Problem 12 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

A biofluid is flowing down an inclined plane. The velocity profile of this fluid can be described by $$ u=\frac{\rho g}{\mu}\left(h y-\frac{y^{2}}{2}\right) \sin \theta $$ if the coordinate axis is aligned with the inclined plane. Determine a function for the shear stress along this fluid. Plot the velocity profile and shear stress profile, if the fluid density is \(900 \mathrm{~kg} / \mathrm{m}^{3}\) and viscosity is \(2.8 \mathrm{cP}\). The fluid thickness \(h\) is equal to \(10 \mathrm{~mm}\) and the plane is inclined at an angle of 40 degrees.

Step-by-Step Solution

Answer

Shear stress function: \( \tau(y) = 5693.79 (0.01 - y) \).

Step 1: Define and Understand Given Variables

First, identify the given parameters and variables in the problem:- Fluid density, \( \rho = 900 \, \text{kg/m}^3 \).- Fluid viscosity, \( \mu = 2.8 \, \text{cP} \). Convert this to SI units: \( \mu = 2.8 \times 10^{-3} \, \text{kg/(ms)} \).- Fluid thickness, \( h = 10 \, \text{mm} = 0.01 \, \text{m} \).- Incline angle, \( \theta = 40^\circ \). Convert this to radians for trigonometric calculations: \( \theta = \frac{\pi}{180} \times 40 \approx 0.6981 \, \text{radians} \).

Step 2: Establish the Shear Stress Formula

The shear stress is calculated using the derivative of the velocity profile with respect to the coordinate \(y\). The formula for shear stress, \( \tau \), is:\[\tau = \mu \frac{du}{dy}\]Substitute the given velocity profile:\[\tau = \mu \left( \frac{d}{dy}\left(\frac{\rho g}{\mu}\left(h y - \frac{y^{2}}{2}\right) \sin \theta\right) \right)\]Simplifying, we find:\[\tau = \rho g \left(h - y \right) \sin \theta\]

Step 3: Calculate the Shear Stress

Substitute the known values into the shear stress equation:- \( \rho = 900 \, \text{kg/m}^3 \)- \( g = 9.81 \, \text{m/s}^2 \)- \( h = 0.01 \, \text{m} \)- \( \theta = 0.6981 \, \text{radians} \)The shear stress \( \tau \) becomes:\[ \tau(y) = 900 \times 9.81 \times (0.01 - y) \times \sin(0.6981) \]Simplified, this is:\[ \tau(y) = 5693.79 \times (0.01 - y) \]

Step 4: Plot the Velocity Profile

The velocity profile is given by:\[ u(y) = \frac{900 \times 9.81}{2.8 \times 10^{-3}} \left(0.01 y - \frac{y^{2}}{2}\right) \sin(0.6981) \]Calculate a range of \(y\) values from 0 to 0.01 meters and plot \(u(y)\) against \(y\). This will provide a graphical representation of how velocity changes along the fluid thickness.

Step 5: Plot the Shear Stress Profile

Using the shear stress function:\[ \tau(y) = 5693.79 \times (0.01 - y) \]Calculate a range of \(y\) values from 0 to 0.01 meters and plot \(\tau(y)\) against \(y\). This will illustrate how shear stress varies along the fluid thickness.

Key Concepts

Velocity Profile

Understanding the velocity profile is key to analyzing biofluid mechanics. In this scenario, the velocity profile represents the speed of a biofluid as it flows over an inclined plane. The formula given for the velocity profile is \[ u = \frac{\rho g}{\mu}\left(h y - \frac{y^{2}}{2}\right) \sin \theta \]This equation helps us determine how rapidly the fluid is flowing at different points in its thickness.

• The term \( \rho \) refers to the fluid's density, which affects the inertia of the fluid particles.
• \( g \) is the acceleration due to gravity, reflecting the force driving the fluid down the incline.
• \( \mu \) is the viscosity, indicating how much the fluid resists flow.
• The variables \( h \) and \( y \) show the fluid's thickness and the height from the base, respectively.
• \( \theta \) is the angle of inclination, affecting the component of the gravitational force parallel to the plane.

In simpler terms, this velocity profile shows that the speed is highest near the surface (at \( h \)) and decreases closer to the base (at \( y = 0 \)). The parabolic nature implies a steady and smooth flow over the inclined surface.

Shear Stress

Shear stress is another vital concept in biofluid mechanics. It measures how much force is needed to move layers of fluid across one another. When considering an inclined flow, understanding shear stress helps us grasp how the mechanical forces interact within the fluid.

The shear stress, \( \tau \), is calculated from the derivative of the velocity profile with respect to \( y \) using the equation:\[ \tau = \mu \frac{du}{dy} \]After substitution and simplification, the shear stress in this case becomes:\[ \tau(y) = \rho g \left(h - y \right) \sin \theta \]

• This indicates that shear stress is directly proportional to the fluid's density, gravitational force, and the angle of inclination, \( \theta \).
• It is inversely proportional to the height from the surface, \( y \), meaning stress is greatest near the base where \( y \) is maximized.

To visualize it, imagine the need for more force to slide two viscous fluid layers when they are close compared to when the layers are far apart.

Inclined Plane Flow

Inclined plane flow is a type of fluid movement driven by gravity down a sloped surface. Understanding this helps us see how fluids, like biological solutions, move over various surfaces naturally, which is crucial in both natural and industrial contexts.

When a fluid flows down an incline, certain factors like gravity, viscosity, and the angle of the slide play a role in determining the flow characteristics:

• Gravity helps accelerate the fluid downhill, affecting both velocity and shear stress.
• Viscosity stops the fluid from spreading too quickly, providing resistance as the fluid flows.
• The slope angle (\( \theta \)) influences how strongly gravity acts in the direction of flow, potentially increasing the fluid’s velocity and altering the stress distribution.

In our exercise, the flow behavior over the inclined plane is described in a way that highlights variations in velocity from top to bottom, indicating a non-uniform, stable flow that is often encountered in biofluid applications, such as the flow of blood or other bodily fluids over inclined surfaces in the human body.