Simple Two- and Three-Dimensional Flow Problems of the Navier-Stokes Equations

Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


This chapter begins with studying steady state layer flows through cylindrical conduits (ellipse, triangle, rectangle) and use of the Prandtl membrane analogy. This study of the Navier-Stokes fluids is important in geophysical fluid dynamics and is manifest in Ekman’s theory and its extensions, where non-inertial effects chiefly influence the details of the fluid flow, evidenced in the Ekman spiral in atmospheric and oceanic boundary flows and in free geostrophic flows as their outer solutions. Extensions of the behavior exhibited by the assumption of constant (turbulent) viscosity are based on influences of depth dependence of the viscosity which influences the circulation pattern of such steady flows. Unsteady flows are analyzed for viscous flows along an oscillating wall and the growth of a viscous boundary layer as a response of an initial tangential velocity jump with time. The chapter closes with the study of an axial laminar jet and viscous flows in a converging two-dimensional channel.


Viscous layer flows Prandtl’s membrane analogy Ekman’s theory – Spiral Ekman’s theory with non-constant viscosity Unsteady viscous layer flows Adjustment of a velocity jump Axisymmetric laminar jets Viscous converging two-dimensional channels 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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