Continuum
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Corresponding Wikipedia article: Continuum mechanics
Continuum
The continuum is a mechanical system described by the physical and scalar fields
\(\rho\) is a density of mass;
\(\mathbf{\overrightarrow{v}}\) is the flow velocity;
\(P\) is a pressure of the continuum;
\(U\) is a volumetric density of a potential energy of any external forces;
\(\mathbf{\overrightarrow{f}}\) is a force expressed by the acceleration, which have any nature except the continuum pressure or the potential energy.
The pressure (static pressure) is a scalar field, which is completely analogous to the potential energy field, but it is a property of the continuum itself. As in a case of energy, the pressure gradient affects the continuum dynamics.
The continuity equation is based on the law of conservation of mass-energy: \[\frac{\partial\rho}{\partial t}+div\;\rho\mathbf{\overrightarrow{v}}=0\tag{1}\] The second equation is the Euler’s equation based on the Newton's second law: \[\rho\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\tag{2}\] The velocity derivative is a material derivative, because of the space-dependent velocity: \[\frac{\mathrm{d}\mathbf{\overrightarrow{v}}}{\mathrm{d}t}=\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}\tag{3}\] The Euler’s equation with the (3) is: \[\rho\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}+\rho(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\tag{4}\] The steady state forms of equations (1) and (4), where \(\frac{\partial\rho}{\partial t}=\frac{\partial\mathbf{\overrightarrow{v}}}{\partial t}=0\), are: \[div\;\rho\mathbf{\overrightarrow{v}}=0\tag{5}\] \[\rho(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{v}}+grad(P+U)=\rho\mathbf{\overrightarrow{f}}\tag{6}\] For an incompressible flow (\(\rho=const\)) the equation (5) is simplified: \[div\;\mathbf{\overrightarrow{v}}=0\tag{7}\] The equation (6) is a sum of the dynamic pressure force, the pressure gradient force, the conservative forces, and other forces \(\mathbf{\overrightarrow{f}}\). The viscous friction force also relates to other forces, which is zero within the ideal or superfluid environments.
When other forces are zero, the pressure within a closed system tends to be equalized over the entire volume, according to the Pascal's law.
Barotropic flow
A barotropic compressible continuum has the property: \[P+U=f(\rho)\] A result is the gradients collinearity: \[grad(P+U)=\frac{\mathrm{d}f(\rho)}{\mathrm{d}\rho}grad\;\rho\tag{8}\] The quadratic dependence \(f(\rho)\) simplifies the Euler's equation, making the density a common factor: \[P+U=k\rho^2\;\;\;\;\;\;\;k=const\] \[grad(P+U)=2k\rho\;grad\;\rho=2\rho\;grad\frac{P+U}{\rho}\]
Bernoulli's law
The one-dimensional steady-state motion equation, which is applicable to a continuum where all parameters are a function of the same single coordinate, is:
\[\rho v\frac{\mathrm{d}v}{\mathrm{d}x}+\frac{\mathrm{d}(P+U)}{\mathrm{d}x}=\rho f\tag{9}\]
The integration of (9) at \(f=0\) gives the well-known Bernoulli's law, as a sum of the dynamic (velocity) pressure, the static (usual) pressure, and the potential energy density \(U\):
\[\frac{\rho v^2}{2}+P+U=const\tag{10}\]
A form of this law for an incompressible fluid within the gravitational field with a potential energy density \(\rho gh\) is more known:
\[\frac{\rho v^2}{2}+P+\rho gh=const\tag{11}\]
\(g\) is a gravitational acceleration;
\(h\) is a hight.
The weight of the stationary incompressible fluid column of height \(H\) is \(\rho gH\), whereby the communicating vessels law exists also.
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