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)
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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 )
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