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# MAE 5130: VISCOUS FLOWS PowerPoint PPT Presentation

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

MAE 5130: VISCOUS FLOWS

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## MAE 5130: VISCOUS FLOWS

Lecture 3: Kinematic Properties

August 24, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

• 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

• Note error in Figure 1-14

• Problem 1-8 should read, ‘Using Eq. (1-3) for inviscid flow past a cylinder…’

### KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION

• Lagrangian Description

• 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

### 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

### 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)

y

+

x

D

A

dy

B

C

dx

y

+

x

D’

A’

D

A

dy

B’

C’

vdt

B

C

dx

udt

y

+

x

D

A

dy

B

C

dx

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’

y

+

x

A’

db

D’

B’

da

C’

y

+

x

D

A

dy

B

C

dx

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

### 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

y

+

x

D

A

dy

B

C

dx

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

### FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT

Note: Mistake in text book Figure 1-14

• 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)

• 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

### 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

### 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

### 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)

### 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)

### 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