Laminar boundary layer problem

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5.1 Key points of boundary layer theory 5.1.1 Introduction of the problem As mentioned earlier, Re∝inertial force/viscous force. Only when Re is greater than the viscous force, but in the extremely thin flow layer right at the solid-fluid boundary where the inertial force≈viscous force, can the drag problem associated with high Re numbers be satisfactorily resolved. Later researchers extended the concept of the flow boundary layer proposed by Prandtl to the heat transfer boundary layer and mass transfer boundary layer in flow systems, thereby determining the rates of heat and mass transfer as well as understanding the relevant influencing factors. Others have also studied chemical reactions in boundary layers, addressing some practical problems. Therefore, boundary layer theory is considered the cornerstone of modern fluid mechanics. 5.1.2 Moving boundary layer (velocity boundary layer) Taking flat-plate flow as an example, for a one-dimensional flow in the x-direction, the flow in the y-direction perpendicular to the wall can be divided into two regions with different properties: (1) yδ (outer main flow region): the influence of the wall is weak, the normal velocity remains essentially constant, and du/dy≈0. Therefore, the viscous force can be ignored (i.e., normal momentum transfer is ignored). When treated as an ideal fluid, the Euler equation applies. These two regions meet at the outer edge of the boundary layer. Since the flow within the layer approaches the external flow gradually rather than abruptly, it is conventionally agreed that at the outer edge of the flow boundary layer (i.e., at y = δ), ux = 0.99u∞, where δ is the thickness of the flow boundary layer, and δ = δ(x). 5.1.3 Heat transfer boundary layer (temperature boundary layer): When a fluid flows past a solid wall with a temperature different from its own (as shown in the figure, in the region where x > x0), a flow boundary layer is formed on the wall. At the same time, a temperature distribution is established due to heat transfer, and this region can be divided into two areas: (1) yδt (the region outside the layer): the normal temperature gradient dt/dy ≈ 0, so normal heat conduction can be ignored. It is generally agreed that at the outer edge of the heat transfer boundary layer (i.e., at y = δt), ts – t = 0.99(ts – t0) ≈ ts – t0/δt – where δt is the thickness of the temperature boundary layer, and δt = f(x) ； ts——wall temperature ； t0 —— The temperature in the region outside the thermal boundary layer (the main fluid). Pr＝ν/α∝momentum transfer capacity/heat transfer capacity. Under normal conditions, Pr>1 (for liquids), δ>δt; Pr≈1 (for gases), δ≈δt. Pr