Exercise

1. Exercise Categorize the following as solid, liquid or gas 1 � air 2 � seawater 3 � wood 4 � Honey 5 � Jello 6 � A bucket of Sand Explain why you have chosen your specific categories 

2. Properties of solids liquids and gases Fluid � A material such that relative positions of nearby elements change significantly when suitable forces are applied. - Resistance properties of the fluid cannot prevent deformation from occurring. - Resistance properties are related to intermolecular forces. 

3. Properties of solids liquids and gases Difference between Fluid and solids - Solid deformation is linearly proportional to applied forces. Difference between liquids and gasses � Both are fluids. The main difference is in the amount of force needed to compress the fluid (change the density of the fluid). Many substances contain properties of both solids and fluids � Jelly or Jello, Paint, Glass, tar For air and water, however, the distinction is clear.

4. Continuum Hypothesis Continuum Hypothesis � Assumes that the macroscopic behavior of fluids is the same as if they were perfectly continuous in structure Figure 1. � Density variations as compared to size of volume being measured[1]. [1] Image taken from � M. E. McIntyre�s lecture notes on fluid dynamics located at: http://www.atm.damtp.cam.ac.uk/people/mem/FLUIDS-IB/

5. Continuum Hypothesis The continuum hypothesis allows us to: 1. Treat all our physical properties as continuous variables and thus allows us to have a clear understanding of a physical property �at a point� 2. Establish equations that are independent of the particle structure of the fluid 3. Derive the Navier-Stokes equation (equations of motion). The continuum hypothesis falls apart when: 1. We examine quantum mechanical scales of materials. 2. Consider fluids with significant concentrations of particles or sediment 3. When we consider extremely low density environments or rapidly spatial variations such as shocks.

6. Behavior of fluids to Volume or Surface forces Volume forces � Long range forces capable of penetrating into the interior of the fluid and act on all elements. - Contact of source of force with material is not necessary. - Also called body forces Examples of body forces in fluid systems: a. Gravity b. Electromagnetic Radiation c. Apparent forces due to coordinate system motion

7. Behavior of fluids to Volume or Surface forces Surface forces � Short range forces that act only on a thin layer adjacent to the boundary of a fluid element. Surface forces are negligible unless there is direct mechanical contact between the interacting elements. -The total force exerted across the surface element will be proportional to its area, . Examples of surface forces in fluids: a. Pressure Fields b. Friction

8. Notation Our standard form for the notation of a vector is If we designate the components of this vector with a set of numerical indices: and a similar notation for the basis vectors, Then the above vector can be represented in index notation as

9. Einstein summation convention "I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." (Kollros 1956; Pais 1982, p. 216). Since the product is summed over a dummy variable, i, we can drop the summation symbol in index notation Using this convention, the del operator is expressed as The divergence is then expressed as and the gradient of a scalar is presented as

10. Kronecker delta function A specific function in index notation that allows us to swap or contract indices is the Kronecker delta function which is defined as Conversion of indices Contraction of indices

11. Some final points Since it is often the case where we are looking at equations, its common convention to drop the basis vector in the use of index vectors. For example is formally expressed in index notation as But since we usually just talk about the three independent equations We drop the basis vector in the index notation

12. Some final points We can see from the previous examples that a quick rule of thumb to use to convert between vector and index form for equations is Be careful though, we will see in the examples that we need to consider what the dummy and vector indices are as well as where the vector operations should be

13. Eulerian framework In this course, unless otherwise stated, we will use the Eulerian framework where scalar or vector fields are presented in the form Where In this notation, time derivatives are material and consist of a local time derivative term and an advection term In our new index notation,

14. Exercise Convert the following to index notation and identify whether it is a vector or scalar 

15. Exercise Convert the following to vector notation and identify whether it is a vector or scalar 

16. Exercise Convert the following to vector notation* 

17. Relative motion near a point Recall the definition of a fluid. The deformation characteristics stated at the beginning of this chapter of a fluid can analyzed quantitatively by considering the difference in the flow field at two �adjacent� points. If we assume that dx is small then we can use Taylor series Expressed in index notation,

18. Relative motion near a point We can express this difference in the velocity field in terms of a symmetric and anti-symmetric part: Strain rate tensor: The strain rate is associated with the rate of compression or expansion of a fluid element along a principal set of axes. Rotational tensor: The rotation tensor is a measure of the non-deformational, rotational aspects of the fluid. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces.

19. Relative motion We can express both of these tensors as matrices Notice that the strain rate is symmetric and the rotational tensor is antisymmetric (swap the indices to see this) The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces.

20. Rotation tensor Recall the definition of vorticity Substitution of these vorticity components into the rotation tensor matrix leads to the result We can see that the rotation tensor is directly related to the vorticity of the fluid and thus a measure of the non- deformational rotational properties of the fluid at that point. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces.

21. Strain rate tensor If the strain rate matrix can be diagonalized, we can observe the deformation effects in the principal axis coordinate system. Notice the strain rate is therefore directly related to the divergence or compressible nature of the fluid. In this CS, we can remove the effects of rotation and just examine the deformational characteristics of the flow. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces. The results of the above analysis are significant. We have shown by considering the relative motion between two nearby points in a fluid medium that all fluid motion is a combination of rotation and deformation. It will be shown in a chapter 5 that the deformational characteristics of the fluid also are intrinsically related to its frictional forces.