The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille's Laws gives the resistance R of the blood as
where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius r branching at an angle into a smaller vessel with radius s. (Note in your answers use t instead of and r and s respectively instead of and .)
(a) Use Poiseuille's Law to find the total resistance of the blood along the path ABC where a and b are the distances shown in the figure. (Your answer may depend on a, b, C, r, s, and t.)
(c) Find the optimal branching angle when the radius of the smaller blood vessel is 3/3 the radius of the larger vessel. =
R=C(L/r^2) is the equation
(a) To find the total resistance of the blood along the path ABC, we need to consider the resistance in each segment of the blood vessels.
Let's start with segment AB. The resistance in segment AB is given by Poiseuille's Law:
R_AB = (8C * L_AB) / (π * r^4)
Next, let's consider segment BC. The resistance in segment BC is given by Poiseuille's Law, but with the radius of the blood vessel being s:
R_BC = (8C * L_BC) / (π * s^4)
To find the total resistance along the path ABC, we can simply add the resistances of the two segments:
Total Resistance = R_AB + R_BC = (8C * L_AB) / (π * r^4) + (8C * L_BC) / (π * s^4)
(b) For this part of the question, there is no given information or specific task mentioned. Please provide more details or clarify your question, and I'll be happy to assist you further.
To find the total resistance of the blood along the path ABC, we need to consider the resistance of each blood vessel segment separately and then sum them up.
Let's start with the main blood vessel (AB) with radius r. According to Poiseuille's Law, the resistance of this segment is given by:
R_AB = (8 * C * L_AB) / (π * r^4)
where L_AB is the length of segment AB.
Next, we consider the smaller blood vessel (BC) with radius s. The resistance of this segment is given by:
R_BC = (8 * C * L_BC) / (π * s^4)
where L_BC is the length of segment BC.
Now, we need to consider the branching point at B. The blood flow splits into two branches - one going towards AC and the other towards BC. The total resistance at this branching point is the sum of the resistances for both branches:
R_B = 1 / (1 / R_AC + 1 / R_BC)
where R_AC is the resistance of the branch going towards AC.
Finally, we can find the total resistance along the path ABC by summing the resistances of the individual segments:
R_ABC = R_AB + R_B + R_BC
To find the optimal branching angle, we need to minimize the total resistance R_ABC with respect to the branching angle theta (θ). The optimal branching angle can be found by taking the derivative of R_ABC with respect to theta and setting it equal to zero:
dR_ABC / dθ = 0
Solving this equation will give us the optimal branching angle.