The purpose of this experiment is to create a meaningful algorithm that converts a patient’s blood pressure, and, from that data, give you an estimated range along the ICA in which the clot will be located.
1.1: Fluid Dynamics
The study of fluids and how they move is known as fluid dynamics, and it is quite different than classical mechanics. Since fluids are not solid objects, there are a lot of factors at play when it comes to determining their momentum, velocity, acceleration, and position. This is because when one is talking about a solid, the object at hand is one that has a definite shape and volume, but liquids and gases, i.e. fluids, take the shape of the container they are in, and change depending on the volume of the container. Since fluids change shape depending on their container, instead of referring to acceleration, velocity, and position, a fluid is described by a multitude of factors including, flow velocity (u), flow acceleration (), viscosity (η), rate of flow (Q), the area of the container (A), sheer stress (τ), and pressure (P). The equations in which fluids are described are numerous, but assuming some of these factors remain constant, there are three main equations used to model fluids; Bernoulli’s Equation, the Cauchy Momentum Equation, and the Navier-Stokes Equations (Principles of Fluid, 2020).
1.2: Flow Velocity and Rate of Flow
Flow velocity is a vector field u, such that which gives the velocity of a fluid at position x and time t. Flow velocity is key in describing a fluid: it determines whether the flow is steady, incompressible, or irrotational. Steady flow occurs when the partial derivative of flow velocity, with respect to time is equal to zero, in other words, flow is steady when . The next thing that flow velocity can tell you is whether the flow is incompressible or compressible. A flow is incompressible when the divergence of the flow velocity is equal to zero, in other words, a flow is incompressible when , where . Finally, the last thing the flow velocity can tell you is if the flow is irrotational. If the flow is irrotational, then the curl of the flow velocity is equal to zero, in other words, Similarly, the curl of the flow velocity is also known as the vorticity of the fluid (ω), which describes the fluid’s tendency to become a vortex (Milrad, 2018).
Another key factor in describing a fluid is its rate of flow, which is like flow velocity, but instead of describing how much distance a fluid will cover in a unit of time, the rate of flow (Q) describes the change in volume per unit of time. The rate of flow and the flow velocity are related by the following equation: rate of flow equals the cross-sectional area of a container (A) times the magnitude of the flow velocity (|u|), in other words, Oftentimes, the velocity at every point in a fluid is not known, the rate of flow is used to describe the fluid, since it is much easier to measure the change in volume per change in time than it is to measure the change in position per change in time of a fluid (Flow Rate, n.d.). This can be worked around when a key observation is made in the relationship of rate of flow to flow velocity; specifically, that , which is important because it gives a direct relationship to rate of flow and flow velocity, meaning that if one is known, than the other is known as well.
1.3: Pressure and Density of a Fluid
Pressure is likely the most important thing in describing a fluid. Defined as , pressure can be used to derive several essential relationships in fluid dynamics. For example, since , according to Newton’s First Law of Motion, . This relationship implies that pressure is a function of time, and in a fluid, it is proportional to the first partial derivative of flow velocity, in other words, . Now the relationship to rate of flow and pressure can be made with (Hodanbosi 1996). Pressure also has a direct relationship to the density of a fluid because density is defined as Substituting mass for density times volume gives the equation , which is a direct relationship to the first derivative of rate of flow, pressure, density, and distance (x). Both density and pressure are essential in forming the equations that describe the motion of fluids. Typically, density and pressure in a given closed system are constant, so knowing these two things and their relationships to rate of flow and flow velocity can be key in modeling the motion of a fluid.
1.4: Bernoulli’s Equation and Poiseuille’s Law
Bernoulli’s equation is one that relates pressure, flow velocity, density, and the elevation of a fluid. Bernoulli’s Equation says that where C is some constant. Since C is a constant and static pressure plus dynamic pressure is equal to the total pressure, Bernoulli’s Equation can be rewritten as assuming that flow is incompressible ( is constant) and the pressures, velocities, and heights observed belong to the same system (Bernoulli’s Equation, n.d.). Bernoulli’s equation is incredibly significant because it relates density, height, pressure, and flow velocity. Therefore, if the density, pressure, and height are known, then the flow velocity can be solved for, and in turn, the rate of flow and other things can be solved for as well.
The next equation pertains specifically to blood flow. Known as Poiseuille’s Law, it relates the viscosity (in Pascal seconds), pressure, length of an artery, and area of the artery to the rate of flow. The equation is as follows: , where r is the radius of the container, P is the pressure, η is the viscosity of the fluid, and l is the length of the container. This equation works very well for blood flow because in this form, it works for a cylindrical container, which is the approximate shape of a human artery, which means this equation can be used to model blood flow in the human circulatory system. A limiting factor of Poiseuille’s Law is that the flow must be laminar; however, in the case of the human circulatory system the blood flow (except for where there is a clot) is laminar (Poiseuille’s Law, n.d.). Knowing this, a good way to estimate where a clot is located within an artery is to study the Reynold’s Number (Re), which predicts whether the flow is laminar or turbulent. If the Reynold’s Number is low, the flow is typically laminar, and if the Reynold’s Number is high, then the flow is typically turbulent. The Reynold’s Number can be calculated using the formula: (Reynold’s Number, n.d.). Thus, the Reynold’s Number is significant because it is used to tell whether a flow is laminar or turbulent.