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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 is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state .
This explains the duality in Darcy's law as a governing equation and a defining equation for absolute permeability. The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equations in classical mechanics exposed to establishing ...
The constitutive relation is expressed as a linear first-order differential equation: = + ˙ This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain.
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 }}'} ):
In one dimension, the constitutive equation of the Herschel-Bulkley model after the yield stress has been reached can be written in the form: [3] [4] ˙ =, < = + ˙, where is the shear stress [Pa], the yield stress [Pa], the consistency index [Pa s], ˙ the shear rate [s], and the flow index [dimensionless].
where is the hydrostatic pressure in addition to the atmospheric one, is the volume at atmospheric pressure, is the volume under additional pressure , and , are experimentally determined parameters. A very detailed historical study on the Tait equation with the physical interpretation of the two parameters A {\displaystyle A} and Π ...