Page 386 - Introduction to Continuum Mechanics

370 Streamline, Pathline, Streakline, Steady, Unsteady, Laminar and Turbulent Flow

Again, this is a straight line parallel to thexi axis.

(iv)Steady and Unsteady Flow
A flow is called steady if at every fixed location nothing changes with time. Otherwise, the
flow is called unsteady. It is important to note, however that in a steady flow, the velocity,
acceleration, temperature etc. of a given fluid particle in general changes with time. In other
=
words, let ¥ be any dependent variable, then in a steady flow, (dW /d*) x~ fixed 0» but
D*F /Dt is in general not zero. For example, the steady flow given by the velocity field

has an acceleration field given by

We note that for steady flow the pathlines coincide with the streamlines and streaklines.
(v) Laminar and Turbulent Flow
A laminar flow is a very orderly flow in which the fluid particles move in smooth layers, or
laminae, sliding over particles in adjacent laminae without mixing with them. Such flow are
generally realized at slow speed. For the case of water flowing through a tube of circular
cross-section, it was found by Reynolds who observed the thin filaments of dye in the tube,
that when the dimensionless parameter NR (now known as Reynolds number) defined by

[where v m is the average velocity in the pipe, d the diameter of the pipe, andp and/* the density
and viscosity of the fluid], is less than a certain value (approximately 2100), the thin filament
of dye was maintained intact throughout the tube, forming straight lines parallel to the axis of
the tube. Any accidental disturbances were rapidly obliterated. As the Reynolds number is
increased the flow becomes increasingly sensitive to small perturbations until a stage is reached
wherein the dye filament broke and diffused through the flowing water. This phenomenon of
irregular intermingling of fluid particle in the flow is termed turbulent. In the case of pipe
flow, the upper limit of the Reynolds number, beyond which the flow is turbulent, is indeter-
minate. Depending on the experimental setup and the initial quietness of fluid, this upper
limit can be as high as 100,000.