The flow of blood in a blood vessel is faster toward the...

The flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the outside. The speed of the blood is given by $$V=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)$$ where $R$ is the radius of the blood vessel, $r$ is the distance of the blood from the center of the vessel, and $p, v,$ and $L$ are physical constants related to the pressure and viscosity of the blood and the length of the blood vessel. II $R$ is constant, we can think of $V$ as a lunction of $r:$ $$V(r)=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)$$ (IMAGES CANNOT COPY) The total blood flow, $Q,$ is given by $Q=\int_{0}^{R} 2 \pi \cdot V(r) \cdot r \cdot d r$ Find $Q$

The flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the outside. The speed of the blood is given by
$$V=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)$$
where $R$ is the radius of the blood vessel, $r$ is the distance of the blood from the center of the vessel, and $p, v,$ and $L$ are physical constants related to the pressure and viscosity of the blood and the length of the blood vessel. II $R$ is constant, we can think of $V$ as a lunction of $r:$
$$V(r)=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)$$
(IMAGES CANNOT COPY)
The total blood flow, $Q,$ is given by $Q=\int_{0}^{R} 2 \pi \cdot V(r) \cdot r \cdot d r$
Find $Q$

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Key Concepts

Parabolic Velocity Profile

This concept involves a velocity distribution that is highest at the center of a vessel and decreases towards the walls, typically resulting in a parabolic profile observed in laminar flow. It explains how variations in speed occur due to the effects of viscosity and boundary conditions in fluid dynamics.

Cylindrical Coordinate Integration

In systems with radial symmetry, such as pipes or blood vessels, integrals are often performed in cylindrical or polar coordinates. This method employs a differential area element, commonly expressed as 2?r dr, to account for contributions from concentric rings in the cross-sectional area.

Volumetric Flow Rate Calculation

The volumetric flow rate quantifies the volume of fluid passing through a given cross-section per unit time. It is calculated by integrating the product of the local velocity and the differential area elements, thereby summing the contributions of flow from all parts of the cross-section.

Definite Integration

This concept refers to the process of integrating a function over a specific interval, which allows for the computation of cumulative quantities, such as total flow. It is a fundamental tool in calculus that aggregates the contributions from small increments across a continuous domain.

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