Problem 21 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Two data points on the rheological diagram of a biofluid are provided. Determine the consistency index and the flow behavior index and the strain rate if the shear stress is increased to 3 dynes \(/ \mathrm{cm}^{2}\). Assume that this is a two-dimensional flow. $$ \begin{aligned} &\frac{d V}{d y}=15 \frac{\mathrm{rad}}{\mathrm{s}}, \tau=0.868 \text { dynes } / \mathrm{cm}^{2} \\ &\frac{d V}{d y}=30 \frac{\mathrm{rad}}{\mathrm{s}}, \tau=0.355 \text { dynes } / \mathrm{cm}^{2} \end{aligned} $$

Step-by-Step Solution

Answer

K is approximately 5.049, n is -1.292, and the new strain rate is 8.73 rad/s.

Step 1: Understand the Problem Context

We are given two data points from a rheological diagram, which relate shear stress (\( \tau \)) and strain rate (\( \frac{dV}{dy} \)). These parameters are typically described by the power-law model for non-Newtonian fluids. We need to find the consistency index (\( K \)), the flow behavior index (\( n \)), and calculate the new strain rate if shear stress is increased.

Step 2: Setup the Power-Law Model Equations

The power-law model is given by the equation: \[ \tau = K \left( \frac{dV}{dy} \right)^n \] We have two equations from the data points:\[ 0.868 = K \cdot 15^n \]\[ 0.355 = K \cdot 30^n \]

Step 3: Solve for the Flow Behavior Index (n)

Divide the first equation by the second to eliminate \( K \):\[ \frac{0.868}{0.355} = \frac{15^n}{30^n} \]Simplifying gives:\[ \left(\frac{15}{30}\right)^n = 2.445 \]This can be expressed as:\[ (0.5)^n = 2.445 \]Taking the natural logarithm of both sides:\[ n \cdot \ln(0.5) = \ln(2.445) \]Solving for \( n \):\[ n = \frac{\ln(2.445)}{\ln(0.5)} \approx -1.292 \]

Step 4: Solve for the Consistency Index (K)

Substitute \( n = -1.292 \) back into one of the original equations:\[ 0.868 = K \cdot 15^{-1.292} \]Solving for \( K \):\[ K = \frac{0.868}{15^{-1.292}} \approx 5.049 \]

Step 5: Determine the Strain Rate for New Shear Stress

Use the power-law model to find the new strain rate \( \frac{dV}{dy} \) when \( \tau = 3 \, \text{dynes/cm}^2 \):\[ 3 = 5.049 \left( \frac{dV}{dy} \right)^{-1.292} \]Solving for \( \frac{dV}{dy} \):\[ \frac{dV}{dy} = \left( \frac{3}{5.049} \right)^{\frac{1}{-1.292}} \approx 8.734 \frac{\text{rad}}{\text{s}} \]

Key Concepts

Power-Law Model

The power-law model is crucial in describing non-Newtonian fluids, often illustrated in rheological studies like this. It predicts how the viscosity of a fluid changes with varying applied forces, which is especially significant for biofluids. The general formula of the power-law model is given by \( \tau = K \left( \frac{dV}{dy} \right)^n \) where:

• \( \tau \) represents the shear stress.
• \( K \) is the consistency index.
• \( \left( \frac{dV}{dy} \right) \) indicates the strain rate.
• \( n \) is the flow behavior index.

In biofluid rheology, the power-law model aids in understanding how fluids behave under different stress conditions. It helps determine how much a fluid will resist flow, relevant when analyzing clinical and physiological applications.

Consistency Index

The consistency index, denoted as \( K \), is a parameter within the power-law model that indicates the fluid's viscosity at a reference shear rate. It depicts how thick a fluid is under normal conditions. For a given shear stress-strain rate relationship, once the flow behavior index \( n \) has been determined, \( K \) can be extracted from the rheological data points. Using the power-law model equation

• Substitute \( n \) into any of the known equation \( \tau = K \left( \frac{dV}{dy} \right)^n \).
• Solve for \( K \) using the provided data. In this exercise, \( K \approx 5.049 \).

The consistency index plays a pivotal role in characterizing the fluid's response to stress, which is vital in designing processes which need predictable fluid performance.

Flow Behavior Index

The flow behavior index \( n \) demonstrates how fluid viscosity changes concerning shear rate, reflecting whether a fluid behaves more like a solid or a liquid under stress. To derive \( n \) in the power-law model, compare shear stress and strain rate from two data points. By eliminating the consistency index \( K \), we can focus solely on \( n \) with the equation

• Calculate the ratio of the shear stresses.
• Take the ratio of the strain rates.
• Use logarithmic equations to solve for \( n \).

In our exercise, \( n \approx -1.292 \), indicating the fluid's shear-thinning nature, meaning its viscosity decreases with an increase in shear rate.

Shear Stress

Shear stress, expressed as \( \tau \), is the force per unit area exerted by the fluid in response to an applied force. It is a critical factor in understanding fluid flow, especially in complex fluids like biofluids. In the equation \( \tau = K \left( \frac{dV}{dy} \right)^n \), shear stress can be evaluated for a known strain rate and fluid properties like \( K \) and \( n \). Understanding shear stress is essential for predicting how fluids behave under different conditions:

• Low shear stress means the fluid resists flow more strongly.
• Higher shear stress indicates that the fluid flows more easily.

Knowing shear stress values helps in tailoring industrial applications and medical therapies, where predicting fluid behavior is necessary.

Strain Rate

The strain rate, denoted by \( \frac{dV}{dy} \), describes the rate at which the fluid particles deform or move in response to applied forces. High strain rates lead to greater deformation as the particles in the fluid slide past each other more quickly. The power-law model defines this relationship as \( \tau = K \left( \frac{dV}{dy} \right)^n \). From known values of shear stress \( \tau \), we can calculate the strain rate if consistency index \( K \) and flow behavior index \( n \) are known. In this problem, given an increased shear stress of 3 dynes/cm², the new strain rate calculates to be approximately

• \( 8.734 \text{ rad/s} \)

Hence, understanding strain rates allows for better predictions of fluid movement, which is essential in applications like fluid transport in biological systems.