Two-dimensional Boundary Layer along a Flat Plate
Introduction: We noted from dimensional considerations that the thickness of the boundary layer for the boundary layer along a flat plate it is possible to devise the same form of relationship by momentum arguments alone.
Consider a thin flat plate, immersed in a uniform steady stream of viscous fluid, whose undisturbed velocity uo is perpendicular to the sharp leading edge and parallel to the plate surface. As any given volume of fluid sweeps over the plate from the leading edge, the effect of the viscous retardation due to the plate surface extends progressively into the fluid column. For immediately downstream of the leading edge only those elements immediately adjacent to the solid surface are perceptibly retarded by it; but as the fluid moves on, these slower-moving elements exert a viscous drag on their neighbors in the stream further from the plate surface, and the frictional retardation is propagated into the fluid by a process akin to diffusion.
The stratum of fluid, influenced by the length x of the plate, loses some fraction, say , of its original average velocity on entering the boundary layer, ie the average velocity in this region is . The loss in momentum per second in the fluid is thus
By Newton's 2nd Law of Motion, this rate of change of momentum must be equal to the frictional drag force at the plate surface, the so-called skin friction, which by Newton's law of viscosity acting on an area of plate of length x and unit width is Where is the velocity gradient at the plate surface, averaged over the distance x.
The assumption that the velocity profile in the boundary layer retains the same form everywhere implies a constant ratio between uo and the average velocity in the boundary layer; i.e.
Where ,a Reynolds number based on x. A rough estimate of can be made. If we assume that the velocity profile is parabolic (as for viscous, streamline flow between parallel plates) then the average velocity in the boundary layer
And k1 = 3. K2 is greater than 1 but probably less than 10. I.e. lies between 2 and 5.5, say then
E.g. for air flowing at 10m/s, with m2/s, at 10cm from leading edge
E.g. for oil flowing at 0.3m/s, with m2/s, at 15cm from leading edge