3-dimensional fluid dynamic equations can be found in the excellent text by Anderson [7] where viscous forces and thermal conduction are included. However, as this
present text deals with fluid motion where shocks occur, the neglect of friction and
thermal conduction in the neighbourhood of the shock must be included and,
accordingly, an artificial viscosity is introduced, as we shall see in due course, into
the set of difference equations that approximate the differential equations for
the flow.
1.2 Eulerian and Lagrangian Form of the Equations
The equations governing a fluid or gas in motion are mathematical expressions of the
laws of conservation of mass, momentum and energy. In the case of an ideal gas, for
example, these equations are supplemented by thermodynamic equations; one called
the equation of state and the other called the caloric equation of state. The equations
describing the motion of the fluid can be written in one of two coordinate systems
[6]; one called the Eulerian system and the other called the Lagrangian system (each
named after the mathematicians, Euler and Lagrange). In the Lagrangian system we
follow the path taken by individual particles of fluid and determine the velocity,
pressure, density etc. as a function of the path taken. Accordingly, the Lagrangian
description of motion is connected with a definite particle or mass element of the
fluid; each particle is assigned a symbol x 0 , for example, indicating its position at
some initial time which is usually taken at time t ¼ 0, while at a later time t the
position of the particle is x(x 0 , t) and, clearly, x(x 0 , 0) ¼ x 0 . The motion of the particle
satisfies Newton’s second law of motion, namely; m(d
2 x/dt
2 ) ¼ F where m is the
mass of this particle of fluid lying between x 0 and x 0 + dx 0 and F is the force acting
on the particle. Suppose, for example, we wish to determine the flow of air over a
fixed surface, we could imagine a weightless soap bubble released into the air. The
path taken by this bubble as it moves over the surface provides a Lagrangian
description of the flow. To indicate the time-rate of change following the fluid
particle the material derivative, D/Dt rather than the usual d/dt has become standard
notation in fluid dynamics. However, in the present text we will adopt a slightly
different notation as we follow specific particles of fluid and we will be returning to
this aspect in Chaps. 4 and 6.
In the Eulerian description, on the other hand, one is not interested in the motion
of individual particles; instead, one is interested in, say, the velocity at points in
space. This is analogous to setting up a fixed and very fine grid throughout space and
noting, for example, the velocity, pressure or density etc. at each grid point, so in the
case of the Eulerian system we are interested in the properties of the fluid, such as,
velocity, pressure, density etc. as they pass fixed points in space.
Another good example, the source of which cannot be recalled, that distinguishes
the Lagrangian and Eulerian systems involves the acceleration of a log on a steadily
flowing river which has a section of rapids on it. By concentrating on the log one
observes that it accelerates as it enters the rapids which is the Lagrangian
2
1 Brief Outline of the Equations of Fluid Flow
present text deals with fluid motion where shocks occur, the neglect of friction and
thermal conduction in the neighbourhood of the shock must be included and,
accordingly, an artificial viscosity is introduced, as we shall see in due course, into
the set of difference equations that approximate the differential equations for
the flow.
1.2 Eulerian and Lagrangian Form of the Equations
The equations governing a fluid or gas in motion are mathematical expressions of the
laws of conservation of mass, momentum and energy. In the case of an ideal gas, for
example, these equations are supplemented by thermodynamic equations; one called
the equation of state and the other called the caloric equation of state. The equations
describing the motion of the fluid can be written in one of two coordinate systems
[6]; one called the Eulerian system and the other called the Lagrangian system (each
named after the mathematicians, Euler and Lagrange). In the Lagrangian system we
follow the path taken by individual particles of fluid and determine the velocity,
pressure, density etc. as a function of the path taken. Accordingly, the Lagrangian
description of motion is connected with a definite particle or mass element of the
fluid; each particle is assigned a symbol x 0 , for example, indicating its position at
some initial time which is usually taken at time t ¼ 0, while at a later time t the
position of the particle is x(x 0 , t) and, clearly, x(x 0 , 0) ¼ x 0 . The motion of the particle
satisfies Newton’s second law of motion, namely; m(d
2 x/dt
2 ) ¼ F where m is the
mass of this particle of fluid lying between x 0 and x 0 + dx 0 and F is the force acting
on the particle. Suppose, for example, we wish to determine the flow of air over a
fixed surface, we could imagine a weightless soap bubble released into the air. The
path taken by this bubble as it moves over the surface provides a Lagrangian
description of the flow. To indicate the time-rate of change following the fluid
particle the material derivative, D/Dt rather than the usual d/dt has become standard
notation in fluid dynamics. However, in the present text we will adopt a slightly
different notation as we follow specific particles of fluid and we will be returning to
this aspect in Chaps. 4 and 6.
In the Eulerian description, on the other hand, one is not interested in the motion
of individual particles; instead, one is interested in, say, the velocity at points in
space. This is analogous to setting up a fixed and very fine grid throughout space and
noting, for example, the velocity, pressure or density etc. at each grid point, so in the
case of the Eulerian system we are interested in the properties of the fluid, such as,
velocity, pressure, density etc. as they pass fixed points in space.
Another good example, the source of which cannot be recalled, that distinguishes
the Lagrangian and Eulerian systems involves the acceleration of a log on a steadily
flowing river which has a section of rapids on it. By concentrating on the log one
observes that it accelerates as it enters the rapids which is the Lagrangian
2
1 Brief Outline of the Equations of Fluid Flow