Visualizing Small Navier Stokes Equation (∇•ū = 0)

AI Thread Summary
The discussion clarifies that the equation ∇•ū = 0 represents the continuity equation for incompressible fluids, not a "small Navier-Stokes equation." It emphasizes that the thickness of line segments in visualizations indicates faster fluid flow. Participants question whether all scenarios would result in positive divergence, noting that directionality affects the interpretation of dilatation. The conversation highlights the importance of sign conventions in defining velocity directions. Overall, the thread focuses on understanding the implications of the continuity equation in fluid dynamics.
James Brady
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i’m trying to picture the divergence in an intentional cube and i’m getting wacky answers.
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the thickness of the line segment represent a faster fluid. wouldn’t all of these be positive divergence just in different directions?
 
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FYI, ##\nabla\cdot\vec{u}=0## is not a "small Navier-Stokes equation." It's the continuity equation for an incompressible fluid. It states that the divergence of the velocity field (i.e., the dilatation of the fluid element passing through that flow field) is zero.

I am not really sure what you are trying to do otherwise. Where's the confusion here? Is it just the sign convention? You have to pick some direction to be positive, and in your example, you've defined all velocities as positive, so in some cases you will be showing positive or negative dilatation based on that.
 
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