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Boundary Layer Theory

The study of the thin layer of fluid immediately adjacent to a solid surface where velocity changes significantly due to viscosity.
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The statement of the theorem

Consider a steady, incompressible flow of fluid with density ρ\rho and dynamic viscosity μ\mu over a surface defined by y=0y=0. We adopt a coordinate system (x,y,z)(x, y, z) where xx is the streamwise direction and yy is the normal direction. The flow field u⃗=(u,v,w)\vec{u} = (u, v, w) must satisfy the continuity equation and the full Navier-Stokes equations:\n\n∂u∂x+∂v∂y+∂w∂z=0\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \nρ(u∂u∂x+v∂u∂y+w∂u∂z)=−∂p∂x+μ(∂2u∂x2+∂2u∂y2+∂2u∂z2)+fx\rho (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + f_x \n(and similar equations for vv and ww)\n\nBoundary Layer Theory posits that if the characteristic length scale of the boundary layer, δ(x)\delta(x), is much smaller than the characteristic length scale of the external flow, Le(x)L_e(x), i.e., δ/Le≪1\delta/L_e \ll 1, then the governing equations can be reduced via an asymptotic expansion. Assuming ∂/∂x≫∂/∂y\partial / \partial x \gg \partial / \partial y and ∂/∂z≪∂/∂x\partial / \partial z \ll \partial / \partial x, the xx-momentum equation simplifies to the boundary layer equation:\n\nρ(u∂u∂x+v∂u∂y)=−∂pe∂x+μ(∂2u∂y2)+fx\rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p_e}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial y^2} \right) + f_x \n\nSubject to the boundary conditions:\n1. No-slip condition at the wall: u(x,0)=0u(x, 0) = 0 and v(x,0)=0v(x, 0) = 0.\n2. Matching condition at the edge of the boundary layer: lim⁡y→δ(x)+(u,v)=(Ue(x),0)\lim_{y \to \delta(x)^+} (u, v) = (U_e(x), 0), where (Ue(x),0)(U_e(x), 0) is the velocity profile of the external, inviscid flow.
Source: Wikipedia