Problem 17 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Use the Cartesian Navier-Stokes equations to approximate the flow through the left ventricle during peak systole. Assume that the gravitational effects on the flow are negligible and that the opening orifice for blood to flow through is \(25 \mathrm{~mm}\) (aorta). The width of the left ventricle can be approximated as \(2 \mathrm{~cm}\) and the total length from the apex of the heart to the aortic valve as \(4 \mathrm{~cm}\). Determine the maximum velocity at both the aorta and within the ventricle. Blood viscosity is \(3.5 \mathrm{cP}\); assume that the pressure gradient across this flow is \(-1 \mathrm{mmHg} / \mathrm{cm}\).

Step-by-Step Solution

Answer

Maximum velocity in the aorta and ventricle can be found using Hagen-Poiseuille's Law with respective dimensions and pressure gradient.

Step 1: Understand the Problem Parameters

Here, we are asked to determine the maximum velocity of blood flow through the aorta and within the left ventricle during peak systole. The effective diameter of the aorta is given as 25 mm, and the left ventricle can be approximated as a cylinder with a width of 2 cm and a length of 4 cm. We also have the blood viscosity (\(3.5\, \text{cP}\) which needs to be converted to \(\text{Pa}\,\text{s}\)) and a pressure gradient of \(-1\, \text{mmHg/cm}\). Gravitational effects are negligible in this case.

Step 2: Conversion and Formula Setup

Convert the pressure gradient from \(\text{mmHg/cm}\) to \(\text{Pa/m}\). Use 1 mmHg = 133.322 Pa. Therefore, the pressure gradient in \(\text{Pa/m}\) is \(-1 \times 133.322 \times 100\). Convert viscosity from \(\text{cP}\) to \(\text{Pa}\,\text{s}\) (1 cP = 0.001 Pa·s), so viscosity is \(3.5 \times 0.001\, \text{Pa}\,\text{s}\).

Step 3: Compute Aortic Flow Using Hagen-Poiseuille's Law

Use Hagen-Poiseuille's Law for a circular orifice: \(Q = \frac{{\pi \Delta P r^4}}{{8 \eta L}}\) where \(\Delta P\) is the pressure difference, \(r\) is the radius of the aorta, \(L\) is the length, and \(\eta\) is the viscosity. The radius \(r = 12.5\,\text{mm} = 0.0125\,\text{m}\). To find the velocity, \(V_{max} = \frac{Q}{A}\) where \(A = \pi r^2\).

Step 4: Apply Hagen-Poiseuille to Ventricular Flow

Approximating the ventricle as a cylinder, apply Hagen-Poiseuille’s Law to compute maximum velocity in a similar manner. Here, the radius \(r = 1\,\text{cm} = 0.01\,\text{m}\) and the length \(L = 4\,\text{cm} = 0.04\,\text{m}\). Similarly, use \(Q = \frac{{\pi \Delta P r^4}}{{8 \eta L}}\) and \(V_{max} = \frac{Q}{A}\) with \(A = \pi r^2\).

Step 5: Solve for Maximum Velocities

Substitute the calculated values of \(\Delta P\), \(r\), \(\eta\), and \(L\) into the Hagen-Poiseuille equation to solve for the volumetric flow \(Q\) and use this to find \(V_{max}\) in both the aorta and the ventricle. Ensure to keep units consistent, using SI units throughout.

Key Concepts

Fluid Mechanics

Fluid mechanics is the branch of physics that studies the behavior of fluids—liquids and gases—in motion and at rest. In the context of our exercise, understanding fluid mechanics is essential for analyzing how blood moves through the cardiovascular system.
Fluid mechanics can be broadly divided into fluid statics and fluid dynamics. Fluid statics deals with fluids at rest, while fluid dynamics considers the forces and motion when fluids are in motion. Our focus here is mainly on fluid dynamics as we consider the flow of blood during peak systole through the heart's structures.
When studying blood flow, we apply the principles of fluid dynamics, notably the Navier-Stokes equations, which describe how the velocity field of fluids evolves in space and time. Through these equations, we can model and understand phenomena such as turbulence, circulation, and viscous flow, which are all critical to assessing blood flow in the heart's chambers and blood vessels.

Cardiovascular Flow

Cardiovascular flow refers to the movement of blood through the heart and the blood vessels. It is vital for the transportation of nutrients, oxygen, and waste products throughout the body.
The heart acts as a pump, maintaining circulation and controlling the blood's flow rate and direction. In our exercise, we specifically focus on how blood flows from the left ventricle to the aorta. The left ventricle is responsible for pumping oxygen-rich blood to the systemic circulation through the aorta.
Blood flow is primarily driven by pressure gradients created by the heart's pumping action. These gradients push the blood through various blood vessels, overcoming resistance due to vessel walls and blood viscosity. Understanding the dynamics of cardiovascular flow, such as during peak systole, can help in determining the health of heart function and in identifying potential cardiovascular issues.

Hagen-Poiseuille's Law

Hagen-Poiseuille's Law provides a model for laminar flow of incompressible fluids through a cylindrical pipe, which is applicable to blood flow in vessels during conditions of steady flow and low Reynolds numbers.
This law is expressed as: \[ Q = \frac{{\pi \Delta P r^4}}{{8 \eta L}} \]
where:

• \(Q\) is the volumetric flow rate
• \(\Delta P\) is the pressure difference
• \(r\) is the inner radius of the pipe
• \(\eta\) is the dynamic viscosity
• \(L\) is the length of the pipe

In our context, this law allows us to estimate the flow rate through the aorta and the ventricle. It highlights how flow rate increases dramatically with the fourth power of the radius, indicating the significant impact small changes in vessel diameter can have on flow. Moreover, the law accentuates how viscosity and tube length affect resistance, providing a predictive framework for cardiovascular flows.

Blood Viscosity

Blood viscosity is a measure of how resistant blood is to flow and is affected by its composition and temperature. It is a crucial factor in determining the strength needed for the heart to pump blood through the circulatory system.
Blood is not a simple fluid; it consists of plasma, red blood cells, white blood cells, and platelets. These components contribute to its non-Newtonian behavior, meaning blood's viscosity can change under different flow conditions.
In our exercise, the viscosity is given as \(3.5\, \text{cP}\), which requires conversion to \(\text{Pa} \cdot \text{s}\) for use in calculations. This conversion is essential for applying fluid mechanics equations accurately.
Understanding blood viscosity helps us to predict how blood flows in different sections of the circulatory system and allows us to evaluate how changes in blood properties, due to health conditions or external factors, can impact cardiovascular performance.