6 - Viscosity
Published online by Cambridge University Press: 05 June 2012
Summary
Shear stresses in Newtonian fluids
Throughout the last four chapters, the fact that real fluids possess viscosity has been almost completely ignored. We have supposed shear stress to be negligible and normal stress to be isotropic, and we have found that in so far as isotropic normal stress – the pressure p – depends upon fluid velocity u, it does so through formulae in which only the local magnitude of u and its rate of change with time appear. We cannot proceed much further on that simple, Eulerian, basis. The principal aims of the present chapter are firstly to establish the Newtonian formulae which relate the components of stress in viscous fluids to gradients of u, secondly to use these formulae to establish a more general equation of motion for fluids than Euler's equation, and thirdly to discuss a variety of relatively simple problems in which the effects of viscosity are dominant – so dominant in most cases that the fluid's inertia is negligible instead. The motion of fluids in such circumstances is sometimes referred to as creeping flow.
Newton himself may have considered only the simple situation illustrated by fig. 6.1, where planar laminae of fluid lying normal to the x2 axis are moving steadily in the x1 direction and sliding over one another, so that there exists a uniform velocity gradient ∂u1/∂x2.
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- Fluid Dynamics for Physicists , pp. 195 - 239Publisher: Cambridge University PressPrint publication year: 1995
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