Special Viscometric Flows

Abstract

If the velocity field has the form

$${{v}^{\left\langle x \right\rangle }}=0,\text{ }{{v}^{\left\langle y \right\rangle }}=v\left( x \right),\text{ }{{v}^{\left\langle x \right\rangle }}=0$$

(14.1)

in a Cartesian coordinate system x, y, z, then we say that the motion is a steady shearing flow. Clearly, such a flow is a curvilineal flow with $u\left( {{x}^{1}} \right)=v\left( x \right),\text{ }w\left( {{x}^{1}} \right)=0,\text{ }{{e}_{1}}\equiv {{e}_{2}}\equiv {{e}_{3}}\equiv 1.$ Hence, by (13.4), the flow is a viscometric flow with a rate of shear x given by

$$x = v'\left( x \right) = \frac{{dv\left( x \right)}} {{dx}}.$$

(14.2)

The basis b^<i> is now just the constant basis, e^<x>, e^<y>, e^<z>, of unit vectors along the coordinate axis.