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, – ) –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 1.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) 2.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 ( , 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: 1.Translation 2.Rotation 3.Shear strain 4.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 1. TRANSLATION dx dy A BC D y x +

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

8 2. ROTATION dx dy A BC D y x +

9 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’ dx dy A BC D A’ B’ C’ D’ y x + dd dd

10 2. ROTATION A’ B’ C’ D’ dd dd y x +

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

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

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

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

15 4. EXTENSIONAL STRAIN (DILATATION) dx dy A BC D A’ B’ C’ D’ Extensional strain in x-direction is defined as the fractional increase in length of the horizontal side of the element y x + 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 d  BA has rotated CW (-d  ) Overall CCW rotation since d  > d  d  and d  both related to velocity derivates through calculus limits Rates of angular rotation (angular velocity) 3 components of angular velocity vector d  dt Very closely related to vorticity Recall: the vorticity, , is equal to twice the local angular velocity, d  /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 2 nd 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 (  ) plus an angular velocity (d  /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: d  /dt d  x /dt, d  y /dt, d  z /dt Also related to vorticity –Shear-strain rate:  xy =  yx,  xz =  zx,  yz =  zy –Rate of dilatation:  xx,  yy,  zz 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