AOE 5104 Class 7 Online presentations for next class: Homework 3 Kinematics 1 Homework 3 Class next Tuesday will be given by Dr. Aurelien Borgoltz daVinci (Aaron Marcus, Justin Ratcliff)

Claude-Louis Navier (February 10, 1785 in Dijon - August 21, 1836 in Paris) Biography After the death of his father in 1793, Navier's mother left his education in the hands of his uncle Emiland Gauthey, an engineer with the Corps of Bridges and Roads (Corps des Ponts et Chaussées). In 1802, Navier enrolled at the École polytechnique, and in 1804 continued his studies at the École Nationale des Ponts et Chaussées, from which he graduated in 1806. He eventually succeeded his uncle as Inspecteur general at the Corps des Ponts et Chaussées. He directed the construction of bridges at Choisy, Asnières and Argenteuil in the Department of the Seine, and built a footbridge to the Île de la Cité in Paris. In 1824, Navier was admitted into the French Academy of Science. In 1830, he took up a professorship at the École Nationale des Ponts et Chaussées, and in the following year succeeded exiled Augustin Louis Cauchy as professor of calculus and mechanics at the École polytechnique. [edit] Contributions Navier formulated the general theory of elasticity in a mathematically usable form (1821), making it available to the field of construction with sufficient accuracy for the first time. In 1819 he succeeded in determining the zero line of mechanical stress, finally correcting Galileo Galilei's incorrect results, and in 1826 he established the elastic modulus as a property of materials independent of the second moment of area. Navier is therefore often considered to be the founder of modern structural analysis. His major contribution however remains the Navier-Stokes equations (1822), central to fluid mechanics.

Pump flow VisEng Ltd http://www.viseng.com/consult/flowvis.html

Equations for Changes Seen From a Lagrangian Perspective Differential Form (for a particle) Integral Form (for a system)

Pump flow O VisEng Ltd http://www.viseng.com/consult/flowvis.html

Conversion from Lagrangian to Eulerian rate of change - Derivative The Substantial Derivative y Time Derivative Convective Derivative x z (x(t),y(t),z(t),t)

Conversion from Lagrangian to Eulerian rate of change - Integral The Reynolds Transport Theorem y Apply Divergence Theorem x z Volume R Surface S

Equations for Changes Seen From a Lagrangian Perspective Differential Form (for a particle) Integral Form (for a system)

Equations for Changes Seen From an Eulerian Perspective Differential Form (for a fixed point in space) Integral Form (for a fixed control volume)

Equivalence of Integral and Differential Forms Cons. of mass (Integral form) Divergence Theorem Conservation of mass for any volume R Then we get or Cons. of mass (Differential form)

Extremely hot (700°C) clouds of gas and ash particles occurring during some explosive volcanic eruptions. They can travel at speeds comparable to a hurricane (100-150 mph). Also known as Nuees Ardentes.

Constitutive Relations - Closing the Equations of Motion Could we solve, in principle, the equations we have derived for a particular flow?

Equations for Changes Seen From an Eulerian Perspective Differential Form (for a fixed point in space) Integral Form (for a fixed control volume)

Constitutive Relations Equations of motion 5 eqns: Mass (1), Momentum (3), Energy (1) 13 unknowns: p, , u, v, w, T, 6 , e Need 8 more equations! Information about the fluid is needed Constitutive relations Thermodynamics: p, , T, e Viscous stress relations p = p(,T) and e = e(,T) Newtonian fluid

Newtonian (Isotropic) Fluid Viscous Stress is Linearly Proportional to Strain Rate Relationship is isotropic (the same in all directions) Stress, is a tensor… …and so has some basic properties when we rotate the coordinate system used to represent it like… Principal axes… axis directions for which all off shear stresses are zero Tensor invariants… combinations of elements that don’t change with the axis directions (Symmetric so yx= xy, yz= zx, xz= zx) But what is strain rate (or rate of deformation?)

Distortion of a Particle in a Flow Physically M>1, accelerating, expanding flow Total change “Cauchy Stokes Decomposition” = rotation Rate of deformation or strain rate + dilation + shear deformation

Distortion of a Particle in a Flow Mathematically Deformation is represented by dV×time so rate of deformation is given by dV V+dV V Total change = rotation + dilation + shear deformation Rate of deformation or strain rate

Newtonian (Isotropic) Fluid So Each stress = Const.× Corresponding strain + Const. × First invariant of comp. rate component strain rate tensor And likewise for y and z. Or

Stokes’ Hypothesis Stokes hypothesized that the total normal viscous stress xx+yy+zz should be zero, so that they can’t behave like an extra pressure (i.e. he wanted to simplify things so that the total pressure felt anywhere in the flow would be the same as the pressure used in the thermodynamic relations). This implies =-⅔µ and remains controversial With this, and in general (non-principal) axes, we finally have and likewise for yy and zz and likewise for yz and xz

The Equations of Motion Differential Form (for a fixed volume element) The Continuity equation These form a closed set when two thermodynamic relations are specified The Navier Stokes’ equations The Viscous Flow Energy Equation

Leonhard Euler 1707-1783

Assumptions made / Info encoded Assumption/Law Mass NS VFEE Conservation of mass Conservation of momentum Conservation of energy Continuum Newtonian fluid Isotropic viscosity Stokes´ Hypothesis Fourier´s Law of Heat conduction No heat addition except by conduction                

Summary Conservations laws Lagrangian and Eulerian perspectives Equations of motion dervied from a Lagrangian perspective The Substantial Derivative and the Reynolds Transport Theorem connect Lagrangian with Eulerian Constitutive Relations provide information about the fluid material Assumptions