Viscous Potential Theory Research Articles

The Resolution of d'Alembert's Paradox and Thermodynamic Admissibility

The d'Alembert paradox, annunciated in 1752, was established after it was shown that the result of a net zero drag, obtained by applying potential theory to the incompressible flow past a sphere, was in contradiction with experimental results which showed a positive drag. Interpreting the result as a flaw in the theory, resulted in the declaration of the paradox. Following d'Alembert, we assume a potential motion, and proceed to analyze the consequences of this assumption using the global principles of continuum mechanics. We show that if the fluid is inviscid, the potential motion is thermodynamically admissible, the drag is zero, and the motion can persist indefinitely. Although no conventional fluid is available to either falsify (or validate), this result experimentally, in principle, the theory could be tested by using a superfluid, such as liquid Helium. If the fluid is viscous, we show that the potential irrotational motion is thermodynamically inadmissible, it is in violation of the second law of thermodynamics, and hence such a motion cannot persist. Indeed, observations show that when a rigid body is impulsively set into motion, an irrotational motion may exist initially but does not persist. Any flow property which is derived from a thermodynamically inadmissible motion cannot be expected to be in agreement with experimental data. As an illustration we show that the drag, calculated from the viscous potential solution of the flow past a sphere, is zero. We submit that since the theory of continuum mechanics predicts that no agreement between results obtained from viscous potential theory and experimental data can be expected, there is no room for a paradox once a contradiction is in fact observed. In nature, or under experimental conditions, the nonpersistence of the thermodynamically inadmissible motion proceeds in a breakup of the irrotational motion which transforms into a rotational and obviously admissible motion. We show that by selecting boundary conditions, required in the solution of the differential equations of motion, such that they satisfy the Clausius–Duhem jump conditions inequality, the thermodynamic admissibility of the solution is a priori assured. We also show the vorticity distribution at the wall associated with the particular choice of boundary condition.

Nonlinear Electrohydrodynamic Kelvin-Helmholtz Instability through Two Porous Layers with Suction / Injection : Viscous Potential Theory

The nonlinear electrohydrodynamic Kelvin-Helmholtz instability of an interface between two porous layers with heat and mass transfer is investigated. The fluid flow is considered to be through semi permeable boundaries above and below at which the fluids may either be flows in or sucked out, in a direction normal to the main streaming direction. Through the linear stability analysis, a general dispersion relation is obtained. The Ginzburg–Landau equation is obtained by employing the method of multiple scale expansion according to the nonlinear stability theorem. The stability of the system is discussed and is presented graphically. It is found that the injection of the two fluids at both boundaries has a stabilizing effect in contrast with the suction at both boundaries. The stable and unstable regions in the nonlinear case are drawn and discussed. Keywords Nonlinear Kelvin-Helmholtz instability ; Viscous potential flow; Interfacial instability; Heat transfer ; Mass transfer; Flow through porous media; Electrohydrodynamic; Multiple scale methods; Suction/Injection.

Kelvin-Helmholtz instability for flow in porous media under the influence of oblique magnetic fields: A viscous potential flow analysis

In this paper, the Kelvin-Helmholtz instability of viscous incompressible magnetic fluid fully saturated porous media is achieved through the viscous potential theory. The flow is considered to be through semi-permeable boundaries above and below the fluids through which the fluid may either be blown in or sucked out, in a direction normal to the main streaming direction of the fluid flow. An oblique magnetic field, mass, heat transfer, and surface tension are present across the interface. Through the linear stability analysis, a general dispersion relation is derived and the natural curves are plotted. Therefore, the linear stability condition is discussed in some depth. In view of the multiple time scale technique, the Ginzburg–Landau equation, which describes the behavior of the system in the nonlinear approach, is obtained. The effects of the orientation of the magnetic fields on the stability configuration in linear, as well as nonlinear approaches, are discussed. It is found that the Darcy's coefficient for the porous layers plays a stabilizing role. The injection of the fluids at both boundaries has a stabilizing effect, in contrast with the suction at both boundaries.

Nonlinear Kelvin–Helmholtz instability of magnetized surface waves on a subsonic gas–viscous potential liquid interface

A nonlinear stability analysis, of magnetic fluids, was carried out for interfacial waves between a subsonic inviscid gas and viscous streaming liquid when a normal constant magnetic field is present. The viscosity term in the problem was carried out using the viscous potential theory. The nonlinear analysis results nonlinear partial differential equations, for different cases, using multiple scales method. Using the modulation concept, we have discussed different numerical examples to show the effects of the system parameters on criteria of the interfacial waves (in)stability.