Chp4_331

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 , y , z ) * shear stress - τ xy , xz , yz There are 6 unknowns: P , , T , U For many flows: =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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25 The fluid particles are different, but the flow properties are the same with respect to time. "... they looked in the depths below them at the waters as they went swiftly past, always new and yet always the same." Ivo Andric The Bridge on the Drina , 1945 Complete differential (Euler): d P = P x d x + P y d y + P z d z + P t d t Total derivative (Lagrange): DP Dt = P x d x d t + P y d y d t + P z d z d t + P t = u P x + v P y + w P z + P t 4.1.2. System Free body, isolated, part of some media confined by definite boundaries There is mutual influence between system and ambient F ( ambient ) = m ( system ) × a ( system )
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26 4.1.3. Control volume Arbitrary volume fixed in space, through which the fluid flows. It is bounded by a closed control surface. At a particular time, this defines the fluid system. It is used as a basis for keeping track of the fluid flow. The problem is how to apply the physical laws, valid for the system, to the control volume. The eq'ns can be formulated in two ways: I - Partial differential equations. * valid for any point in the flow, thus providing detailed description. * hard to solve analytically, difficult to solve numerically. II - Integral form of the conservation laws. * simpler, and physically correct. * frequently involves approximations Can derive one system from the other!
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27 The application of conservation laws to control volume. I
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This note was uploaded on 09/08/2010 for the course AME 331 taught by Professor Zohar during the Fall '08 term at Arizona.

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

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