Problem 5 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Flow in a two-dimensional channel of width \(W\) has a velocity profile defined by $$ v_{x}(y)=k\left[\left(\frac{W}{2}\right)^{2}-y^{2}\right] $$ where \(y=0\) is located at the center of the channel. Sketch the velocity distribution (with \(k=1\) and \(W=10\) ) and find the shear stress/unit width of the channel at the wall.

Step-by-Step Solution

Answer

The velocity profile is parabolic; at the wall, the shear stress is \(-10 \times \mu\).

Step 1: Interpret the Velocity Profile

The velocity profile is given as \(v_{x}(y)=k\left[\left(\frac{W}{2}\right)^{2}-y^{2}\right]\). Here, \(k\) is a constant, \(W\) is the channel width, and \(y\) represents the position from the center line (\(y=0\)) of the channel to the wall (at \(y=\pm \frac{W}{2}\)). The profile indicates a parabolic distribution, maximum at the center and zero at the walls.

Step 2: Substitute Given Values into the Equation

Set \(k=1\) and \(W=10\) in the equation: \[v_{x}(y)=1\left[\left(\frac{10}{2}\right)^{2}-y^{2}\right]=25-y^{2}\]The velocity profile simplifies to \(v_{x}(y)=25-y^{2}\).

Step 3: Sketch the Velocity Distribution

Plot \(v_{x}(y) = 25-y^2\) against \(y\) for \(y\) ranging from \(-5\) to \(5\). The graph is a downward-facing parabola centered at \(y=0\), with the maximum velocity at the center (\(y=0\)) being 25 and decreasing to zero at \(y=\pm5\).

Step 4: Differentiate to Find Velocity Gradient at the Wall

The shear stress is related to the velocity gradient. First, differentiate the velocity profile with respect to \(y\):\[\frac{dv_{x}}{dy} = \frac{d}{dy}(25 - y^{2}) = -2y\]The surface of the wall is located at \(y=\pm5\). Substitute \(y=5\) (since both walls are symmetric, choose one for calculation):\[\frac{dv_{x}}{dy}\bigg|_{y=5} = -2(5) = -10\]

Step 5: Calculate Shear Stress at the Wall

The shear stress \(\tau\) is given by the equation \(\tau = \mu \frac{dv_{x}}{dy}\), where \(\mu\) is the dynamic viscosity. Given that \(\frac{dv_{x}}{dy} = -10\) at \(y=5\), the shear stress per unit width is:\[\tau = \mu \times -10\]Since \(\mu\) is not provided, the exact numerical value of shear stress is \(-10 \times \mu\). Complete shear stress would require value for \(\mu\).

Key Concepts

Velocity Profile

In fluid mechanics, the velocity profile defines how the velocity of fluid particles varies across a section of a channel. For our given problem, the velocity profile of fluid flowing in a two-dimensional channel is described by the equation \[v_{x}(y)=k\left[\left(\frac{W}{2}\right)^{2}-y^{2}\right]\]Here, the profile is parabolic, which is typical in laminar flow through channels. This means that the velocity is highest at the channel center (where the value of \(y\) is zero) and reduces to zero at the walls of the channel (where \(y\) equals ±\(W/2\)).

• The term \(k\) is a scaling constant that affects the overall magnitude of the velocity.
• \(W\) is the total width of the channel.
• \(y\) represents the transverse position from the center to the wall.

When substituting \(k = 1\) and \(W = 10\), the equation simplifies, resulting in a parabolic velocity profile indicative of steady, uniform flow.

Shear Stress

Shear stress in fluid mechanics arises due to the fluid flow across surfaces such as the channel walls. This stress is essentially a force per unit area exerted parallel to the wall. It is crucial because it determines how forcefully the fluid layers interact at various points in the flow.
The shear stress \(\tau\) at the wall can be found using the formula:\[\tau = \mu \frac{dv_{x}}{dy}\]Where:\

\
• \(\mu\) is the dynamic viscosity, which quantifies the fluid's resistance to flow.
\
• \(\frac{dv_{x}}{dy}\) is the velocity gradient at the channel wall.
\

In this example, the shear stress per unit width depends on the velocity gradient at the wall. The problem indicates that the wall is y = \(±5\). Thus, by calculating \(\frac{dv_{x}}{dy}\) at the wall, the shear stress per unit width is \(-10 \times \mu\), where \(\mu\) is unknown unless provided.

Velocity Distribution

The velocity distribution refers to how velocity values are spread across the cross-section of the flow. It provides insight into how momentum is distributed within the fluid. By examining the graph of the velocity distribution, termed as v_{x}(y) = 25-y^2, we observe the following characteristics:

• The maximum velocity occurs at the channel center (\(y = 0\)), in this case, 25 units.
• Velocity decreases symmetrically towards the channel walls, reaching zero at \(y = \pm5\).

This typical parabolic distribution suggests a steady, uniform flow with no abrupt changes in velocity, making it a classical representation in fluid mechanics studies of laminar flows in channels. The shape of the velocity curve directly impacts how forces such as shear stress interact with the channel walls.

Differentiation

In our context, differentiation is a mathematical technique used to obtain the derivative of the velocity profile with respect to the position \(y\). This operation allows us to compute the velocity gradient, a crucial factor for evaluating shear stress.
The differentiation of the velocity profile \(v_{x}(y) = 25-y^2\) results in:\[\frac{dv_{x}}{dy} = \frac{d}{dy}(25 - y^{2}) = -2y\]This derivative describes how rapidly velocity changes with respect to \(y\). At the wall (\(y=\pm5\)), substituting \(y = 5\) gives a gradient of -10. This negative value indicates that the flow velocity decreases as one approaches the wall, reflecting how layered fluid particles within the channel slow down relative to one another.
Differentiation, therefore, is an essential operation in pinpointing velocity gradients, instrumental in evaluating mechanical stresses within fluid systems.