2: Introduction and Fundamental Concepts in Mechanics and Fluid Mechanics

1. 2: Introduction and Fundamental Concepts in Mechanics and Fluid Mechanics • Mechanics • Force-Body-Motion • Fluid Mechanics • Fluid as A Continuum / Continuum Assumption • Fluid as A Continuum • Property at A Point • Property Fields • Methods of Description of Motion: Lagrangian VS Eulerian Descriptions • Scalar, Vector, and Tensor Fields • Classification of Fields • Steady VS Unsteady Fields [Does f at any one point change with time t?] • Uniform VS Non-Uniform Fields [Does f at any one time t change with the spatial location in the region R?] • Classification of Fluid Flows

2. Mechanics

3. Fundamental ConceptMechanics: Force-Body-Motion • Mechanics: Three main components: Force - Body - Motion. • System-Surroundings-Interactions: Investigate the resulting effects of the mechanical interactions between the system and its surroundings. • Interactions: Forces (in FBD) and Work • Effects: Motion (and other related quantities) Mechanics Motion of aBody under the action of Forces. (Effects of Forces on the Motion of a Body.)

4. Thermodynamics Thermal Energy and Thermodynamic Properties of Substance Force Body Motion Linear Motion Translation Angular Motion (Rigid-Body-Like) Rotation Deformation • Force: An effortto move a body against itsinertia. (in Newton’s second law aspect.)

5. Fundamental ConceptThree Efforts of Force, Three Types of Body, and Three Types of Motion Forces (3 Efforts) Body (3 Types): (~ according to the degree of idealization of permitted motions.) Motion (3 Components): Particle Linear Motion (Translation) Force Rigid Body Angular Motion (Rigid-Body-Like Rotation) Moment of Force Deformable Body Deformation Intensity of Force

6. Force-Body-Motion in Equations of Mechanics 1. Force and linear motion (translation). 2. (Moment of) Force and angular motion (rotation). 3. (Intensity of) Force and deformation. (Constitutive Relation) Strain Stress Hooke’s law Newton’s law of viscosity

7. Fluid Mechanics

8. Fundamental Concept: Fluid Mechanics Fluid Mechanics = Mechanics (Force and Motion) and Thermodynamics (Energy and Energy Transfer) of Fluid Motion

9. Heater Exit Inlet air Mass flowrate Density profile Mass flowrate Density profile Velocity profile Velocity profile Temperature profile Temperature profile Fluid Mechanics = Mechanics (Force and Motion) and Thermodynamics (Energy and Energy Transfer) of Fluid Motion System (Control Volume): Fluid stream only, excluding the solid heater Given: At inlet, Question: At exit,

10. a a (t) Solid finite deformation under constant shear Fluid continuous deformation under constant shear no matter how small shear is Fundamental Concept: Definition of Fluids Simple models for simple solid and fluid. Definition of Fluid: A fluid is a substance that deforms continuouslyunder the application of a shear (tangential) stress no matter how small the shear stress may be. (Fox, et al., 2004)

11. d m, dV Continuum Assumption y x z Fundamental ConceptFluid as A Continuum: Continuum Assumption, Property at A Point dV dV’ • Fluctuation due to random molecular motion • Continuum assumption breaks down. • Macroscopic spatial variation Random motion of molecules has little effect on macroscopic mean density. Random motion of molecules significantly affects macroscopic mean density. • Above this limit: • Continuum:A fluid is assumed to be a continuum. • Property at a point: Its property is assumed at a point, e.g., density at a point. • Field Function: A fluid property f is assumed as a continuous field function of space and time: • Below this limit, the continuum assumption breaks down, and the molecular motion of molecules must be taken into account.

12. Two Methods of Description of Fluid Motion

13. Eulerian Description Lagrangian Description y y y x x x z z Reference time to Current time t z Current time t Time ‘Particle name’ Time Spatial position Time evolution of a property fat a fixed point in space is given by Time evolution of a property f of a material particle is given by Fundamental ConceptTwo Methods of Description of Fluid Motion: Lagrangian VS Eulerian Descriptions Eulerian Description Lagrangian Description

14. Lagrangian Description reads the current (at time t) value of the property of the material particle is equal to . Eulerian Description reads the value of the property at position and time is equal to .

15. Fundamental Concept: Lagrangian VS Eulerian Views • To put simply Eulerian view: • We watch an identified region in space of interest, and • see what happens in the region. [Here, we watch a block in the street and see New York cabs passing in and out of our block.] Lagrangian view: • We watch/follow the identified mass of interest, and • see what happens to the identified mass. [Here, we watch and follow, say, one taxicab wherever it goes.]

16. Space-time point Timet Eulerian/Field Descriptions Scalar, Vector, and Tensor Fields: Velocity and Property Fields • Common Property Fields: Scalar Field: density: pressure: temperature: Vector Field: velocity: Note that it is customary to use (u, v, w) for the (x,y,z) components of velocity, respectively. Tensor Field: stress tensor:

17. t t t+dt t+dt Steady and Uniform Property Field fSteadiness:Doesf at any one point change with time t ? • Steadiness: Doesfat any one point change with time t ? Shading represents the value of the property f , say light = high value, dark = low value changes with time t does not change with time t

18. t t does not change with spatial location changes with spatial location Steady and Uniform Property Field fUniformity in a region R: Doesf at any one time t change with spatial location ? • Uniformity in a region R: Doesfat any one time t change with spatial location ?

19. t t t+dt t+dt t t+dt t t+dt Example: Steadiness and Uniformity Steady ? Uniform at time t? Uniform at time t + dt? ? ? ?

20. Vorticity (related to angular velocity) Classification of Fluid Flows r = constant • M < 1 - Subsonic • M = 1 - Sonic; • M ~ 1 - Transonic • M > 1 - Supersonic • M > ~ 5 - Hypersonic m = 0, or effect of viscous stress can be neglected. The number of spatial coordinates x, y, z, that is required to specify the velocity field. Laminar: Smooth and orderly (non-random), can be steady or unsteady. Turbulent Flow: Random fluctuation of velocity field, inherently unsteady. Internal: Flow that is bounded by solid surfaces. External: Flow over body immersed in unbounded fluid.

21. Laminar: orderly motion Turbulent: random motion Laminar-Transition-Turbulence: Subsonic Jet Transition from laminar flow to turbulent flow via instability in subsonic jet. From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

22. Laminar-Transition-Turbulence: Subsonic Jet Transition from laminar flow to turbulent flow via instability in subsonic jet. From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

24. Example: Classification and Velocity Field Question: 1. Classify the following velocity fields by stating whether it is • steady or unsteady? • if unsteady, also find • 1-, 2-, or 3-dimensional? • Then, write down the functional form of the field, e.g., • Also find • the divergence of the velocity field • NOTE: The divergence of the velocity field is related to the compressibility of the flow. • the vorticity, which is defined as the curl of the velocity field • NOTE: The vorticity is a measure of the angular velocity of a fluid element.

25. Example: Classification and Velocity Field • In addition, using MLtT as the set of primary dimensions, also state the dimension of the constant a, b, c, etc.

26. Example: Classification and Velocity Field

27. Representation of A Vector Field: A Vector Plot Let a velocity field be given by where the field is defined in the region and , x and y are given in meters, and a = 1 s-1 and b = -1 s-1. Sketch a vector plot for this field.

28. Example: A Vector Plot • Sketch a vector plot for the following velocity fields. x and y are given in meters.