What is the Radial Component of Velocity on the Surface of a Solid Sphere?

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SUMMARY

The discussion centers on the determination of the radial component of velocity in hydrodynamic flows around a solid sphere, specifically using the equation v = -grad φ, where φ is the velocity potential. The participant successfully calculated the fluid velocity and confirmed that the radial component of velocity is zero on the sphere's surface, indicating that flow is parallel to the surface rather than penetrating it. This conclusion is essential for understanding fluid dynamics in spherical geometries.

PREREQUISITES
  • Understanding of hydrodynamic principles and fluid flow.
  • Familiarity with the concept of velocity potential in fluid mechanics.
  • Knowledge of vector calculus, particularly gradient operations.
  • Basic principles of flow around solid bodies, specifically spheres.
NEXT STEPS
  • Study the derivation of velocity potential in fluid dynamics.
  • Learn about the implications of boundary conditions in fluid flow around solid objects.
  • Explore the mathematical techniques for calculating flow fields using potential functions.
  • Investigate the behavior of fluid flow at different geometrical boundaries, including cylinders and irregular shapes.
USEFUL FOR

This discussion is beneficial for students and professionals in fluid mechanics, particularly those studying hydrodynamics, as well as engineers involved in designing systems where fluid flow around solid objects is critical.

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Homework Statement



In certain hydrodynamic flows the velocity of a liquid is given by v = -grad φ,
where φ is referred to as the velocity potential. The potential given by:
'equation'
corresponds to flow around a solid sphere of radius a. U is a constant.
Determine the velocity of the fluid throughout the flow. Confirm that the
radial component of velocity is zero on the surface of the sphere.

I've successfully found the velocity of the fluid, as my answer is in agreement with the answer given. However, I'm not sure this part of the question means: '
Confirm that the
radial component of velocity is zero on the surface of the sphere.'

Could anyone help me?
 
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A radial component of the velocity would mean something flows into or out of the sphere. This should not happen, and you are supposed to verify this.
In other words, the flow at the surface has to be parallel to the surface.
 

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