Continuity equation in Lagrangian coordinates

In summary, the continuity equation in Eulerian coordinates, which describes the conservation of mass in a fluid, can be derived from the Lagrangian form by considering the incremental change in mass and the specific volume. The final form of the continuity equation in Lagrangian coordinates involves the initial density, the current density, and the Lagrangian coordinates at the initial and current moments.
  • #1
c0der
54
0
From the Eulerian form of the continuity equation, where x is the Eulerian coordinate:

[itex] \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0[/itex]

The incremental change in mass is, where m is the Lagrangian coordinate:

[itex] dm = \rho dx[/itex]

The specific volume is:

[itex] V = \frac{1}{\rho}[/itex]

How does one get the final form of the continuity equation in Lagrangian coordinates as follows:

[itex]\frac{\partial V}{\partial t} = \frac{\partial u}{\partial m} [/itex]
 
  • #3
$$\rho (t,\xi)\mathrm{det}\,\Big(\frac{\partial \xi}{\partial\hat \xi}\Big)=\hat\rho(\hat\xi),$$ here ##\hat \xi## are the Lagrangian coordinates in the initial moment ##t=0## and ##\xi=\xi(t,\hat\xi)## are the current Lagrangian coordinates; ##\hat\rho## is the initial desity and ##\rho (t,\xi)=\hat\rho(\xi(t, \hat\xi))## is the current density
 

1. What is the Continuity Equation in Lagrangian Coordinates?

The continuity equation in Lagrangian coordinates is a mathematical expression that describes the conservation of mass in a fluid flow. It relates the rate of change of mass in a specific fluid parcel to the flow velocity and the density of the fluid.

2. How is the Continuity Equation used in Fluid Mechanics?

The continuity equation is a fundamental concept in fluid mechanics and is used to determine the behavior of fluids in motion. It helps to understand how the flow of a fluid is affected by changes in the flow velocity and density.

3. What is the difference between the Continuity Equation in Lagrangian and Eulerian Coordinates?

The continuity equation in Lagrangian coordinates follows a specific fluid parcel, while the continuity equation in Eulerian coordinates describes the flow at a fixed point in space. In Lagrangian coordinates, the density of the fluid parcel remains constant, while in Eulerian coordinates, the density can change with time and space.

4. What is the significance of the Continuity Equation in Lagrangian Coordinates in Real-World Applications?

The continuity equation in Lagrangian coordinates is essential in many real-world applications, such as weather forecasting, oceanography, and aerodynamics. It allows scientists and engineers to predict the behavior of fluids in various scenarios, which is crucial for designing efficient and safe systems.

5. How is the Continuity Equation derived in Lagrangian Coordinates?

The continuity equation in Lagrangian coordinates is derived from the conservation of mass principle, which states that mass cannot be created or destroyed. By considering a small fluid parcel and applying the principle, the continuity equation can be derived in Lagrangian coordinates.

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