Fluid mechanics, equations vorticity

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

The Lattice Boltzmaim method (LBM) is an approach to obtaining solutions of fluid mechanics problems. Instead of solving the Navier-Stokes equation or equivalent formulations such as the streamfunction-vorticity equations, one finds solutions of a mesoscopic kinetic theory equation. The LBM developed from an earlier method called the lattice gas automaton (LGA). Rothman and Zaleski [107] and Chen and Doolen [108] discuss the LBM. 

The equation of motion given by Maxey and Riley is valid provided that two Reynolds numbers based on the radius of the sphere are small compared to unity. The two Reynolds numbers are uqRIv and R uol(Lv), where uq is a velocity that is characteristic of the undisturbed fluid, wq is a velocity that is characteristic of the relative motion between the particle and the undisturbed fluid, and T is a characteristic length of the undisturbed flow. These conditions imply that the time required for a significant change in the relative velocity is large compared to the timescale for viscous diffusion, and that viscous diffusion remains the dominant mechanism for the transfer of vorticity away from the sphere. 

Consider, for instance, classical electrodynamics. When Maxwell developed the classical field equations he had in his mind a rather concrete model of the ether (See for instance Nersessian 1992). The ether was considered as a fluid and magnetism was conceived as vortices in that fluid, and electric currents consisted of small particles that flowed between the vortices. This mechanical model of the ether was the basis and source of inspiration for his derivation of the equations. The full scope of his equations could, at that time, only be understood relative to this or similar models of the ether. Only much later when the special theory of relativity was introduced, it became possible to get rid of a conaete ether model. 

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