0.3 The kinematics of fluid motion  (Page 9/9)

The fact that the rate of deformation tensor is linear in the velocity field has an important consequence. Since we may superimpose two velocity fields to form a third, it follows that the deformation tensor of this is the sum, of the deformation tensors of the fields from which it was superimposed. If $v_i=\lambda \left(i\right)x_i$ (no summation on $i$ ) , we have a deformation which is the superposition of three stretching parallel to the three axis. However, if $v_1=f\left(x_2\right)$ , $v_2=0$ , $v_3=0$ so that only nonzero component of the deformation tensor is $e_{1}2=½f^{\text{'}}\left(x_2\right)$ , the motion is one of pure shear in which elements parallel to the coordinate axis is not stretched at all. Note however that in pure stretching an element not parallel or perpendicular to the direction of stretching will suffer rotation. Likewise in pure shear an element not normal to or in the plane of shear will suffer stretching.

Principal axis of deformation

The rate of deformation tensor is a symmetric tensor and the principal axis of deformation can be found. They correspond to the eigenvalues of the matrix and the eigenvalues are the principal rates of strain. A set of particles that is originally on the surface of a sphere will be deformed to an ellipsoid whose axes are coincident with the principal axis.

Vorticity, vortex lines, and tubes

We have frequently reminders of rotating bodies of fluid such as tropical storms, hurricanes, tornadoes, dust devils, whirlpools, eddies in the flow behind objects, turbulence, and the vortex in draining bathtubs. The kinematics of these fluid motions is described by the vorticity.

The antisymmetric part of the rate of strain tensor ${\Omega }_{ij}$ represents the local rotation, ${\omega }_k$ . Recall

The curl of velocity is known as the vorticity ,

Thus the vorticity and the antisymmetric part of the rate of strain tensor is a measure of the rotation of the velocity field. An irrotational flow field is one in which the vorticity vanishes everywhere. The field lines of the vorticity field are called vortex lines and the surface generated by the vortex lines through a closed curve $C$ is a vortex tube . The strength of a vortex tube is defined as the surface integral of the normal component. It is equal to the circulation around the closed curve $C$ that bounds the cross-section $S$ by Stokes' theorem.

We observe that the strength of a vortex tube at any cross-section is the same, as $\mathbf{w}$ is a solenoidal vector. The surface integral of the normal component of a solenoidal vector vanishes over any closed surface. The surface integral on the surface of a vortex tube is zero because the sides are tangent to the vorticity vector field. Thus the surface integral across any cross-section must be equal in magnitude. The magnitude of the vorticity field can be visualized from the relative width of the vortex tubes in the same manner that the magnitude of the velocity field can be visualized by the width of the stream tubes.

Because the strength of the tube does not vary with position along the tube, it follows that the vortex tubes are either closed, go to infinity or end on solid boundaries of rotating objects. In a real fluid satisfying the no-slip boundary condition, vortex lines must be tangential to the surface of a body at rest, except at isolated points of attachment and separation, because the normal component of vorticity vanishes on the stationary solid.

When the vortex tube is immediately surrounded by irrotational fluid, it will be referred to as a vortex filament . A vortex filament is often just called a vortex, but we shall use this term to denote any finite volume of vorticity immersed in irrotational fluid. Of course, the vortex filament and the vortex require the fluid to be ideal (zero viscosity) to make strict sense, because viscosity diffuses vorticity, but they are useful approximations for real fluids of small viscosity.

Helmholtz gave three laws of vortex motion in 1858. For the motion of an ideal (zero viscosity) barotropic (density is a single valued function of pressure) fluid under the action of conservative external body forces (gradient of a scalar), they can be expressed as follows:

1. Fluid particles originally free of vorticity remain free of vorticity.
2. Fluid particles on a vortex line at any instant will be on a vortex line at subsequent times. Alternatively, it can be said that vortex lines and tubes move with the fluid.
3. The strength of a vortex tube does not vary with time during the motion of the fluid.

The equations for the dynamics of vorticity will be developed later.

P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. C. Truesdell, The Kinematics of Vorticity, Indiana University Press, 1954.

Assignment 4.1

Assignment 4.2

1. divergence
2. curl
3. rate of deformation tensor
4. antisymmetric tensor
5. Which fields are solenoidal or irrotational?