# 0.3 The kinematics of fluid motion

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## Topics covered in this chapter

• Particle paths and material derivatives
• Streamlines
• Streaklines
• Dilatation
• Reynolds' transport theorem
• Conservation of mass and the equation of continuity
• Deformation and rate of strain
• Physical interpretation of the deformation tensor
• Principal axis of deformation
• Vorticity, vortex lines, and tubes

Reading assignment: Chapter 4 of Aris

Kinematics is the study of motion without regard to the forces that bring about the motion. Already, we have described how rigid body motion is described by its translation and rotation. Also, the divergence and curl of the field and values on boundaries can describe a vector field. Here we will consider the motion of a fluid as microscopic or macroscopic bodies that translate, rotate, and deform with time. We treat fluids as a continuum such that the fluid identified to be at a specific point in space at one time with neighboring fluid will be at another specific point in space at a later time with the same neighbors, with the exception of certain bifurcations. This identification of the fluid occupying a point in space requires that the motion is deterministic rather than stochastic, i.e., random motions such as diffusion and turbulence are not described. Central to the kinematics of fluid motion is the concept of convection or following the motion of a "particle" of fluid.

## Particle paths and material derivatives

Fluid motion will be described as the motion of a "particle" that occupies a point in space. At some time, say $t=0$ , a fluid particle is at a position $\xi =\left({\xi }_{1},{\xi }_{2},{\xi }_{3}\right)$ and at a later time the same particle is at a position $\mathbf{x}$ . The motion of the particle that occupied this original position is described as follows.

$\mathbf{x}=\mathbf{x}\left(\xi ,t\right)\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{x}_{i}=x\left({\xi }_{1},{\xi }_{2},{\xi }_{3},t\right)$

The initial coordinates $\xi$ of a particle will be referred to as the material coordinates of the particles and, when convenient, the particle itself may be called the particle $\xi$ . The terms convected and Lagrangian coordinates are also used. The spatial coordinates $\mathbf{x}$ of the particle may be referred to as its position or place . It will be assumed that the motion is continuous, single valued and the previous equation can be inverted to give the initial position or material coordinates of the particle which is at any position $\mathbf{x}$ at time $t$ ; i.e.,

$\xi =\xi \left(\mathbf{x},t\right)\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{\xi }_{i}={\xi }_{i}\left({x}_{1},{x}_{2},{x}_{3},t\right)$

are also continuous and single valued. Physically this means that a continuous arc of particles does not break up during the motion or that the particles in the neighborhood of a given particle continue in its neighborhood during the motion. The single valuedness of the equations mean that a particle cannot split up and occupy two places nor can two distinct particles occupy the same place. Exceptions to these requirements may be allowed on a finite number of singular surfaces, lines or points, as for example a fluid divides around an obstacle. It is shown in Appendix B that a necessary and sufficient condition for the inverse functions to exist is that the Jacobian

$J=\frac{\partial \left({x}_{1},{x}_{2},{x}_{3}\right)}{\partial \left({\xi }_{1},{\xi }_{2},{\xi }_{3}\right)}$
should not vanish.

The transformation $\mathbf{x}=\mathbf{x}\left(\xi ,t\right)$ may be looked at as the parametric equation of a curve in space with $t$ as the parameter. The curve goes through the point $\xi$ , corresponding to the parameter $t=0$ , and these curves are the particle paths . Any property of the fluid may be followed along the particle path. For example, we may be given the density in the neighborhood of a particle as a function $\rho \left(\xi ,t\right)$ , meaning that for any prescribed particle $\xi$ we have the density as a function of time, that is, the density that an observer riding on the particle would see. (Position itself is a "property" in this general sense so that the equations of the particle path are of this form.) This material description of the change of some property, say $\Im \left(\xi ,t\right)$ , can be changed to a spatial description $\Im \left(\mathbf{x},\mathbf{t}\right)$ .

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