The resistance of blood flow, R, in a blood vessel is dependent on the length of the blood

vessel, the radius of the blood vessel, and the viscosity of the blood. This relationship
is given by R = 8Lη/πr^4 where r is the radius, L is the length, and the positive value η is the viscosity (the larger the value of η the more viscous the blood).
(a) Find the derivative of R with respect to r. Is the derivative positive or negative? What is the physical interpretation of this? Does this result make sense?
(b) Find the derivative of R with respect to η. Is the derivative positive or negative? What is the physical interpretation of this? Does this result make sense?
(c) As body temperature increases, the blood vessel will widen (i.e. radius increases) and the blood becomes less viscous. Similarly as the body temperature lowers, the blood vessel will constrict and the blood will also become more viscous. We see then that the radius of the blood vessel and the viscosity of the blood are dependent on temperature. However, the length of the blood vessel essentially remains constant with temperature changes. Find the rate of change of resistance with respect to temperature, T. That is, find an expression for dR/dT

(a) To find the derivative of R with respect to r, we differentiate the formula R = 8Lη/πr^4 with respect to r, treating L and η as constants:

dR/dr = 8Lη/π * (-4) * r^(-4-1)
dR/dr = -32Lη/πr^5

The derivative of R with respect to r is -32Lη/πr^5. Since this derivative is negative, it means that as the radius of the blood vessel increases, the resistance of blood flow decreases.

The physical interpretation of this is that a larger radius allows for a greater cross-sectional area for blood flow, leading to less resistance. This makes sense intuitively since a wider blood vessel can accommodate more blood and reduce the force required for blood to flow through it.

(b) To find the derivative of R with respect to η, we differentiate the formula R = 8Lη/πr^4 with respect to η, treating L and r as constants:

dR/dη = 8L/πr^4

The derivative of R with respect to η is 8L/πr^4. Since this derivative is positive, it means that as the viscosity of the blood (η) increases, the resistance of blood flow also increases.

The physical interpretation of this is that more viscous or thicker blood requires more force to flow through the blood vessel, leading to higher resistance. This result makes sense as it aligns with our understanding of fluid dynamics.

(c) To find the rate of change of resistance with respect to temperature (T), we need to differentiate the formula R = 8Lη/πr^4 with respect to T, treating L, r, and η as constants. Since the formula does not explicitly depend on temperature, we can conclude that dR/dT = 0.