Conservation and Component Form
In differential form, the equations are:
where
- ρ is the fluid mass density,
- u is the fluid velocity vector, with components u, v, and w,
- E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
- p is the pressure,
- denotes the tensor product, and
- being the zero vector.
These equations may be expressed in subscript notation. The second equation includes the divergence of a dyadic product, and may be clearer in subscript notation:
where the i and j subscripts label the three Cartesian components: ( x1, x2, x3 ) = ( x, y, z ) and ( u1, u2, u3 ) = ( u, v, w ). These equations may be more succinctly expressed using Einstein notation, in which matched indices imply a sum over those indices and and :
Note that the above equations are expressed in conservation form, as this format emphasizes their physical origins (and is often the most convenient form for computational fluid dynamics simulations). By subtracting the velocity times the mass conservation term, the second equation (momentum conservation), can also be expressed as:
or, in vector notation:
but this form for the momentum conservation equation obscures the direct connection between the Euler equations and Newton's second law of motion. Similarly, by subtracting the velocity times the above momentum conservation term, the third equation (energy conservation), can also be expressed as:
or
Read more about this topic: Euler Equations (fluid Dynamics)
Famous quotes containing the words conservation, component and/or form:
“A country grows in history not only because of the heroism of its troops on the field of battle, it grows also when it turns to justice and to right for the conservation of its interests.”
—Aristide Briand (1862–1932)
“... no one knows anything about a strike until he has seen it break down into its component parts of human beings.”
—Mary Heaton Vorse (1874–1966)
“During the long ages of class rule, which are just beginning to cease, only one form of sovereignty has been assigned to all men—that, namely, over all women. Upon these feeble and inferior companions all men were permitted to avenge the indignities they suffered from so many men to whom they were forced to submit.”
—Mary Putnam Jacobi (1842–1906)