Fluid Dynamics: An Introduction by Michel Rieutord (auth.)

By Michel Rieutord (auth.)

This publication is devoted to readers who are looking to study fluid dynamics from the start. It assumes a easy point of arithmetic wisdom that may correspond to that of so much second-year undergraduate physics scholars and examines fluid dynamics from a physicist’s standpoint. As such, the examples used basically come from our surroundings in the world and, the place attainable, from astrophysics. The textual content is prepared in a innovative and academic structure, geared toward prime readers from the easiest fundamentals to extra advanced issues like turbulence and magnetohydrodynamics. routines on the finish of every bankruptcy aid readers to check their figuring out of the topic (solutions are supplied on the finish of the book), and a distinct bankruptcy is dedicated to introducing chosen elements of arithmetic that newbies will not be accustomed to, for you to make the ebook self-contained.

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50) This expression may be used to show that the Second Principle of Thermodynamics implies the positivity of transport coefficients like the viscosities and the thermal conductivity. For this purpose, we need considering a volume attached to the fluid particles. The Second Principle says that the entropy of this mass increases more than the entropy produced either by the internal heat sources or by the external heat flux. t / dV T 0 As it must be verified for any velocity or temperature field, it implies that 0; 0; 0 These inequalities show that the irreversibility of thermodynamic transformations is intimately associated with the diffusion phenomena that are represented by these coefficients.

For small values of the rate of strain (T ! 9 More About Rheological Laws: Non-Newtonian Fluids 43 Fig. 4 Flow corresponding to a plane extension Fig. 1 D4 • The uniaxial extension (see Fig. T /=T z=2T One then sets xx yy xx zz D In the Newtonian limit, it turns out that E D 3 . • The biaxial extension flow (see Fig. 79) 44 1 The Foundations of Fluid Mechanics Fig. T /=T One may notice that this latter motion can be obtained from the uniaxial extension flow by changing the sign of the rate of strain.

Let us consider an elementary surface d S on which an elementary force d f is applied. These two vectors are related in a very general way by d f D Œ d S or with indices df i D ij dSj We thus define the stress tensor Œ  and at the same time the stress T D Œ n applied on a given point of the surface whose normal is n. Thus defined, the stress is a force per unit surface. 3 3 This implies in particular that the stress tensor is independent of the surface on which the stress is computed. It is independent of its orientation n and its curvature radii.

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