MAE 5130: VISCOUS FLOWS

1. MAE 5130: VISCOUS FLOWS Lecture 3: Kinematic Properties August 24, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. CHAPTER 1: CRITICAL READING • 1-2 (all) • Know how to derive Eq. (1-3) • 1-3 (1-3.1 – 1-3.6, 1-3.8, 1-3.12 – 1-3.17) • Understanding between Lagrangian and Eulerian viewpoints • Detailed understanding of Figure 1-14 • Eq. (1-12) use of tan-1 vs. sin-1 • Familiarity with tensors • 1-4 (all) • Fluid boundary conditions: physical and mathematical understanding • Comments • Note error in Figure 1-14 • Problem 1-8 should read, ‘Using Eq. (1-3) for inviscid flow past a cylinder…’

3. KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION • Lagrangian Description • Follow individual particle trajectories • Choice in solid mechanics • Control mass analyses • Mass, momentum, and energy usually formulated for particles or systems of fixed identity (ex., F=d/dt(mV) is Lagrandian in nature) • Eulerian Description • Study field as a function of position and time; not follow any specific particle paths • Usually choice in fluid mechanics • Control volume analyses • Eulerian velocity vector field: • Knowing scalars u, v, w as f(x,y,z,t) is a solution

4. KINEMATIC PROPERTIES • Let Q represent any property of the fluid (r, T, p, etc.) • Total differential change in Q • Spatial increments • Time derivative of Q of a particular elemental particle • Substantial derivative, particle derivative or material derivative • Particle acceleration vector • 9 spatial derivatives • 3 local (temporal) derivates

5. 4 TYPES OF MOTION • In fluid mechanics we are interested in general motion, deformation, and rate of deformation of particles • Fluid element can undergo 4 types of motion or deformation: • Translation • Rotation • Shear strain • Extensional strain or dilatation • We will show that all kinematic properties of fluid flow • Acceleration • Translation • Angular velocity • Rate of dilatation • Shear strain are directly related to fluid velocity vector V = (u, v, w)

6. y + x 1. TRANSLATION D A dy B C dx

7. y + x 1. TRANSLATION D’ A’ D A dy B’ C’ vdt B C dx udt

8. y + x 2. ROTATION D A dy B C dx

9. y + x 2. ROTATION • Angular rotation of element about z-axis is defined as the average counterclockwise rotation of the two sides BC and BA • Or the rotation of the diagonal DB to B’D’ D A A’ db D’ dy B’ da B C dx C’

10. y + x 2. ROTATION A’ db D’ B’ da C’

11. y + x 3. SHEAR STRAIN D A dy B C dx

12. y + x 3. SHEAR STRAIN • Defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC) D A db dy da B C dx Shear-strain increment Shear-strain rate

13. COMMENTS: STRAIN VS. STRAIN RATE • Strain is non-dimensional • Example: Change in length DL divided by initial length, L: DL/L • In solid mechanics this is often given the symbol e, non-dimensional • Recall Hooke’s Law: s = Ee • Modulus of elasticity • In fluid mechanics, we are interested in rates • Example: Change in length DL divided by initial length, L, per unit time: DL/Lt gives units of [1/s] • In fluid mechanics we will use the symbol e for strain rate, [1/s] • Strain rates will be written as velocity derivates

14. y + x 4. EXTENSIONAL STRAIN (DILATATION) D A dy B C dx

15. y + x 4. EXTENSIONAL STRAIN (DILATATION) • Extensional strain in x-direction is defined as the fractional increase in length of the horizontal side of the element A’ D’ D A dy B’ C’ B C dx Extensional strain in x-direction

16. FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT Note: Mistake in text book Figure 1-14

17. COMMENTS ON ANGULAR ROTATION • Recall: angular rotation of element about z-axis is defined as average counterclockwise rotation of two sides BC and BA • BC has rotated CCW da • BA has rotated CW (-db) • Overall CCW rotation since da > db • da and db both related to velocity derivates through calculus limits • Rates of angular rotation (angular velocity) • 3 components of angular velocity vector dW/dt • Very closely related to vorticity • Recall: the vorticity, w, is equal to twice the local angular velocity, dW/dt (see example in Lecture 2)

18. COMMENTS ON SHEAR STRAIN • Recall: defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC) • Shear-strain rates • Shear-strain rates are symmetric

19. COMMENTS ON EXTENSIONAL STRAIN RATES • Recall: the extensional strain in the x-direction is defined as the fractional increase in length of the horizontal side of the element • Extensional strains

20. STRAIN RATE TENSOR • Taken together, shear and extensional strain rates constitute a symmetric 2nd order tensor • Tensor components vary with change of axes x, y, z • Follows transformation laws of symmetric tensors • For all symmetric tensors there exists one and only one set of axes for which the off-diagonal terms (the shear-strain rates) vanish • These are called the principal axes

21. USEFUL SHORT-HAND NOTATION • Short-hand notation • i and j are any two coordinate directions • Vector can be split into two parts • Symmetric • Antisymmetric • Each velocity derivative can be resolved into a strain rate (e) plus an angular velocity (dW/dt)

22. DEVELOPMENT OF N/S EQUATIONS: ACCELERATION • Momentum equation, Newton • Concerned with: • Body forces • Gravity • Electromagnetic potential • Surface forces • Friction (shear, drag) • Pressure • External forces • Eulerian description of acceleration • Substitution in to momentum • Recall that body forces apply to entire mass of fluid element • Now ready to develop detailed expressions for surface forces (and how they related to strain, which are related to velocity derivatives)

23. SUMMARY • All kinematic properties of fluid flow • Acceleration: DV/Dt • Translation: udt, vdt, wdt • Angular velocity: dW/dt • dWx/dt, dWy/dt, dWz/dt • Also related to vorticity • Shear-strain rate: exy=eyx, exz=ezx, eyz=ezy • Rate of dilatation: exx, eyy, ezz are directly related to the fluid velocity vector V = (u, v, w) • Translation and angular velocity do not distort the fluid element • Strains (shear and dilation) distort the fluid element and cause viscous stresses