MrsDrPoe: Fluid Kinematics

It's another Thesis Tuesday on the blog, which means a fun filled day of fluid mechanics! (Well, I'm having fun anyway...) Today we'll be investigating Fluid Kinematics, or the ways a fluid particle can move or deform in flow field. Check out the images here as you read.

Linear Motion/Deformation
The easiest type of motion or deformation to visualize is pure linear motion (called translation) or pure linear deformation. If a particle is translated, it is simply moved from point A to point B without rotating or changing shape. If a particle undergoes linear deformation, it is squished or stretched, changing the volume of the fluid particle. The rate at which a particle is linearly deformed is called the volumetric dilatation rate. Note that for incompressible fluids, the volume of a particle cannot be changed; therefore, linear deformation is impossible (since the density is constant), and the volumetric dilatation rate is equal to zero. Note that at any given time, a particle can be just translated or translated while being linearly deformed.

Angular Motion/Deformation
The second category of motion or deformation undergone by a fluid particle is angular motion (called rotation) or angular deformation. Rotation is simply the revolving of a fluid particle around an axis while the particle maintains its shape; a flow field that does not cause a particle to rotate is called an irrotational field. Note that vorticity is defined mathematically as twice the rotation rate. If a particle is angularly deformed, the angles of a the particle are changed. If we picture a fluid particle as a square, it would become more diamond shaped through angular deformation. Again, at any time a particle can be undergoing one or both of these (angular motion or deformation).

The Underlying Math
If we are given a 3D velocity field, that means we know the x-velocity (denoted u), y-velocity (denoted v), and z-velocity (denoted w) components. Each component can be a constant or a function of position (x, y, and/or z), and derivatives with respect to each direction (x, y, and z) my be taken for each component. This gives us a strain rate matrix of nine total derivatives:

du/dx, du/dy, du/dz
dv/dx, dv/dy, dv/dz
dw/dx, dw/dy, dw/dz

These velocity field derivatives (or strain rates) provide us with information about the forces placed on a particle due to normal stress and strain in the flow field. Linear deformation can be determined from examining du/dx, dv/dy, and dw/dx; angular deformation can be determined from examining the cross derivatives, i.e. the remaining six. Translation in any direction is occurring if any of the velocity components are non-zero; rotation occurs if the curl of the velocity field is non-zero.

Example
Given a velocity field, V = (5 , 2x+y, 0):

u = 5, v = 2x+y, and w = 0 ...thus a fluid particle in this field is undergoing translation; it has both x- and y-velocity components

du/dx = 0, dv/dy = 1, and dw/dz = 0 ...the volumetric dilatation rate = du/dx + dv/dy + dw/dz = 1; therefore, the particle is being linearly deformed. Since dy/dv is the only non-zero component, it is easy to see that the particle is actually being stretched in the y-direction as it moves. (Note: if dv/dy = -1, the particle would be compressed in the y-direction.)

dv/dx = 2, du/dy = dw/dy = dv/dz = du/dz = dw/dx = 0 ...the rotation rate about the x-axis = 0.5(dw/dy - dv/dz) = 0, about the y-axis = 0.5(du/dz - dw/dz) = 0, and about the z-axis = 0.5(dv/dx - du/dy) = 1; thus, the particle is rotating. The vorticity is twice the rotation rate, or 2*1 = 2. Furthermore, the shear strain rate = dv/dx + du/dy = 2 for this case, so the particle is also undergoing angular deformation.