Comptes Rendus. Mécanique (Mar 2021)
Correspondence between de Saint-Venant and Boussinesq 5: Viscosity and hydraulic resistance
Abstract
Fluid viscosity is a main feature of fluids; an inviscid fluid does not exist even though a large number of theories has been advanced for flows of such fluids. The velocity of fluid flow may considerably be reduced due to the presence of fluid viscosity both for laminar and turbulent flows. The letters exchanged between de Saint-Venant (dSV) and Boussinesq mainly refer to laminar flow. The most peculiar statement of dSV is that, if the flow in a typical lowland river would be laminar, its velocity would be larger than the speed of sound. It is evident that this statement is wrong because laminar flow has stringent limitations typically expressed by the Reynolds number.The hydraulic resistance is another peculiar feature in hydrodynamics, given that several theories have been advanced which do not at all reflect the everyday experience. One of these is the d’Alembert paradox, stating that the resistance is equal to zero for a steady flow of an inviscid and incompressible fluid. Typically, a body suspended in a large pipe is considered. Applying the momentum equation in the axial direction sufficiently up- and downstream of the body, the resistance would indeed become zero, as occurs for potential flows. Again, this is far away from everybody’s experience, such as for a swimmer or a walker under heavy wind.This paper mainly explores the letters exchanged between the two late 19th century scientists dSV and Boussinesq. Their considerations indirectly advanced the turbulence theory later developed by Boussinesq, based on the Boussinesq turbulent exchange coefficient. It also rectifies the role of dSV in improving the fundamental equations currently referred to as the Navier–Stokes equations. Both aspects of turbulence and hydraulic resistance are by now still under considerable research activity.
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