Page 302 - Intro to Tensor Calculus

296

Boundary Conditions

Fluids problems can be classified as internal flows or external flows. An example of an internal flow
problem is that of fluid moving through a converging-diverging nozzle. An example of an external flow
problem is fluid flow around the boundary of an aircraft. For both types of problems there is some sort of
boundary which influences how the fluid behaves. In these types of problems the fluid is assumed to adhere
r
to a boundary. Let ~ b denote the position vector to a point on a boundary associated with a moving fluid,
r
r
and let ~ denote the position vector to a general point in the fluid. Define ~v(~) as the velocity of the fluid at
r
r
the point ~ and define ~v(~ b ) as the known velocity of the boundary. The boundary might be moving within
the fluid or it could be fixed in which case the velocity at all points on the boundary is zero. We define the
boundary condition associated with a moving fluid as an adherence boundary condition.

Definition: (Adherence Boundary Condition)
An adherence boundary condition associated with a fluid in motion
is defined as the limit lim ~v(~r)= ~v(~r b )where ~r b is the position
r
r ~→~ b
vector to a point on the boundary.

Sometimes, when no finite boundaries are present, it is necessary to impose conditions on the components
of the velocity far from the origin. Such conditions are referred to as boundary conditions at infinity.

Summary and Additional Considerations
Throughout the development of the basic equations of continuum mechanics we have neglected ther-
modynamical and electromagnetic effects. The inclusion of thermodynamics and electromagnetic fields adds
additional terms to the basic equations of a continua. These basic equations describing a continuum are:

Conservation of mass
The conservation of mass is a statement that the total mass of a body is unchanged during its motion.
This is represented by the continuity equation
∂% k D%
~
+(%v ) ,k =0 or + %∇· V =0
∂t Dt
k
where % is the mass density and v is the velocity.
Conservation of linear momentum
The conservation of linear momentum requires that the time rate of change of linear momentum equal
the resultant of all forces acting on the body. In symbols, we write

Z Z Z n
D i i i X i
%v dτ = F (s) i dS + %F (b) dτ + F (α) (2.5.43)
n
Dt
V S V α=1
∂x k v is the material derivative, F
where Dv i = ∂v i + ∂v i k i are the surface forces per unit area, F i are the
Dt ∂t (s) (b)
body forces per unit mass and F i represents isolated external forces. Here S represents the surface and
(α)
V represents the volume of the control volume. The right-hand side of this conservation law represents the
resultant force coming from the consideration of all surface forces and body forces acting on a control volume.