Question

The speed of blood in the aorta is much higher than the speed of blood through a capillary bed. How can this fact be explained using the continuity equation, assuming that we are interested in average flow and that there is no net fluid loss? A. The aorta is located higher than the capillary bed. B. The pressure in the aorta is the same as the pressure in the capillary bed. C. The cross-sectional area of all the capillaries added together is much greater than the cross-sectional area of the D. aorta. D. The cross-sectional area of a capillary is much smaller than the cross- sectional area of the aorta.

Short answer

Option C: The cross-sectional area of all the capillaries added together is much greater than the cross-sectional area of the aorta.

Step-by-step solution

Understanding the Continuity Equation

The continuity equation states that for an incompressible fluid, the product of the cross-sectional area (A) and the velocity (v) at any two points along a streamline must remain constant: In this context, we analyze the equation A₁v₁ = A₂v₂.

Comparing Cross-Sectional Areas

Compare the cross-sectional areas of the aorta and the capillaries. The aorta has a relatively large cross-sectional area, but the total cross-sectional area of all capillaries combined is much larger. Use Aₐ and A_c to represent the area of the aorta and the total cross-sectional area of all capillaries, respectively. Since A_c > Aₐ, consider the impact on velocity.

Using the Continuity Equation

Apply the continuity equation to relate the blood flow speeds in the aorta and capillaries. Since A₁v₁ = A₂v₂ and A_c > Aₐ, solve for the velocity: Since A_c > Aₐ and the flow rate must remain consistent, the velocity v in the aorta (vₐ) must be higher than the velocity in capillaries (v_c) to balance the equation Aₐvₐ = A_cv_c.

Conclusion

Identify the correct explanation for why the speed of blood in the aorta is higher. Considering the continuity equation and the different cross-sectional areas, the explanation is that the total cross-sectional area of all the capillaries added together is much greater than the cross-sectional area of the aorta.

Key concepts

Aorta Blood Flow

The aorta is the main artery that carries oxygenated blood from the heart to the rest of the body. Since it needs to transport large volumes of blood quickly, the velocity of blood flow within the aorta is very high. This is essential for the efficient delivery of oxygen and nutrients.
The properties of blood flow within the aorta can be understood using the principles of fluid dynamics. In particular, the continuity equation helps explain why blood flow velocity is significantly higher in the aorta compared to smaller vessels.
The aorta's relatively large diameter allows it to handle the high flow rates generated by the heart without requiring extremely high pressure.

• High-speed flow ensures quick distribution.
• Larger diameter balances load without excess pressure.

Capillary Cross-Sectional Area

Unlike the aorta, capillaries are tiny blood vessels with an extremely small diameter. However, what's interesting is the combined cross-sectional area of all the capillaries in the body. If we sum up the total cross-sectional area of these small vessels, we find that it is much larger than that of the aorta.
Due to this vast collective cross-sectional area, the flow velocity of blood through capillaries is much slower. This slow speed is advantageous for several reasons:

• It allows time for the exchange of oxygen, carbon dioxide, nutrients, and waste products between blood and tissues.
• It reduces the risk of damage to delicate tissues by high-pressure blood flow.

In summary, although individual capillaries have a tiny diameter, their cumulated area is extensive, reducing the blood flow velocity in these vessels.

Incompressible Fluid Dynamics

In fluid dynamics, blood is considered an incompressible fluid, meaning its volume does not change under pressure. The continuity equation tells us that for an incompressible fluid like blood, the product of cross-sectional area (A) and velocity (v) must remain constant along any stretch of a vessel network.
This relationship can be mathematically stated as:
where A₁ and v₁ are the area and velocity at one point, and A₂ and v₂ are area and velocity at another point.

• This principle means if the flow rate stays the same, a decrease in the vessel's cross-sectional area must result in an increase in flow velocity.

Understanding this helps in grasping why blood speeds up in the narrower aorta and slows down as it moves into the vastly greater cross-sectional area of capillary beds.

Velocity and Area Relationship

The relationship between the velocity of a fluid and the cross-sectional area of the vessel it flows through is inversely proportional. This means:

• When the cross-sectional area of a vessel increases, the flow velocity decreases.
• Conversely, when the cross-sectional area decreases, the flow velocity increases.

This can be particularly illustrated with blood flow:
In the case of the aorta, since the cross-sectional area is relatively small compared to the combined area of all the capillaries, the velocity of blood must be high to maintain a consistent flow rate.
In capillaries, due to the massive combined cross-sectional area, the velocity of blood flow is significantly reduced. This relationship is crucial for maintaining effective and efficient circulation in the body, ensuring that each organ receives the necessary nutrients and oxygen without overburdening the smaller vessels with too high pressures.