Bifurcation Problems in Nonlinear Elasticity by R.W. Dickey

By R.W. Dickey

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Extra resources for Bifurcation Problems in Nonlinear Elasticity

Sample text

As mentioned before these phenomenological definitions differ slightly from the usual definitions of the Navier-Stokes transport coefficients. 56c) These equations were postulated long before the development of linear irreversible thermodynamics. The first equation is known as Fourier's law of heat conduction. It gives the definition of the thermal conductivity λ. 3). It gives a definition of the shear viscosity coefficient η. The third equation is a more recent development. It defines the bulk viscosity coefficient ηV.

These 24 Linear Irreversible Thermodynamics equations are known collectively as linear constitutive equations. When they are substituted into the conservation equations they yield the Navier-Stokes equations of hydrodynamics. The conservation equations relate thermodynamic fluxes and forces. They form a system of equations in two unknown fields - the force fields and the flux fields. The constitutive equations relate the forces and the fluxes. By combining the two systems of equations we can derive the Navier-Stokes equations which in their usual form give us a closed system of equations for the thermodynamic forces.

The system is maintained at a fixed thermodynamic state (say temperature) by placing it in contact with a reservoir, with which it exchanges energy (heat) in such a manner as to keep the temperature of the system of interest fixed. The heat capacity of the reservoir must be much larger than that of the system, so that the heat exchanged from the reservoir does not affect the reservoir temperature. Classical mechanics is an awkward vehicle for describing this type of system. The only way that thermodynamic systems can be treated in Newtonian or Hamiltonian mechanics is by explicitly modelling the system, the reservoir and the exchange processes.

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