Reynolds transport theorem

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Reynolds Transport Theorem

Reynolds transport theorem is a fundamental theorem used in formulating the basic laws of fluid dynamics. The basic laws are the continuity equation, the momentum equations and the energy equation. In physics and engineering, these laws are also known, respectively, as the law of conservation of mass, Newton's second law, and the laws of thermodynamics.

Imagine a system with a control volume and a control surface. Reynolds' transport theorem states that the rate of change of an extensive property N within the system is equal to the rate of change of N within the control volume and the net rate of change of N through the control surface. For an example, the law of conservation of mass states that rate of change of the property, mass, is equal to the sum of the rate of accumulation of mass within a control volume and the net rate of flow of mass across the control surface. Given that mass is indestructible and a system has fixed mass, the rate of change of mass is zero.

The differential forms of these equations with additional assumptions are commonly known as the Navier-Stokes equations. The additional assumptions are the Newton's viscosity law and the Fourier's conduction law.

1 General Form
2 Momentum Formulation
3 Energy Formulation
4 References

[edit] General Form

The Reynolds transport theorem refers to any extensive property, N, of the fluid in a particular control volume. It is expressed in terms of a substantive derivative on the left-hand side.

$\frac{DN_{sys}}{Dt} = \int_{c.v.}^{} \frac{\partial}{\partial t} (\rho \eta) dV + \int_{c.s.}^{} \rho \eta \overrightarrow{\upsilon_b}\cdot \widehat{n} dA+\int_{c.s.}^{} \rho \eta \overrightarrow{\upsilon_r}\cdot \widehat{n} dA$ ,

where η is N per unit mass, t is time, c.v. refers to the control volume, c.s. refers to the control surface, ρ is the fluid density, V is the volume, $υ b$ is the velocity of the boundary of the control volume (the control surface), $υ r$ is the velocity of the fluid with respect to the control surface, n is the outward pointing normal vector on the control surface, and A is the area.

[edit] Momentum Formulation

The momentum equation is found by substituting momentum in for N. From this, η is found to be velocity. The time rate of change of momentum (now the left hand side of the equation) is, from Newton's second law, equal to the net force. Note that this is a vector equation.

$\sum_{} \overrightarrow{F}=\int_{c.v.}^{}\frac{\partial}{\partial t}(\rho \overrightarrow{\upsilon}) dV + \int_{c.s.}^{} \rho \overrightarrow{\upsilon}(\overrightarrow{\upsilon_b} \cdot \widehat{n})dA + \int_{c.s.}^{} \rho \overrightarrow{\upsilon}(\overrightarrow{\upsilon_r} \cdot \widehat{n})dA$ ,

where F is force, $υ$ is the velocity of fluid in a coordinate system attached to the control surface, and all other variables are defined as in the general formulation.

[edit] Energy Formulation

The energy equation is found by substituting energy in for N. From this, η is found to be energy per unit mass.

$\dot{Q}-\sum{} \dot{W}=\int_{c.v.}^{} \frac{\partial}{\partial t} \left [ \rho \left ( \frac{\upsilon^2}{2}+gz+\tilde{u}\right ) \right ] dV + \int_{c.s.}^{} \left [\frac{\upsilon^2}{2}+gz+\tilde{u}+\frac{p}{\rho}\ \right]\rho \overrightarrow{\upsilon_b}\cdot\widehat{n}dA + \int_{c.s.}^{} \left [\frac{\upsilon^2}{2}+gz+\tilde{u}+\frac{p}{\rho}\ \right]\rho \overrightarrow{\upsilon_r}\cdot\widehat{n}dA$

where Q is the heat transfer into the control volume, W is the work done by the system, g is the acceleration due to gravity, z is the vertical distance from an arbitrary datum, $\tilde{u}$ is the specific internal energy of the fluid, p is the pressure and all other variables are defined as in the general formulation.