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No-Slip Boundary Condition

The fundamental assumption that the fluid velocity at the solid surface is zero. Mathematically, this is expressed as u(y=0)=0u(y=0) = 0, where uu is the velocity component normal to the surface and yy is the coordinate perpendicular to the surface.
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The statement of the theorem

Let u(x,y)=(u,v)\mathbf{u}(x, y) = (u, v) be the velocity vector field in the domain Ω\Omega. On the solid surface Γ\Gamma defined by y=0y=0, the no-slip boundary condition mandates that the fluid velocity matches the wall velocity uwall\mathbf{u}_{wall}. Assuming a stationary wall, this is expressed as:\nu(x,0)=(u(x,0) v(x,0))=(0 0 0) \mathbf{u}(x, 0) = \begin{pmatrix} u(x, 0) \ v(x, 0) \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}