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# Boundary Layer Governing Equations.

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