Unformatted text preview: blood as a tube of
radius R and length L, then the cost function F is
.
From the first derivative of F with respect to R, determine the relationship between Q and
the vessel radius. Using the second derivative, show that this is a maximum.
Solution
Beginning with the given equation and inserting the result form (a)
becomes, Taking the first derivative with respect to R we find when the cost function is minimized: Solving for Q, which illustrates that the flow rate varies with the cube of the radius.
To establish whether Q is a minimum of a maximum value, we compute the second
derivative of the equation for
above, Since the second derivative is positive, the cost function is minimized. Q is a
monotonically increasing function of radius to the third power. Therefore, when F is
minimized, Q reaches its maximum value. (c) Relate the shear str...
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 Spring '12
 Leach
 Fluid Dynamics, dimensionless groups

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