The behavior of flows away from boundaries can be understood with simplified descriptions, where we can neglect the effects of fluid viscosity.  Where needed, such as to explain lift generation, we use some overall effect such as the "Kutta condition" instead of worrying in detail about flow behavior near surfaces.  Now we consider what really happens at the boundaries of a moving fluid. We can think of two kinds of boundaries:  solid walls, and free boundaries.

At a solid wall, the relative velocity between the fluid and the wall must come to zero: this is called the "no-slip condition". Basically, this occurs because fluid molecules, moving along with random thermal motion plus the mean fluid velocity, hit the solid-wall molecules, and eventually, the relative momentum along the flow direction is all lost, transferred to the solid surface. As these molecules bounce back into the stream, like cars swerving out of a stopped right lane, they slow down the fluid in the neighboring layers. Thus, a "boundary layer" is created, where the stream velocity increases rapidly from zero at the wall, to the "external flow" velocity.

Consider the highway analogy and see what a fast-moving car would do as the (rather impatient) driver sees the right lane stopping, and cars swerving out into the second lane: s(he) will move to the left to keep going at high speed, along a path where s(he) sees minimum obstruction ahead.  Likewise, if we take a streamline in the "external flow" , starting very close to the surface at the leading edge, and follow it around an airfoil and see where it goes, we will see that it moves further and further away from the surface, as the region of slow-moving fluid thickens. This is the "growth" of the boundary layer. As distance downstream increases, the boundary layer gets thicker. If we place some device downstream, to "pull" the fluid, such as a suction device which decreases the pressure downstream, the flow near the surface speeds up, so the boundary layer gets thinner. Conversely, if we increase the downstream pressure, the slow-moving region gets thicker and thicker, and the fast-moving streamline of the external flow moves further and further away from the surface. Eventually, if we increase the downstream pressure enough, substantial portions of the boundary layer may just stop. The external flow will then move sharply away from the surface. This is "flow separation".

In the following sections, we'll quickly see:

1. How to obtain simplified equations describing the boundary layer, starting from the conservation equations of fluid mechanics.
2. How to solve these equations and obtain boundary layer properties.
3. Simplified engineering methods to obtain "overall effects" of the boundary layer ("Integral Methods")
4. The phenomenon of "transition", where the boundary layer behavior changes from being smooth and laminar, to chaotic and turbulent.
5. Effect of turbulence on drag
6. How to describe turbulence using the conservation equations
7. How to estimate the drag and other effects of turbulence.
8. How to control separation and transition.
9. What happens when the external flow is fast enough to experience "compressibility".

We start from Newton's 2nd Law of Motion, and obtain the equation describing conservation of momentum for fluid passing through a control volume. We derive this first in integral form. To obtain detailed relations between pressure, velocity and shear forces at each point, we need the differential form of the momentum equation. To get this, we must first bring the integral equation under a single type of integral, so that we can set the integrand to zero. In turn, this is accomplished by converting surface integrals to volume integrals and vice versa, using the appropriate theorems of vector analysis: the Divergence Theorem in this case. (Note: a similar transformation between line integrals and surface integrals can be obtained using Stokes' Theorem).

The differential form of the momentum equation basically says:

"Unsteady rate of change of momentum per unit volume,  at a point + convective rate of change of momentum per unit volume = pressure force + shear force + body force, per unit volume".

If the inertial forces are much greater than the body forces, we can neglect the body forces. Examples where we cannot neglect body forces are: flow of liquids (gravitational force must generally be considered); or flow of ions in a magnetic field (electromagnetic forces must be considered).

If the inertial forces are far greater than the shear forces, we can also neglect the shear force terms. This occurs at high Reynolds number,  in flows away from surfaces.

If the shear forces are neglected (whether or not the body forces are neglected), the resulting equation is called the Euler equation. The Euler equation is used in many sophisticated calculations of aerodynamics, of rotating machines, and compressible flow over wings.

If we further apply conditions of "irrotationality", we can rewrite the momentum equation, combined with the equation of mass conservation, as a "potential equation", as we have seen before in low-speed aerodynamics, and compressible aerodynamics.

Lets step back to the situation where we cannot neglect the shear forces: flows near boundaries of the fluid. These are generally thin layers, which we will call "boundary layers" (what else?? :-) In these regions, we must retain the full momentum equation including the shear force terms

We must relate the shear stress to the velocity, density etc., which are the flow variables of interest. In the case of solids, the shear stress is related to shear strain. If the relation is linear, the constant of proportionality is called the shear modulus.

In the case of fluids, there is no resistance to mere shear strain: fluids cannot resist change of shape. However, there is resistance to the rate of strain. In the ideal case of a Newtonian Fluid ( not to be confused with Newtonian Flow!!!) the relation is linear, and the constant of proportionality is called the "absolute viscosity", a property of the particular fluid in question. And example of a "non-Newtonian" fluid is blood, where the resistance is much less if the rate of strain is applied parallel to the platelets, than perpendicular to them.

So, we assume that the fluid is Newtonian. Next, we also assume that "bulk viscosity" is zero. Bulk viscosity is the constant of proportionality between normal stress (such as pressure) and the rate of normal strain. In the case of a very rapid compression, such as that in the leading edge of a shock wave, the bulk viscosity maybe significant, and is a cause of the loss of stagnation pressure through a shock. In most other situations, the rate of normal strain is far too small for us to worry about bulk viscosity.

This is called the Stokes' hypothesis.

The momentum equation with this hypothesis applied, is called the Navier-Stokes equation. It is highly nonlinear. We must seek simplified forms to use in the boundary layer. To do so, we use some facts that we observe about bounday layers:

1. The spatial rate at which properties change across a boundary layer is very large, compared to the rate at which things change along the flow direction. In other words, derivatives with respect to y are much higher than derivatives with respect to x.

2. The static pressure is constant across a boundary layer.

These two facts are used to simplify the Navier-Stokes equations into a set of "boundary layer equations". The whole character of the equation changes, from the elliptic nature of the N-S equns, to the parabolic nature of the B-L equations. It is possible to march downstream and solve a boundary layer.
 

Links to Material on Viscous Flow
 
 

Title	r	theta	Type	Author(s)
Boundary Layers	3	54	9	nk
Introduction to Viscous Flow	2	54	9	ls
Navier-Stokes equations Derivation and Solutions	2	54	1	ls
Derivation of Boundary layer equations	2	54	1	ls
Exact Solutions to the Boundary Layer equations	2	54	1	ls
Boundary Layer Profile	2	54	1	ls
Thwaites Method	2	54	1	ls
Transition	2	54	1	ls
Fluid Stresses; Stokes Relation; Stokes hypothesis	2	54	1	ls
Nondimensionalized equations for viscous flow	2	54	1	ls
Hagen-Poseuille Flow;   Axisymmetric laminar fully developed  pipe flow with pressure gradient	2	54	1	ls
Prandtl's Boundary Layer Theory	2	54	1	ls
Exact solutions of N-S eqns' Couette Flow; parallel plates	2	54	1	ls