An introduction to non-newtonian mechanics.pptx

An introduction to non newtonian mechanics

1. Continuum Continuum A mathematical concept A material body consists of a set of infinitesimal points with an associated measure for mass & volume Continuously varying fields can be defined/assumed to exist for all points in the body Continuum Mechanics Originated ~1827 with A. L. Cauchy Extended principles of Newtonian mechanics (for discrete bodies) to the continuum Applicability Knudsen number ( Kn ) = Mean free path/Length scale Kn ≤ 0.01, continuum approximation safely holds Kn ≥ 1, continuum approximation is not valid: need to use statistical mechanics

2. Concept of Continuum - Illustrated Continuum is a mathematical concept that allows properties of a body to be defined at individual points Conceptualized using density variation of increasingly small volumes in a given domain ρ ( x,y,z ) = lim ( Δ V→ δ V) ( Δ m/ Δ V) Valid when length scale of domain >> mean free path of molecules Knudsen number << 1 Kn ≤ 0.01 Source: http://pillars.che.pitt.edu/student/... Accessed: 17/10/2016

3. Properties defined in a Continuum Concept of Continuum leads to the definition of field variables Field Variable has a specific value at a given point and instant in time (Deterministic versus Probabilistic): e.g. ρ ( x,y,z,t ) for density Consider as Fluid Property any that can be described as field variable Density Velocity Stress Viscosity Surface Tension Temperature Pressure

4. Balance equation (Mass) Motion: x = χ( X ,t )   Balance of Mass (Cauchy)   X x κ 1 Κ 2 (t)

5. Balance equation (Momentum) Closed form Net force: Momentum change Equating Angular momentum →   Stress Components based Body force: Momentum change: Equating (Force= MassxAccn ) we get   σ xx +(∂ σ xx /∂x).(∆x/2) σ xx - ∆ σ xx σ yx + (∂ σ yx /∂y).(∆y/2) σ yx - (∂ σ yx /∂y).(∆y/2)

6. Balance equation (Energy) Work done by external forces Use t = Tn → Equates to (Use Work done = = + Stress-Power Final equation  

7. ( Im )balance equation (Entropy) In addition, the Second law of thermodynamics is required to hold for processes where the model is used Using divergence identity & substituting from energy balance Therefore, where ξ is dissipation or rate of entropy production  

8. Balance equations (Final) Description: Lagrangian (D/Dt) vs Eulerian ( ∂/∂x) Balance of Mass (Cauchy) = 0 Balance of Linear Momentum (Cauchy) Balance of Angular Momentum (Cauchy) Balance of Energy (First Law of Thermodynamics) Entropy Inequality (Second Law of Thermodynamics)  

9. Axioms of Constitutive Modeling See: Ch. 3, Truesdell & Rajagopal (2000) Determinism Stress at a point determined by history of motion of the body Local Action Particles far away do not affect stress at a given point Frame-Indifference (also Oldroyd’s criteria for invariance) Material properties do not change b/w inertial frames Note: Change of frame= x * = c (t) + Q (t)( x - x ); t* = t - a Internal Constraints Incompressibility Inextensibility Rigidity Information on the microstructure is used to motivate the form for the stress experienced by the material

10. Non-Newtonian Fluid Models Generalized Newtonian models Rivlin-Ericksen tensors (Example of a frame-invariant derivative) Differential models (Only Creep) Rate-type models (Both Creep & Stress-Relaxation) Can write in explicit form involving stretch from “natural configuration” (see Rajagopal 2003) Integral models  

11. Summary Continuum hypothesis allows extending Newtonian mechanics to a point (Mean free path/Domain length) ≡ Kn ≤ 0.01 Balance equations Mass Momentum (Cauchy’s 1 st Law = Newton’s 2 nd Law in a Continuum) Energy Entropy Inequality Constitutive Modeling Use microstructural information to motivate material model Non-Newtonian fluid models Time Independent: Generalized Newtonian Time Dependent: Differential, Rate-type, Integral