Problem 64 - Calculus for Biology and Medicine

Question

Some worms swim by passing an undulatory wave along their bodies. The force that small worms apply to the water by passing this wave can be modeled using a formula derived by Lamb (1911) $$ F=\frac{4 \pi L \mu U}{\left(-0.077-\ln \left(\frac{\rho U a}{4 \mu}\right)\right)} $$ where \(U\) is the velocity of undulation, \(L\) is the length of the worm, \(a\) is the radius of the worm's body, and \(\mu\) and \(\rho\) respectively the viscosity (or "stickiness") and density of the water through which the worm swims. Calculate \(d F / d U\), the rate of change of the force with increasing undulation velocity.

Step-by-Step Solution

Answer

The rate of change \( \frac{dF}{dU} \) is given by the expression \( \frac{4 \pi L \mu (0.923 - \ln (\frac{\rho U a}{4 \mu}))}{\left(-0.077 - \ln (\frac{\rho U a}{4 \mu})\right)^2} \).

Step 1: Identify Components

First, identify all components involved in the formula. We have:- The force applied, \( F \),- Velocity of undulation, \( U \),- Length of the worm, \( L \),- Radius of the worm's body, \( a \),- Viscosity, \( \mu \),- Density of water, \( \rho \). The given formula is \( F=\frac{4 \pi L \mu U}{\left(-0.077-\ln \left(\frac{\rho U a}{4 \mu}\right)\right)} \). We need to find \( \frac{dF}{dU} \).

Step 2: Apply Quotient Rule

The function \( F \) involves a quotient. Use the quotient rule for differentiation:\[ \frac{d}{dU} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dU} - u \cdot \frac{dv}{dU}}{v^2} \]where \( u = 4 \pi L \mu U \) and \( v = -0.077 - \ln \left( \frac{\rho Ua}{4\mu} \right) \).

Step 3: Differentiate Numerator

Differentiate the numerator \( u = 4 \pi L \mu U \):\( \frac{du}{dU} = 4 \pi L \mu \)since \( 4 \pi L \mu \) are constants.

Step 4: Differentiate Denominator

Differentiate the function \( v = -0.077 - \ln \left( \frac{\rho Ua}{4\mu} \right) \):First find \( \frac{d}{dU} \left( \ln \left( \frac{\rho Ua}{4\mu} \right) \right) = \frac{1}{\frac{\rho Ua}{4\mu}} \times \frac{\rho a}{4\mu} \). This simplifies to \( \frac{1}{U} \).Thus, \( \frac{dv}{dU} = - \frac{1}{U} \).

Step 5: Substitute into Quotient Rule

Substitute the derivatives back into the quotient rule:\[ \frac{dF}{dU} = \frac{\left(-0.077 - \ln \left( \frac{\rho U a}{4 \mu} \right)\right) \cdot 4 \pi L \mu - 4 \pi L \mu U \left( -\frac{1}{U} \right)}{\left(-0.077 - \ln \left( \frac{\rho U a}{4 \mu} \right)\right)^2} \]Notice that the terms simplify as the second term actually turns into \( 4 \pi L \mu \).

Step 6: Simplify the Expression

Combine and simplify the expression:\[ \frac{dF}{dU} = \frac{4 \pi L \mu (-0.077 - \ln \left( \frac{\rho U a}{4 \mu} \right) + 1)}{\left(-0.077 - \ln \left( \frac{\rho U a}{4 \mu} \right)\right)^2} \]This is the rate of change of force with respect to undulation velocity.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us understand how a function changes. In biological contexts, such as the motion of organisms, differentiation allows us to analyze the rate at which something is changing with respect to time or another variable. A derivative represents this rate of change.
For example, in our problem, the derivative \( \frac{dF}{dU} \) indicates how the force (\( F \)) exerted by the worm changes as the velocity of its undulation (\( U \)) increases. By examining this derivative, biologists or mathematicians can predict how altering the undulation speed affects the force applied to the surrounding water. This insight is crucial in understanding how organisms interact with their environments and optimize their movements.

Quotient Rule

The quotient rule is a technique used in differentiation when dealing with functions that are ratios of other functions. When we have a function expressed as \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of a variable like \( U \), the quotient rule tells us how to find its derivative.
The rule is as follows:

• Differentiate \( u \), which we call \( \frac{du}{dU} \),
• Differentiate \( v \), yielding \( \frac{dv}{dU} \).
• The derivative of \( \frac{u}{v} \) is then \( \frac{v \cdot \frac{du}{dU} - u \cdot \frac{dv}{dU}}{v^2} \).'

This formula helps us differentiate complicated factional expressions like the one in the force problem for undulatory locomotion. By applying the quotient rule correctly, one can break down the complexities of interdependent biological processes. Understanding these intricacies leads to better models of motion and mechanics in nature.

Biological Modelling

Biological modelling is a powerful technique used to represent real-world biological processes through mathematical equations and simulations. By using models, scientists and researchers can predict behaviors and outcomes in biological systems which would be difficult to analyze otherwise.
Our exercise involves modeling the force generated by undulating worms in water, a problem with real-world application in understanding locomotion. The formula used in the problem is a reflection of the underlying dynamics of how force, velocity, viscosity, density, and other biological parameters interact. Here are a few points to ponder:

• Biological models enable scientists to simulate different scenarios, such as varying undulation speeds, to observe potential effects on biological functions.
• Models need validation with real-world data to confirm their accuracy.
• They offer a way to test hypotheses about natural phenomena in a controlled, risk-free environment.

Effective models can lead to breakthroughs in medicine, environmental science, and understanding evolutionary behavior.

Undulatory Locomotion

Undulatory locomotion is a type of movement exhibited by some organisms in which waves of motion pass along the body to generate propulsion. This method is common in aquatic environments, where it is used by worms, eels, and other elongate organisms.
In the given exercise, understanding this locomotion is crucial as it informs the development of the mathematical model. The model describes how force changes due to velocity, giving insights into:

• The physical principles driving this type of motion,
• How changes in bodily wave patterns affect propulsion efficiency,
• The interaction between the organism and its fluid environment.

By using calculus to analyze these factors, researchers can make predictions about energy expenditure and the physical adaptations necessary for efficient movement in water. This insight enhances our comprehension of evolutionary adaptations and biological efficiency across diverse life forms.