Problem 4 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

The velocity vector for a steady incompressible flow in the \(x y\) plane is given by \(\vec{v}=\) \(\frac{4}{x} \vec{i}+\frac{4 y}{x^{2}} \vec{j}\), where the coordinates are measured in centimeters. Determine the time it takes for a particle to move from \(x=1 \mathrm{~cm}\) to \(x=4 \mathrm{~cm}\) for a particle that passes through the point \((x, y)=(1,4)\).

Step-by-Step Solution

Answer

The particle takes 1.875 seconds to move from \(x=1\) to \(x=4\).

Step 1: Understanding the Problem

We need to find the time it takes for a particle to move along the vector field described by \( \vec{v}= \frac{4}{x} \vec{i}+\frac{4y}{x^2} \vec{j} \) from the point \((1, 4)\) to \(x = 4\). The particle crosses the line \(x=1\) at \(y=4\).

Step 2: Integrating Velocity to Find Time

The velocity component along the \(x\)-direction is given by \(\frac{dx}{dt} = \frac{4}{x}\). To find the time it takes for the particle to go from \(x = 1\) to \(x = 4\), we integrate with respect to \(t\): \[dt = \frac{x}{4} dx\] Integrating both sides from \(x = 1\) to \(x = 4\): \[t = \int_{1}^{4} \frac{x}{4} \, dx\]

Step 3: Calculating the Integral

Carry out the definite integral from Step 2: \[t = \frac{1}{4} \int_{1}^{4} x \, dx\] Evaluating the integral, we get: \[t = \frac{1}{4} \left[ \frac{x^2}{2} \right]_{1}^{4} = \frac{1}{4} \left( \frac{4^2}{2} - \frac{1^2}{2} \right)\]

Step 4: Simplifying the Result

Simplify the result from Step 3: \[t = \frac{1}{4} \left( \frac{16}{2} - \frac{1}{2} \right) = \frac{1}{4} \left( 8 - 0.5 \right) = \frac{1}{4} \times 7.5 = \frac{7.5}{4}\] Which simplifies to: \[t = 1.875 \, \text{seconds}\]

Key Concepts

Velocity Vector

In fluid mechanics, velocity vectors help describe the movement of particles within a fluid flow. A velocity vector such as \(\vec{v}= \frac{4}{x} \vec{i} + \frac{4y}{x^2} \vec{j}\) includes components that indicate the speed and direction of the particle at any point in space. In this particular exercise, the \(\vec{i}\) component \(\left(\frac{4}{x}\right)\) represents motion in the x-direction, while the \(\vec{j}\) component \(\left(\frac{4y}{x^2}\right)\) describes the motion in the y-direction.

Velocity vectors are crucial because they provide a comprehensive way to analyze and calculate how particles will move through a given flow field. In our problem, understanding these components allows us to find out how long it takes the particle to move from one point to another, an essential part of analyzing fluid dynamics.

Definite Integral

The definite integral is a mathematic technique used to determine the accumulated values, such as time, given a rate represented by a function over an interval. In this context, it helps calculate the time a particle takes to traverse certain distances in a velocity field.

By integrating the function \(dt = \frac{x}{4} dx\) over the interval from \(x = 1\) to \(x = 4\), we could find the amount of time \(t\). This provides a precise, numeric measure for the time needed to traverse these points. The resulting integral is:

• \(\int_{1}^{4} \frac{x}{4} \; dx\)
• Which gives \(\frac{1}{4} \int_{1}^{4} x \; dx\)

Integrals like these are pivotal because they show how accumulation—from velocity to time—occurs across a space or time.

Particle Motion

Particle motion in fluid mechanics is the study of how individual particles move through a fluid field based on forces and conditions applied to them. In this exercise, we're concerned with the motion of a particle in a steady incompressible flow. The particle starts at \((x, y) = (1, 4)\) and moves until \(x = 4\).

This motion is influenced by the velocity of the fluid at each point and is described by the velocity vector given. Knowing how to manipulate these vectors, and perform integrals, allows us to determine not only the path of the particle but also the time it takes to move from one point to another. The beauty of analyzing particle motion in this way lies in the ability to predict and understand the behavior of particles and fluid in dynamic environments, which is critical in fluid mechanics.

Fluid Mechanics

Fluid mechanics deals with the behavior of fluids (liquids and gases) and how they interact with forces and surfaces. It's a broad field that incorporates principles from physics and engineering to solve complex problems related to fluid flow. In the scope of the exercise given, fluid mechanics principles help define the velocity field, depicted by a velocity vector, that allows the determination of a particle's motion over time.

Fluid mechanics is often applied in a multitude of areas such as designing water distribution systems, predicting weather patterns, and understanding ocean currents. This exercise demonstrates a small part of the expansive principles of fluid mechanics, illustrating how mathematical tools like integration are used to solve real-world problems involving fluid flow.