## Laminar Boundary Layer Theory

**Introduction: **The boundary layer flow and heat transfer of stretching flat plates or cylinders are of practical importance in fiber technology and extrusion processes, and of theoretical interest. The production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. Examples are numerous and they include the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film in condensation processes, paper production, glass blowing, metal spinning and drawing plastic films, and polymer extrusion.

The quality of the final product depends on the rate of heat transfer at the stretching surface. Sakiadis (1961) was the first to consider the boundary layer flow on a moving continuous solid surface. Crane (1970) extended this concept to a stretching sheet with linearly varying surface speed and presented an exact analytical solution for the steady two-dimensional stretching of a surface in a quiescent fluid. Since then, many authors have considered various aspects of this problem and obtained similarity solutions.

The boundary layer flow due to a stretching vertical surface in a quiescent viscous and incompressible fluid when the buoyancy forces are taken into consideration have been considered in the papers by Daskalakis (1993), Chen (1998, 2000), Lin and Chen (1998), Ali (2004), Partha et al.(2005), and Ishak et al. (2007, 2008a). Lin and Shih (1980, 1981) considered the laminar boundary layer and heat transfer along horizontally and vertically moving cylinders with constant velocity and found that the similarity solutions could not be obtained due to the curvature effect of the cylinder. In the present paper, we show that the similarity solutions may be obtained by assuming that the cylinder is stretched with linear velocity in the axial direction. The present study may be regarded as the extension of the papers by Grubka and Bobba (1985) and Ali (1994), from a stretching sheet to a stretching cylinder. Thus, the results obtained can be compared with those of Grubka and Bobba (1985) and Ali (1994), if the curvature parameter is neglected.

This is a course in the theory of the thin boundary layer that forms when ﬂuid ﬂows past a solid body at high Reynolds number

**Section 1**: The basic equations of ﬂuid dynamics. We start by deriving the basic equations of viscous ﬂuid ﬂow. Most of the material in this section should be revision (but I’ve assumed no prior knowledge, in case it isn’t).

**Section 2:**Introducing the boundary layer. Using the equations derived in Section 1, we consider the relative importance of inertial and viscous forces in a ﬂuid. This is expressed by a dimensionless quantity called the Reynolds number. We consider ﬂow past a solid obstacle at high Reynolds number, and give simple arguments to show that a thin boundary layer must form, over which the ﬂuid velocity decreases precipitously from its bulk ﬂow value to zero (“no slip”) at the solid surface.

**Section 3:**The boundary layer equations. Here we put the qualitative discussion of Section 2 on a more formal footing by deriving the equations to be used in the analysis of various boundary layer phenomena in section 4.

**Section 4:**Exact solutions of the boundary layer equations. We start with the simplest exact solution of the equations derived in Section 3: the boundary layer that forms when ﬂuid ﬂows past a ﬂat plate at zero angle of incidence. Using this example, we motivate the existence of self-similar solutions to the boundary layer equations and apply this idea to more general ﬂow geometries. We discuss ﬂow round blunt obstacles and show that the boundary layer can often separate from the surface, causing a turbulent wake that leads to large drag. We discuss techniques for eliminating this undesirable phenomenon, such as the careful streamlining of aero foils. Finally, we apply boundary layer theory to ﬂuid jets.

**Section 5:**Compressible ﬂows. In sections 1 to 4 we made the simplifying assumption that the ﬂuid is incompressible. We now discuss the conditions under which this assumption breaks down, and present the equations necessary to describe the way in which the thermodynamics of compression interacts with the mechanics of ﬂow.