# 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.