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In developing a mathematical theory of boundary layers, the first step is to show the existence, as the Reynolds number R tends to infinity, or the kinematic viscosity
tends to zero, of a limiting form of the equations of motion, different from that obtained by putting
in the first place. A solution of these limiting equations may then reasonably be expected to describe approximately the flow in a laminar boundary layer for which R is large but not infinite. This is the basis of the classical theory of laminar boundary layers.
The full equation of motion for for a two-dimensional flow are:
|
(1) |
|
(2) |
|
(3) |
where the x and y variables are, respectively, the horizontal and vertical coordinates, u and v are, respectively, the horizontal and vertical fluid velocities and p is the fluid pressure. A wall is located in the plane y = 0. We consider non-dimensional variables
|
(4) |
|
(5) |
|
(6) |
|
(7) |
|
(8) |
|
(9) |
where L is the horizontal length scale,
is the boundary layer thickness at x = L, which is unknown. We will obtain an estimate for it in terms of the Reynolds number R. U is the flow velocity, which is aligned in the x-direction parallel to the solid boundary. The non-dimensional form of the governing equations is:
|
(10) |
|
(11) |
|
(12) |
where the Reynolds number for this problem is
|
(13) |
Inside the boundary layer, viscous forces balance inertia and pressure gradient forces. In other words, inertia and viscous forces are of the same order, so
|
(14) |
Now we drop the primes from the non-dimensional governing equations and with equation (2.14) we have
|
(15) |
|
(16) |
|
(17) |
In the limit
,
the equations above reduce to:
|
(18) |
|
(19) |
|
(20) |
Notice that according to equation (2.19), the pressure is constant across the boundary layer. In terms of dimensional variables, the system of equations above assume the form:
|
(21) |
|
(22) |
|
(23) |
To solve the system of equations above we need to specify initial and boundary conditions.
Next: Steady State Laminar Boundary
Up: No Title
Previous: Introduction.
Karl P Burr
2003-03-12