematical Molecular Nuclear Particle Phenomenology Plasma Polymer Statistical
Biophysics Virophysics Biomechanics Medical physics Cardiophysics Health physics Laser medicine Medical imaging Nuclear medicine Neurophysics Psychophysics
Agrophysics Soil Atmospheric Cloud Chemical Econophysics Geophysics Physical chemistry v t e Isaac Newton Publications
Other writings Notes on the Jewish Temple
Quaestiones quaedam philosophicae
Discoveries and inventions Calculus Fluxion Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant Theory expansions Kepler's laws of planetary motion Problem of Apollonius Newtonianism Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics
Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Newton's quotient Newton–Puiseux theorem Solar mass Dynamics Absolute space and time Finite difference Table of Newtonian series Impact depth Structural coloration Inertia Spectrum Phrases "Hypotheses non fingo" "Standing on the shoulders of giants" Life Cranbury Park Woolsthorpe Manor Early life Later life Religious views Occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution Friends and family Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill Cultural depictions Newton (Blake) Newton (Paolozzi) In popular culture Related Writing of Principia Mathematica List of things named after Newton
Elements of the Philosophy of Newton Isaac Ne |