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
To find the derivative of a function, we can differentiate each term with respect to the given variable and then combine the results. Let's start with part (a).
(a) To find the derivative of R with respect to r, we need to differentiate the expression R = 8Lη/πr^4 with respect to r.
Using the power rule of differentiation, the derivative of r^n (where n is a constant) with respect to r is given by n * r^(n-1).
Differentiating the expression R = 8Lη/πr^4 with respect to r, we get:
dR/dr = (8Lη/π) * (-4) * r^(-4-1)
= -32Lη/πr^5
This derivative is negative because of the negative sign in front of the term. The physical interpretation of this is that as the radius of the blood vessel increases, the resistance to blood flow decreases. This makes sense because a larger vessel allows for easier blood flow, resulting in lower resistance.
(b) To find the derivative of R with respect to η, we differentiate the expression R = 8Lη/πr^4 with respect to η.
Since η is a constant, differentiating with respect to η treats it as a constant term, which means its derivative is zero.
dR/dη = 8L/πr^4
The derivative is positive because it does not contain any negative terms. The physical interpretation of this is that as the viscosity of blood increases, the resistance to blood flow also increases. This makes sense because more viscous blood encounters greater resistance, resulting in higher overall resistance.
(c) To find the rate of change of resistance with respect to temperature, T, we need to differentiate the expression R = 8Lη/πr^4 with respect to T.
Since the length of the blood vessel essentially remains constant with temperature changes, its derivative with respect to T is zero.
dR/dT = (δR/δr) * (dr/dT) + (δR/δη) * (dη/dT)
Since the radius of the blood vessel changes with temperature, we need to find the derivative of r with respect to T. Similarly, since the viscosity of the blood changes with temperature, we need to find the derivative of η with respect to T.
The expressions for dr/dT and dη/dT will depend on the specific relationship between the temperature and the changes in the radius and viscosity of the blood vessel. To obtain dR/dT, you need this information.