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Let us consider the common equation for drag force for any body.

$F_D = \frac{1}{2}\rho v^2C_dA$

Here the A is the representative area which is the so called area of cross section of the body for most shapes under conditions of stable velocity (that is the angle of attack/velocity/viscosity of the medium is not so much).

Now my question is about the distribution of these forces.

  1. Given an extended body, how will the drag force be distributed across the points on the surface of the body. That is given say a sphere, how is this force distributed throughtout the surface of the body? Say there are 60 points uniformly distibuted in the sphere. The entire sphere is moving forward with a velocity of $v$. So each point in the sphere has a velocity of $v$. In that case, will the drag force at each point be equal to $F_D$ ?

  2. Next if we consider an extended body where the velocities at each point is not the same, then will the same equation be applied to calculate the drag force at each point ? Because my viewing of the drag force is kind of like a "whole body thing" and this contradicts with this notion.

I will post more clarification if required.

EDIT(1) : Posted more clarity on the questions.

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  • $\begingroup$ $A$ does not have a 'value at each point'. It's an area. $\endgroup$
    – Danu
    Commented Oct 29, 2013 at 19:09
  • $\begingroup$ Right. That is my concern. Also this is not a homework. I am a CS Grad student working on the simulation of a rigid body through a medium and I cam across this problem. $\endgroup$
    – rajaditya_m
    Commented Oct 29, 2013 at 19:36
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    $\begingroup$ The homework tag does not just apply to actual homework assignments: it's a type of question. $\endgroup$
    – Danu
    Commented Oct 29, 2013 at 19:38
  • $\begingroup$ This question seems to be "For a spherical body moving through a fluid how is the net drag force distributed across the surface?" $\endgroup$
    – Dave
    Commented Oct 29, 2013 at 19:41
  • $\begingroup$ Edited the question for more clarity of what is asked. $\endgroup$
    – rajaditya_m
    Commented Oct 29, 2013 at 19:50

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Each point on the surface will have a pressure (force normal to surface) and drag (force tangential to surface). By integrating over the entire surface you get the overall effect which is sometimes expressed in force/moments as

$$ F_D = \frac{1}{2} \rho v_{body}^2 A_{body} C_D \\ M_D = \frac{1}{2} \rho v_{body}^2 \ell A_{body} C_M $$

The $v_{body}$ used is just a convenient scaling factor to get things in the right units. The same with $A_{body}$. You will not use these equations to get the forces on a small area of a body. They work only for the entire body.

To get into the details of the forces in each infinitesimal surface area patch ${\rm d}A$ you will need to solve the fluid dynamics equations for continuity and momentum which will give you the velocity vector and pressure at each location.

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