The interdependence of friction, pressure gradient, and flow rate in unsteady laminar parallel flows

Solutions to the Laplace-transformed Navier–Stokes equations are developed that describe transients in fully developed channel and pipe flow. The relative ease with which inverse Laplace transforms can be carried out numerically makes it straightforward to find the form of new expressions relating flow rate, pressure gradient, and wall friction, for flows of arbitrary unsteadiness in time. In particular, expressions for flow rates and cumulative throughflows are derived in terms of wall shear stress and pressure–gradient histories, together with a channel-flow counterpart to Zielke’s pipe-flow friction law [J. Basic Eng. 90, 109 (1968)] expressing wall shear stress as a functional of flow-rate history. It is shown how these results can be expressed as the unsteady counterparts to well-known steady-flow relationships. The relative importance of unsteady effects to quasisteady ones is determined by a dimensionless parameter of the form (1/U)(∂U/∂t)R2/ν, where R is the span of the duct. When departures from quasisteady forms of these relations exist, the memory of the Navier–Stokes equations of earlier transients, comparable in size to present ones, extends roughly 0.2 R2/ν seconds backward in time. The derived relationships are used to illustrate how path-dependent quantities such as flow work vary with different unsteady flow histories.