MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 6

1. MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 6 FLUID KINETMATICS

2. Description of Fluid Motion • We are interested in the evolution of fluid particle in a space • When we are interested on some properties of a group objects in certain domain, Eulerian description is more simple and convenient to express our interest information • In fluid mechanics it is usually easier to use the Eulerian method to describe a flow in either experimental or analytical investigation.

3. Conservation EquationIntegral Vs. Differential Forms (Text p.272) • Integral Form of Mass and Momentum Conservation Equations under Eulerian description • They are finite volume approaches: • Mass Conservation Equation (Example 5.6): • Rate of change of mass over the whole volume = Net mass across the whole volume surface + Rate of change of density over the whole volume

4. Integral Forms Conservation Equation • Momentum Conservation Equation (Example 5.15): • Rate of change of momentum over the whole volume + Net momentum across the whole volume surface = Surface Force on the whole volume surface + Body Force on the whole volume

5. Integral Forms Conservation Equation • If we don’t consider the spatial variation of variable, e.g. velocity distribution, we can use the integral form conservation equations to obtain the overall values under our specified control volume, just like the above examples • If we use the integral form conservation equation in this way, there are infinite variable distribution functions, which satisfy the integral form conservation equations. Normally, we assume the variable distribution is a uniform one (Text book p.195)

6. Integral Forms Conservation Equation

7. Integral Forms Conservation Equation • The spatial distribution function of the variable in the conservation equation is necessary to investigate the fluid motion, e.g. it may affect the dynamic of the control volume like example 5.15. In other words, we need to know the each point velocity or velocity field in our interested domain. • In order to do so, the differential form conversation equations are used instead of integral form equations for obtaining the spatial variables function of the conversation equations.

8. Differential Forms Conservation Equation • In fact that, the differential form of the conservation equations is derived from integral form of the conservation equation. So, they are equivalent to each other in mathematical point of view • But integral form equations of the balance equations can not be reduced further. To address this issue, the conservation of equations are expressed in differential from integral form by invoking the integral theorems of Chapter 3

9. Differential Forms Conservation Equation • In fact that, the differential form of the conservation equations is derived from integral form of the conservation equation. So, they are equivalent to each other in mathematical point of view • But integral form of the balance equations can not be reduced further. To address this issue, the conservation of equations are expressed in differential from integral form by invoking the integral theorems of Chapter 3

10. Differential Forms Conservation Equation • Differential Form of Mass and Momentum Conservation Equations under Eulerian description • They are infinitesimal volume approaches: • Mass Conservation Equation: • Rate of change of mass = Divergence of momentum + Rate of change of density

11. Differential Forms Conservation Equation • Momentum Conservation Equation: • Rate of change of momentum + Divergence of momentum flux = Gradient of Surface Force + Gradient of Body Force

12. Differential Forms Conservation Equation • By solving the above partial differential equations (PDE) with suitable boundary condition, e.g. initial valve or boundary value of the corresponding variables of the equations, variable function of the conservation equations are obtained.

13. Elementary Fluid Motions (Text p.273) • We are interested in variable function of the conservation equations, i.e. spatial velocity function. • Spatial velocity function provides the relative motion between neighbor fluid particles, which is called fluid kinematics. • We define four elementary fluid particle relative motion to quantify and describe the fluid motions

14. Elementary Fluid Motions

15. 0 z y x z 0 y x Translation

16. Linear Deformation (Strain Rate)

17. Rotation (Vorticity)

18. Angular Deformation (Strain Rate)

19. Example 1

20. Solution 1

21. Example 2

22. Solution 2