Variational Theory of Mixtures in Continuum Mechanics

Abstract

In continuum mechanics, the equations of motion for mixtures are derived through the use of Hamilton's extended principle which regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of densities, entropies and successive spatial gradients of each constituent. We first write the equations of motion for each constituent of an inviscid miscible mixture of fluids without chemical reactions or diffusion. Our work leads to the equations of motion in an universal thermodynamic form in which interaction terms subject to constitutive laws, difficult to interpret physically, do not occur. For an internal energy function of densities, entropies and spatial gradients, an equation describing the barycentric motion of the constituents is obtained. The result is extended for dissipative mixtures and an equation of energy is obtained. A form of Clausius-Duhem's inequality which represents the second law of thermodynamics is deduced. In the particular case of compressible mixtures, the equations reproduce the classical results. Far from critical conditions, the interfaces between different phases in a mixture of fluids are layers with strong gradients of density and entropy. The surface tension of such interfaces is interpreted.

24 pages

References in corpus (1)

• Noether theorem in fluid mechanics

Cited by in corpus (18)

• Continuum thermodynamics of chemically reacting fluid mixtures
• Symmetric form for the hyperbolic-parabolic system of fourth-gradient fluid model
• A new form of governing equations of fluids arising from Hamilton's principle
• Mixture of Fluids involving Entropy Gradients and Acceleration Waves in Interfacial Layers
• A variational principle for two-fluid models
• The watering of tall trees - Embolization and recovery
• Dynamics of liquid nanofilms
• Hamiltonian Principle in Binary Mixtures of Euler Fluids with Applications to the Second Sound Phenomena
• A variational derivation of the nonequilibrium thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation
• Travelling waves near a critical point of a binary fluid mixture
• Bubble number in a caviting flow
• The Hamilton principle for fluid binary mixtures with two temperatures
• Fluid mixtures in nanotubes
• Dissipative Two-Fluid Models
• Dynamics effects in gradient theory for fluid mixtures
• Thermodynamically Consistent Hydrodynamic Models of Multi-Component Fluid Flows
• Dynamical pressure for fluid mixtures with several temperatures
• On the number of droplets in aerosols