The velocity field in a drag-induced Couette Flow between parallel flat plates remains constant on plane surfaces which are parallel to the flat plates. Individual plane laminae are sheared and thus slide over each other, the velocity being purely axial everywhere. Thus an element of fluid with a rectangular section at time t = 0 in a plane spanned by the direction of flow and the direction of shear is distorted into a parallelogram at time t > 0 (see Figure 1). Similarly, the velocity field in a pressure-induced Poiseuille Flow in a long pipe of circular croscs-section normal to its axis remains constant on cylindrical surfaces which are concentric to the axis of the pipe. Individual cylindrical laminae slide over each other, the velocity again being purely axial everywhere. Smooth flows of such kinds are called laminar flows [see Richardson (1989), Rosenhead (1963), Shapiro (1964) and White (1974)] or, in the older literature, Streamline Flows and generally occur at low values of an appropriate dimensionless number, usually a Reynolds Number. It is to be distinguished from a rough or erratic Turbulent Flow, which is a high Reynolds number flow.

The origin of turbulence appears to be instability of the associated low Reynolds number laminar flow. Indeed, all laminar flows seem to be susceptible to instabilities leading eventually to turbulence (see Drazin & Reid (1981) for a discussion of the instability of several basic laminar flows).

In a turbulent flow, flow variables such as velocity and pressure fluctuate in space and time in an apparently random manner. No such fluctuations are observed in a laminar flow, though it should be noted that a laminar flow need not be steady, nor necessarily very smooth.

The details of the transition from laminar to turbulent flow are not well understood. For flow by a wall, however, the main steps as the Reynolds number is increased appear to comprise:

an initial, often two-dimensional, instability which leads to:

a Secondary Flow which is generally three-dimensional and itself unstable which leads to:

a succession of increasingly complex secondary flows which are again themselves unstable and which lead to:

further, almost without exception three-dimensional, flows which are themselves unstable, and so on, until:

intense, local, three-dimensional fluctuations are produced which grow both in size and in number, merge and eventually:

the flow becomes fully turbulent.

The exact point at which the flow ceases to be laminar is debatable and probably irrelevant.

It is important to note that an increase in Reynolds number occurs not only through an increase in speed. It also occurs through an increase in dimension or size. Thus, laminar flows tend to involve relatively slow motions of small objects. Turbulent flows, in contrast, tend to involve relatively fast motions of large objects. Some flows, particularly those associated with Boundary Layers, Jets and *wakes*, are laminar upstream and turbulent downstream. This is because the characteristic dimension involved in the Reynolds number is then the axial distance from the start of the flow. A typical example of this is smoke from a lighted cigarette. Near the cigarette, there is a smooth column of smoke: the flow is laminar. Further away, the column breaks down: the flow is turbulent.

#### REFERENCES

Drazin, P. G. and Reid, W. H. (1981) *Hydrodynamic Stability*. Cambridge University Press. Cambridge.

Richardson, S. M. (1989) *Fluid Mechanics*. Hemisphere. New York.

Rosenhead, L. Ed. (1963) *Laminar Boundary Layers*. Clarendon Press. Oxford.

Shapiro, A. H. (1964) *Shape and Flow*. Heinemann. London.

White, F. M. (1974) *Viscous Fluid Flow*. McGraw-Hill. New York.

#### References

- Drazin, P. G. and Reid, W. H. (1981)
*Hydrodynamic Stability*. Cambridge University Press. Cambridge. - Richardson, S. M. (1989)
*Fluid Mechanics*. Hemisphere. New York. - Rosenhead, L. Ed. (1963)
*Laminar Boundary Layers*. Clarendon Press. Oxford. - Shapiro, A. H. (1964)
*Shape and Flow*. Heinemann. London. - White, F. M. (1974)
*Viscous Fluid Flow*. McGraw-Hill. New York.