4.3 Lagrangian Equations in Plane Geometry
with Artificial Viscosity
Let us now consider an alternative derivation of the equations of fluid flow in
Lagrangian form with artificial viscosity included. Here, we will adopt the Lagrangian notation used by Zel’dovich and Raizer [3] and in the article by C. F. Sprague [2].
4.3.1 Continuity Equation
Let us consider a one-dimensional model of the fluid flow in, say, the x-direction. We
will concentrate our attention on a small infinitesimal mass of fluid moving with the
flow at some instant in time which we take to be at t ¼ 0. At this instance let us
assume that this small element of fluid is located at x 0 , has length dx 0 , cross-sectional
A and density ρ(x 0 , 0). The mass of this fluid element is ρ(x 0 , 0)Adx 0 . As this fluid
element moves its density and volume may change but its mass remains constant.
After a time t the fluid element is located at x ¼ x(x 0 , t) with a length, dx, where
dx ¼ x(x 0 + dx 0 , t) À x(x 0 , t) and with density ρ(x 0 , t). Conservation of mass implies
that
ρ x 0 , 0
ð
ÞAdx 0 ¼ ρ x 0 , t
ð
ÞAdx
Hence,
1
1
ρ x 0 , t
ð
Þ
¼
1
ρ x 0 , 0
ð
Þ
∂x
∂x 0
:
Writing the left-hand side in terms of the specific volume υ rather than the density
ρ we have
υ x 0 , t
ð
Þ ¼
1
ρ x 0 , 0
ð
Þ
∂x
∂x 0
,
differentiating with respect to time yields the following continuity equation,
1 Partial derivatives are used here to indicate the changes in position and time of specific particles;
nonetheless, it should be understood that these partial derivatives imply that we are in fact following
the path taken by specific particles of fluid according to the Lagrangian description.
4.3 Lagrangian Equations in Plane Geometry with Artificial Viscosity
133
with Artificial Viscosity
Let us now consider an alternative derivation of the equations of fluid flow in
Lagrangian form with artificial viscosity included. Here, we will adopt the Lagrangian notation used by Zel’dovich and Raizer [3] and in the article by C. F. Sprague [2].
4.3.1 Continuity Equation
Let us consider a one-dimensional model of the fluid flow in, say, the x-direction. We
will concentrate our attention on a small infinitesimal mass of fluid moving with the
flow at some instant in time which we take to be at t ¼ 0. At this instance let us
assume that this small element of fluid is located at x 0 , has length dx 0 , cross-sectional
A and density ρ(x 0 , 0). The mass of this fluid element is ρ(x 0 , 0)Adx 0 . As this fluid
element moves its density and volume may change but its mass remains constant.
After a time t the fluid element is located at x ¼ x(x 0 , t) with a length, dx, where
dx ¼ x(x 0 + dx 0 , t) À x(x 0 , t) and with density ρ(x 0 , t). Conservation of mass implies
that
ρ x 0 , 0
ð
ÞAdx 0 ¼ ρ x 0 , t
ð
ÞAdx
Hence,
1
1
ρ x 0 , t
ð
Þ
¼
1
ρ x 0 , 0
ð
Þ
∂x
∂x 0
:
Writing the left-hand side in terms of the specific volume υ rather than the density
ρ we have
υ x 0 , t
ð
Þ ¼
1
ρ x 0 , 0
ð
Þ
∂x
∂x 0
,
differentiating with respect to time yields the following continuity equation,
1 Partial derivatives are used here to indicate the changes in position and time of specific particles;
nonetheless, it should be understood that these partial derivatives imply that we are in fact following
the path taken by specific particles of fluid according to the Lagrangian description.
4.3 Lagrangian Equations in Plane Geometry with Artificial Viscosity
133