The boundary-layer approximation for laminar flows with chemical reactions

THE BOUNDARY-LAYER APPROXIMATION FOR LAMINAR FLOWS WITH CHEMICAL REACTIONS 

Let us consider a steady flow for which a Cartesian coordinate system (x, y, z) can be established such that -h x is the principal flow direction and all flow properties are independent of z. In this two-dimensional (x, y) flow, it will further be assumed that except in a layer extending parallel to the principal flow direction, all flow properties vary so slowly that transport effects are negligibly small. For convenience, the viscous, diffusive, and heat-conducting layer will be placed in the vicinity of the plane y = 0 (which, for example, may represent a stationary flat plate, or may divide two parallel 

FIGURE 12.1. Schematic illustration of the profile of a representative flow property in a boundary layer. 

The boundary-layer approximation states that the orders of magnitude of the gradients are 

Equations (5)-(8) may easily be used to obtain the estimates of the orders of magnitude of the terms in equations (2)-(4) indicated beneath these equations. Since S/l 1, the indicated orders of magnitude imply that the last term in equation (2) is large compared with the other terms on the right-hand side, the last term in equation (3) is small compared with the other terms on the right-hand side, and the third term in L(a) [equation (4)] is small compared with the fourth (last) term. Equation (2) then shows that the dominant (last) viscous term will be of the same order as the convective terms appearing on the left-hand side only if the Reynolds number. 

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