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Denote the velocity of a fluid ##\vec u## and define a potential ##\vec u = -\nabla \phi##. Let ##\hat n ## be an outward-oriented surface normal to a solid boundary. I would express no penetration at the boundary as $$ u \cdot \hat n = 0 \implies \nabla \phi \cdot \hat n = 0.$$

However, the text writes $$\frac{\partial \phi}{\partial n} = 0$$ where ##n## is the direction of the outward-oriented normal. Can someone explain this result?

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# A Fluids and the no-penetration condition

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