Problem 7 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Considering one-dimensional fluid (density, \(\rho\); viscosity, \(\mu\) ) flow in a tube with an inlet pressure of \(p_{i}\) and outlet pressure of \(p_{o}\) and tube radius of \(r\) and length \(l\), the density of the fluid can be represented as \(\rho\). Express the wall shear stress as a function of these variables (hint: balance Newton's second law of motion).

Step-by-Step Solution

Answer

The wall shear stress is given by \( \tau_w = \frac{(p_i - p_o) \cdot r}{2l} \).

Step 1: Identify Relevant Variables and Equations

In a tube with fluid flow, wall shear stress \( \tau_w \) is the force per unit area exerted by the fluid on the wall. The flow is driven by the pressure difference \( p_i - p_o \) along the length \( l \). The density \( \rho \) and viscosity \( \mu \) of the fluid also influence the flow dynamics. In this context, we should apply Newton's second law to balance forces in terms of shear stress and pressure drop.

Step 2: Apply Newton's Second Law

According to Newton's second law, the change in momentum of the fluid per unit time is equal to the net force acting on the fluid. For a fluid element of length \( l \), radius \( r \), and cross-sectional area \( A = \pi r^2 \), the pressure force on one side is \( p_i \cdot A \) and on the other side is \( p_o \cdot A \). The pressure drop force is \( F_{pressure} = (p_i - p_o) \cdot \pi r^2 \).

Step 3: Relate Shear Stress to Viscous Forces

The shear force due to viscosity acts tangentially at the wall surface. This force is distributed along the circumference of the tube, i.e., \( F_{shear} = \tau_w \cdot 2 \pi r \cdot l \) where \( \tau_w \) is the wall shear stress. By balancing the forces (pressure drop and shear force), we set \( F_{pressure} = F_{shear} \).

Step 4: Derive the Expression for Wall Shear Stress

Equating the pressure force and shear force gives:\[(p_i - p_o) \cdot \pi r^2 = \tau_w \cdot 2 \pi r \cdot l\]Solving for \( \tau_w \), we specify\[\tau_w = \frac{(p_i - p_o) \cdot r}{2l}\]This is the expression for wall shear stress in terms of the given variables.

Key Concepts

Wall Shear Stress

Wall shear stress is all about how the fluid rubs against the wall of a tube as it flows. Think of it like the friction you feel when you slide your hand across a table. In the context of fluid dynamics, wall shear stress, denoted by \( \tau_w \), is a measure of how much force the moving fluid exerts against the walls of its container, in this case, a tube.

When we look at the problem provided, the primary force that drives the fluid through the tube is caused by a pressure difference, \( p_i - p_o \), between the inlet and outlet. This pressure "drop" causes the fluid to flow along the tube. The wall shear stress is not just a trivial force, it's what eventually resists the fluid's flow due to the stickiness or internal friction provided by the walls of the tube.

To make it simpler, the greater the wall shear stress, the harder it is for the fluid to keep moving at its current speed. The formula from the exercise gives us a way to calculate this stress:

• The pressure difference \( (p_i - p_o) \)
• The radius of the tube \( r \)
• The length of the tube \( l \)

These are integral to understanding how much stress is exerted by the flowing fluid on the walls.

Newton's Second Law

Newton's Second Law is a rock-solid principle of physics that connects force, mass, and acceleration. But when dealing with fluids, it's all about pressure and changes in momentum. This law, often written as \( F = ma \) in solid mechanics, translates to a balance of forces in fluid flow situations.

In the case of fluid dynamics within a tube, we consider the movement of the fluid as a series of tiny particles or molecules each experiencing forces. The fluid's momentum changes due to the pressure difference \( (p_i - p_o) \) across the tube. Here, the force in action primarily results from the driving pressure difference and the counterbalancing force from the wall shear stress.

According to Newton's second law, for the fluid to achieve a steady flow, these forces need to be balanced. This means:

• The push from the pressure difference must equal the resistive shear forces, \( F_{pressure} = F_{shear} \).
• The formula representing this balance helps us determine the wall shear stress as \( \tau_w = \frac{(p_i - p_o) \cdot r}{2l} \).

The law ensures that there is a perfect equilibrium, ensuring that the fluid particles are in harmonious motion.

Viscosity

Viscosity deals with how thick or sticky a fluid is, affecting how it moves. Think of it as the internal friction within the fluid. Imagine honey pouring out of a jar versus water; honey is way slower and thicker because it has higher viscosity.

In fluid dynamics, viscosity is a crucial player. Fluids with high viscosity (like honey or syrup) resist flow more than low-viscosity fluids (like water or air). This property is vital in determining how the fluid behaves when moving through different environments, such as tubes.

For our exercise, viscosity impacts how the fluid slides along the tube wall, contributing to the wall shear stress \( \tau_w \). A higher viscosity means more significant shear stress, requiring more force or pressure to move the fluid at a constant speed.

• Low viscosity equals less internal friction and easier flow.
• High viscosity equals more internal friction and slower, resistant flow.

Viscosity essentially dictates how fluid "sticky" the interaction is, affecting energy loss as the fluid moves.

Pressure Difference

Pressure difference is the powerhouse behind the fluid's motion in the tube. It's the difference in pressure between two points in the flow, like the pressure at the inlet \( p_i \) versus the outlet \( p_o \). This difference is what propels the fluid forward.

In the given problem, the pressure difference \( p_i - p_o \) is the main driving force that causes the fluid to move along the tube. The higher the pressure at the inlet compared to the outlet, the faster and more forcefully the fluid will move. This is because pressure is essentially the force applied over an area.

• A large pressure difference results in a stronger force, pushing the fluid more vigorously.
• A small pressure difference means less force and a slower moving fluid.

In our fluid flow analysis, this pressure difference is counteracted by the wall shear stress \( \tau_w \) that's trying to resist the flow. When set into the formula \( \tau_w = \frac{(p_i - p_o) \cdot r}{2l} \), you can see how the difference directly influences the wall shear stress needed to maintain steady flow.