Blood Flow Velocity After Bifurcation: Continuity Equation Numerical

Correct Answer: 60 cm/s

Understanding the Blood Vessel Bifurcation Problem

This problem is based on the principle of conservation of volume flow rate, which is mathematically expressed through the continuity equation. Since blood is assumed to be incompressible and the vessel walls are rigid, the amount of blood flowing through the original vessel per unit time must be equal to the total amount of blood flowing through the two smaller vessels per unit time.

The important point is that the original blood vessel does not simply become one smaller vessel. Instead, it bifurcates into two identical smaller vessels. Therefore, the total outgoing flow rate is the sum of the flow rates through both smaller vessels.

Continuity Equation for Incompressible Fluid Flow

For an incompressible fluid flowing steadily through a vessel, the volume flow rate is given by:

Q = Av

where Q is the volume flow rate, A is the cross-sectional area of the vessel, and v is the velocity of the fluid.

Because blood is incompressible, the total volume of blood entering the bifurcation per unit time must equal the total volume leaving through the two smaller vessels. Therefore:

A₁v₁ = A₂v₂ + A₃v₃

Since the two smaller blood vessels have the same radius, their cross-sectional areas are equal. The velocity of blood in each smaller vessel is also the same. Therefore, the equation becomes:

A₁v₁ = 2A₂v₂

Given Values

The radius of the original blood vessel is:

r₁ = 2.0 cm

The velocity of blood in the original vessel is:

v₁ = 30 cm/s

The radius of each smaller blood vessel is:

r₂ = 1.0 cm

Let the velocity of blood in each smaller vessel be v₂.

Step-by-Step Calculation of Blood Velocity

Step 1: Write the Area of a Circular Blood Vessel

The cross-sectional area of a circular blood vessel is:

A = πr²

Therefore, the cross-sectional area of the original vessel is:

A₁ = π(2.0)² = 4π cm²

The cross-sectional area of each smaller vessel is:

A₂ = π(1.0)² = π cm²

Step 2: Apply Conservation of Volume Flow Rate

Since one original vessel bifurcates into two smaller vessels, the continuity equation is:

A₁v₁ = 2A₂v₂

Substituting the known values:

(4π)(30) = 2(π)(v₂)

This gives:

120π = 2πv₂

Dividing both sides by 2π:

v₂ = 60 cm/s

Final Answer

The velocity of blood in each of the smaller vessels is 60 cm/s.

Why Does the Blood Velocity Increase?

At first, it may appear that the velocity should decrease because one blood vessel divides into two vessels. However, the correct result depends on the total cross-sectional area before and after bifurcation.

The original vessel has a radius of 2.0 cm, so its cross-sectional area is 4π cm². Each smaller vessel has an area of π cm². Since there are two smaller vessels, their combined cross-sectional area is:

Total outgoing area = 2 × π = 2π cm²

Thus, the combined cross-sectional area after bifurcation is only half the cross-sectional area of the original vessel. For the same volume of incompressible blood to pass through a total area that is reduced by half, the velocity must become twice as large.

Therefore:

New velocity = 2 × 30 = 60 cm/s

Role of the Incompressibility and Rigid-Wall Assumptions

The statement that blood is incompressible means its density remains essentially constant during the flow. The assumption of rigid vessel walls means that the radii and cross-sectional areas of the vessels do not change because of pressure. These assumptions allow the direct application of the continuity equation without considering changes in density or vessel dimensions.

Under these conditions, the total volume flow rate remains conserved. Therefore, any decrease in the total cross-sectional area must be compensated by an increase in the flow velocity.

Key Concept Behind the Numerical

The central principle used in this question is the conservation of volume flow rate. For an incompressible fluid, the volume entering a junction per unit time must equal the total volume leaving the junction per unit time.

For one vessel dividing into two identical vessels:

A₁v₁ = 2A₂v₂

Since the area of a circular vessel depends on the square of its radius, reducing the radius from 2.0 cm to 1.0 cm reduces the area of each smaller vessel to one-fourth of the original area. Two such vessels together provide only half of the original total cross-sectional area. Consequently, the blood velocity doubles from 30 cm/s to 60 cm/s.

Conclusion

Using the continuity equation for incompressible fluid flow, the volume flow rate in the original blood vessel must equal the combined volume flow rate through the two smaller vessels. The original vessel has an area of 4π cm², while the two smaller vessels together have a total area of 2π cm². Since the total cross-sectional area becomes half, the velocity becomes twice the original value. Therefore, the velocity of blood in each smaller vessel is 60 cm/s.