As briefly discussed in Chapter 4, the motion of Newtonian fluids is described by the Navier-Stokes equations. Due to the non-linear nature of these equations and the general complexity of the flow geometry, analytical solutions of Navier-Stoke’s equations has been exhibiting a major problem in fluid mechanics. The continuous development in the area of computer technology and the introduction of powerful numerical methods in the last two decades have brought a breakthrough in the area of Computational Fluid mechanics (CFD). Using CFD-methods, viscous flow problems within arbitrary channel geometries can be solved numerically regardless the complexity of the geometry. This requires significant computational efforts. An adequate treatment of CFD-methods is beyond the scope of this book. However, in the context of this course, in Chapter 9, we present the essential features of the computational fluid mechanics that are necessary for the basic understanding of the physics behind CFD. This includes a rather detailed introduction into turbulence and its modeling.
In this chapter, we introduce a class of exact solutions of the Navier-Stokes equations for the two-dimensional laminar flow, a special case of viscus flows, where the velocity does not exhibit a random characteristic. Exact analytical solutions are found only for few cases, where the flow can be assumed unidirectional. This implies that the velocity vector has a component in longitudinal direction only that may change in lateral direction. A general overview of a class of exact solutions for viscous laminar flows through two-dimensional channels is found in Schlichting . In a few curved channels, where the velocity vector of a two-dimensional flow has generally two components, the coordinate system can be transformed such that the velocity vector has only one direction in a curvilinear coordinate system. In the following sections, several cases are presented that are of fundamental significance for understanding the motion of viscous flows.
KeywordsVelocity Distribution Laminar Flow Wall Shear Stress Couette Flow Outer Cylinder
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- 2.Jeffery, G.B.: The two-dimensional steady motion of a viscous fluid. Phil. Mag. 6th Ser., 455 (1915)Google Scholar
- 7.Milsaps, K., Pohlhausen, K.: Thermal distribution in Jeffery-Hamel Flow. Journal of Aeronautical Sciences 20, 187 (1953)Google Scholar
- 9.Oseen, C.W.: Über die Stokes’sche Formel und über eine verwandte Aufgabe in Hydrodynamik. Ark. Nath. Astron. Fys. 6(29) (1910)Google Scholar