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

In summary, the conversation discusses the velocity of a liquid in certain hydrodynamic flows, which is given by v = -grad φ with φ representing the velocity potential. The potential corresponds to flow around a solid sphere of radius a, with U being a constant. The task at hand is to determine the velocity of the fluid throughout the flow and confirm that the radial component of velocity is zero at the surface of the sphere. This means that the flow at the surface should be parallel to the surface and not entering or exiting the sphere.
  • #1
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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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  • #2
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.
 

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

What is the radial component of velocity?

The radial component of velocity is the component of an object's velocity that is directed towards or away from a fixed point, also known as the center of rotation. It is perpendicular to the tangential component of velocity, which is parallel to the object's motion along a curved path.

How is the radial component of velocity calculated?

The radial component of velocity can be calculated using the following formula: Vr = V * cos(theta), where V is the total velocity and theta is the angle between the velocity vector and the radial direction. Alternatively, it can also be calculated using the equation Vr = r * omega, where r is the distance from the center of rotation and omega is the angular velocity.

What is the significance of the radial component of velocity?

The radial component of velocity is important in understanding the motion of objects in circular or curved paths. It helps determine the direction and magnitude of the object's velocity at any given point along its path, allowing for the prediction and analysis of its motion.

How does the radial component of velocity affect acceleration?

The radial component of velocity does not directly affect acceleration, as it is perpendicular to the tangential component of velocity. However, changes in the radial component of velocity can lead to changes in the tangential component, which in turn can result in acceleration or deceleration along the curved path.

Can the radial component of velocity be negative?

Yes, the radial component of velocity can be negative. A negative value indicates that the object is moving in the opposite direction of the radial vector, either away from the center of rotation or towards it, depending on the specific scenario. A positive value indicates that the object is moving in the same direction as the radial vector.

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