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Prandtl's Boundary Layer Equations

The simplified set of equations derived by Prandtl, assuming steady, incompressible flow. The key momentum equation is ∂u∂x+∂v∂y=0\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 and ρ(u∂u∂x+v∂u∂y)=−∂p∂x+μ∂2u∂y2\rho (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}) = -\frac{\partial p}{\partial x} + \mu \frac{\partial^2 u}{\partial y^2}.
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The statement of the theorem

Assuming steady, incompressible flow (ρ=constant\rho = constant) and neglecting the yy-momentum equation, Prandtl's boundary layer equations are:\n\n1. **Continuity Equation (2D):**\n∂u∂x+∂v∂y=0 \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \n2. **xx-Momentum Equation:**\nρ(u∂u∂x+v∂u∂y)=−∂p∂x+μ∂2u∂y2 \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu \frac{\partial^2 u}{\partial y^2} \nThese equations govern the velocity components (u,v)(u, v) within the boundary layer, subject to the no-slip condition at y=0y=0.
Source: Wikipedia