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For completion, one must make hypotheses on the forms of τ and p, that is, one needs a constitutive law for the stress tensor which can be obtained for specific fluid families and on the pressure. Some of these hypotheses lead to the Euler equations (fluid dynamics) , other ones lead to the Navier–Stokes equations.
The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
This constitutive equation is also called the Newton law of viscosity. The total stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} can always be decomposed as the sum of the isotropic stress tensor and the deviatoric stress tensor ( σ ′ {\displaystyle {\boldsymbol {\sigma }}'} ):
The definition of the Gibbs function is = + where H is the enthalpy defined by: = +. Taking differentials of each definition to find dH and dG, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes): = where S is the entropy, V is volume, (minus sign due to reversibility, in which dU = 0: work other than pressure-volume may be done and is equal ...
The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often be modeled by considering a substance distributed throughout some region of space.
According to van der Waals, the theorem of corresponding states (or principle/law of corresponding states) indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree.
The task is to solve the combined set of differential-algebraic (DAE) equations, instead of just the ordinary differential equations (ODE) of Newton's second law. This problem was studied in detail by Joseph Louis Lagrange , who laid out most of the methods for solving it. [ 1 ]
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations , though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation .