Chp4_331 - FLUID DYNAMICS 4 Integral form of the basic laws The properties which define a flow are density(viscosity temperature velocity U Ux=U Uy=V

# Chp4_331 - FLUID DYNAMICS 4 Integral form of the basic laws...

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23 FLUID DYNAMICS 4. Integral form of the basic laws The properties which define a flow are: * density (viscosity) - ρ , μ * temperature - Τ * velocity - U : U x = U , U y = V , U z = W * normal stress, pressure - P ( σ x , σ y , σ z ) * shear stress - τ xy , τ xz , τ yz There are 6 unknowns: P , ρ , T , U For many flows: ρ =const., T =const. 4 unknowns Shear stresses are proportional to velocity gradients, so they don't constitute additional unknowns. Thus, 6 equations are required! The tools: mechanics (Newton's) thermodynamics electromagnetics (Maxwell's)
24 4.1. The fundamental laws 1. Conservation of mass (1) 2. Conservation of momentum (3) 3. Conservation of energy (1) 4. Equation of state (1) 5. Entropy (1) 4.1.1. There are 2 approaches to derive the theoretical model: I - Lagrange , history of each particle Follow a specific particle and find its location and properties at every instant. Therefore, x ( t ), y ( t ), and z ( t ). Then, P = f [ x ( t ), y ( t ), z ( t ), t ] T = g [ x ( t ), y ( t ), z ( t ), t ] etc. II - Euler , state of the flow at a point ( x , y , z ) P = f [ x , y , z ; t ] , T = g [ x , y , z ; t ] etc. Variations at a fixed point with time, random particles. P / t , T / t . . . . ( x , y , z ; t )