Problem 11 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

The velocity distribution for laminar flow between two parallel plates can be represented as $$ \frac{u}{u_{\max }}=1-\left(\frac{2 y}{h}\right)^{2} $$ where \(h\) is the separation distance between the two flat plates and the origin is located half-way between the plates. Consider the flow of blood at \(37^{\circ} \mathrm{C}(\mu=3.5 \mathrm{cP})\) with maximum velocity of \(25 \mathrm{~cm} / \mathrm{s}\) and a separation distance of \(10 \mathrm{~mm}\). Calculate the force on a \(0.25 \mathrm{~m}^{2}\) section of the lower plate.

Step-by-Step Solution

Answer

The force on the lower plate is 8.75 N.

Step 1: Identify the Known Variables

The problem provides several key variables:- Maximum velocity, \(u_{\max} = 25 \text{ cm/s}\)- Viscosity of blood, \(\mu = 3.5 \text{ cP} = 3.5 \times 10^{-3} \text{ Pa}\cdot\text{s}\)- Separation distance, \(h = 10 \text{ mm} = 0.01 \text{ m}\)- Area of the plate, \(A = 0.25 \text{ m}^2\).From these values, we can proceed to calculate the velocity gradient and the shear stress on the plate.

Step 2: Determine Velocity Gradient at the Plate

The velocity distribution is given by \[ \frac{u}{u_{\max}} = 1 - \left( \frac{2y}{h} \right)^2 \]The velocity gradient at the plate, which is located at \(y = -\frac{h}{2}\), is determined by deriving the velocity distribution function:\[ \frac{du}{dy} = \frac{d}{dy}\left[ u_{\max}\left(1 - \left(\frac{2y}{h}\right)^2\right) \right] \]\[ = \frac{d}{dy}\left( u_{\max} - u_{\max}\frac{4y^2}{h^2} \right) \]\[ = -u_{\max}\frac{8y}{h^2} \]At the plate \(y = -\frac{h}{2}\):\[ \left.\frac{du}{dy}\right|_{y = -\frac{h}{2}} = -u_{\max} \cdot \frac{8(-h/2)}{h^2} = \frac{4u_{\max}}{h} \]

Step 3: Calculate Shear Stress on the Plate

Shear stress \(\tau\) on the plate is defined as\[ \tau = \mu \left( \frac{du}{dy} \right) \]Substituting the viscosity and the velocity gradient:\[ \tau = 3.5 \times 10^{-3} \times \frac{4 \times 25}{0.01} \]\[ \tau = 3.5 \times 10^{-3} \times 10000 \]\[ \tau = 35 \text{ Pa} \]

Step 4: Calculate Force on the Lower Plate

The force \(F\) exerted over the plate area \(A\) is given by:\[ F = \tau \cdot A \]Substituting the obtained shear stress and the area:\[ F = 35 \times 0.25 \]\[ F = 8.75 \text{ N} \]

Step 5: Conclusion

The calculated force on the lower plate is \(8.75 \text{ N}\). This result considers the given parameters and the laminar flow of blood between the plates.

Key Concepts

Laminar Flow

Laminar flow is a type of fluid motion where the fluid moves in smooth, parallel layers, without any disruption between them. In this mode, each layer slides over the one below it with very little mixing. This kind of flow usually occurs at lower velocities and is predictable, making it easier to model mathematically. In our exercise, blood flowing between two parallel plates represents laminar flow. The formula given for velocity distribution in such a scenario shows that the flow profile is parabolic. This means that the velocity is highest in the middle and decreases towards the plates. This predictable nature of laminar flow is crucial in various medical and engineering applications, where precise fluid modeling is required.

Shear Stress

Shear stress is a measure of how hard the fluid is pushing or pulling on a surface, causing it to deform. For the fluid moving over a surface, like blood over the lower plate in our exercise, shear stress depends on how fast the fluid layers are moving relative to each other. Mathematically, it is defined using the velocity gradient at the wall and the fluid's viscosity:

• Velocity gradient: The change in velocity as you move perpendicular to the flow direction.
• Fluid viscosity: A measure of the fluid's resistance to flow.

In the context of biofluid mechanics, understanding shear stress is vital since it can affect blood cell health and vessel walls. In our case, the calculated shear stress on the plate helps determine the force exerted by the moving blood.

Velocity Distribution

Velocity distribution refers to how the speed of the fluid varies within the flow field. In laminar flow between two plates, the velocity distribution is given by a parabolic equation. This equation models how the velocity changes with distance from the center, where it is at its peak. The distribution is crucial for determining how the fluid interacts with the surfaces, impacting both pressure and shear forces. The given formula, \[\frac{u}{u_{\max}} = 1 - \left(\frac{2y}{h}\right)^2\]shows that at the center, farthest from the exerting boundaries, blood moves fastest, and slows down as it nears the plates. This concept is important for engineers and medical professionals working with fluid transport systems and vessel flows.

Blood Flow

Blood flow is a vital aspect of both biofluid mechanics and physiological function. It concerns how blood moves through vessels and various body compartments. In the context of laminar flow, as seen in this exercise, understanding blood flow can help in designing medical devices, predicting blood pressure, and assessing cardiovascular health. The behavior of blood, a non-Newtonian fluid with varying viscosity, adds complexity to modeling its movement accurately. In our example, the maximum velocity and fixed temperature simulate real-world conditions where predictions about blood behavior can support medical interventions and device optimizations.

Fluid Viscosity

Fluid viscosity is a measure of a fluid's resistance to deformation or flow. It represents how "thick" or "sticky" a fluid is. In biofluid mechanics, viscosity is a critical property, especially for fluids like blood. Blood viscosity must be accurately considered to predict how it behaves under various flow conditions. It affects velocity distribution and shear stress, both of which we analyzed in the exercise. The exercise highlights blood's specific viscosity value at body temperature, which is lower compared to many other substances. This lower viscosity facilitates smoother flow but still affects various physiological and mechanical processes within the body.