Chapter 6: Fluids And Motion
Chapter 6 Fluids And Motion
About how fast can a small fish swim before experiencing turbulent flow around its body? How much higher must your blood pressure get to compensate for 5% narrowing in your blood vessels? (The pressure difference across your blood vessels is essentially equal to your blood pressure.) If someone replaced the water in your home plumbing with olive oil, how much longer would it take you to fill a bathtub? You are trying to paddle a canoe silently across a still lake and know that turbulence makes noise. How quickly can the canoe and the paddle travel through water without causing turbulence? The pipes leading to the showers in your locker room are old and inadequate. Although the city water pressure is 700,000 Pa, the pressure in the locker room when one shower is on is only 600,000 Pa. Use the volume equation: volume = (π pressure difference pipe diameter^4) / (128 pipe length fluid viscosity) to calculate the approximate pressure if three showers are on. If the plumbing in your dorm carried honey instead of water, filling a cup to brush your teeth could take a while. If the faucet takes 5 seconds to fill a cup with water, how long will it take with honey, assuming all other conditions remain unchanged? How quickly would you have to move a 1-cm-diameter stick through olive oil to reach a Reynolds number of 2000, so that turbulence begins to appear? Olive oil has a density of 918 kg/m^3. The effective obstacle length of a blimp is its width—the amount of airflow separation around it. How slowly would a 15-m wingspan blimp have to move to keep the airflow laminar? (Air density is 1.25 kg/m^3.)
Paper For Above instruction
Fluid dynamics plays a crucial role in understanding various natural and engineered systems, from the swimming speed of small fish to the design of aeronautical vehicles and blood flow in arteries. Critical parameters such as Reynolds number, pressure, viscosity, and flow regime (laminar or turbulent) determine how fluids move within different contexts. This paper explores these concepts, applying them to common scenarios like aquatic locomotion, plumbing systems, and aerodynamics, to elucidate the principles governing fluid motion and transition to turbulence.
Flow Around Small Fish and Reynolds Number
The speed at which a small fish can swim without experiencing turbulence depends on the Reynolds number, which characterizes the flow regime around the fish. The Reynolds number (Re) is given by Re = (ρ v L) / μ, where ρ is fluid density, v is flow velocity, L is characteristic length (such as body length), and μ is dynamic viscosity. For small fish, the threshold for laminar flow typically occurs at Re
Blood Pressure and Vascular Narrowing
When blood vessels narrow by 5%, the increased resistance necessitates a higher pressure difference to maintain flow. According to Poiseuille’s law, ΔP = (8 μ L Q) / (π r^4), where Q is flow rate, and r is vessel radius. A 5% decrease in radius results in roughly a 22% increase in resistance. Therefore, blood pressure must increase by a similar proportion to compensate, which could amount to a rise of approximately 120 mm Hg in typical systolic pressures, highlighting the cardiovascular impact of minor arterial narrowing (Guyton & Hall, 2015).
Filling a Bathtub with Olive Oil
Replacing water with olive oil significantly affects flow rate due to the difference in viscosity; olive oil has a viscosity approximately ten times that of water (~0.1 Pa·s vs. 0.001 Pa·s). Using the Hagen-Poiseuille equation adapted for orifice flow, the time to fill the bathtub increases proportionally with viscosity. If filling with water takes about 10 minutes, filling with olive oil would take roughly 100 minutes under identical pressure conditions, illustrating how viscosity drastically impacts flow rate (White, 2011).
Silent Canoeing and Turbulence
To paddle silently, the canoe and paddle must move at speeds that avoid turbulence. The critical Reynolds number for flow in water is around 2000. Using the same Reynolds number calculation, with water density at 998 kg/m^3, a typical paddle diameter of 0.3 m, and the critical Reynolds number of 2000, the maximum speed before turbulence is approximately 2 m/s. Maintaining below this speed ensures quiet, laminar flow and minimal noise from turbulence (Munson et al., 2013).
Plumbing Pressure Calculations
Given the pressure drop in the locker room plumbing system when one shower is on (from 700,000 Pa to 600,000 Pa), the pressure is reduced by 100,000 Pa. Assuming a linear relation and using the flow equation provided, with three showers operating, the pressure can be estimated with proportional scaling. This calculation indicates the pressure drops further to about 500,000 Pa when multiple fixtures are active, emphasizing the need for adequate pipe design to maintain sufficient pressure (Cengel & Boles, 2015).
Filling a Cup with Honey
The flow rate of honey through the faucet depends directly on viscosity; honey's viscosity (~10 Pa·s) is much higher than water (~0.001 Pa·s). The time to fill a cup, initially 5 seconds with water, scales proportionally with viscosity, resulting in approximately 50 seconds for honey. This example underscores the importance of viscosity in practical fluid transport (Schlichting & Gersten, 2017).
Reynolds Number and Turbulence in Olive Oil
A 1-cm-diameter stick moved through olive oil with a density of 918 kg/m^3 needs to reach a speed that results in Re = 2000. Rearranging the Reynolds number formula: v = (Re μ) / (ρ D). With μ ≈ 0.1 Pa·s and D=0.01 m, the required speed is about 2.18 cm/s. Moving the stick faster than this induces turbulent flow, demonstrating how fluid properties and object size determine flow regime transitions (Brennen, 2013).
Flow around a Blimp
To keep airflow laminar around a large blimp, the airflow speed must be sufficiently low. Using the Reynolds number criterion Re
Conclusion
Understanding the principles of fluid dynamics, including the effects of viscosity, pressure, and flow velocity, is essential in diverse fields spanning biology, engineering, and environmental sciences. From the swimming speed of fish to the design of quiet plumbing and aerodynamically efficient vehicles, the interplay of these parameters governs the behavior of fluids in practical applications. Recognizing the thresholds for laminar and turbulent flow enables optimized system designs, reducing energy consumption, noise, and wear while improving performance.
References
- Brennen, C. E. (2013). Hydrodynamics of Swimming and Flying. Princeton University Press.
- Cengel, Y. A., & Boles, M. A. (2015). Thermodynamics: An Engineering Approach. McGraw-Hill Education.
- Guyton, A. C., & Hall, J. E. (2015). Textbook of Medical Physiology. Elsevier.
- Milne-Thomson, L. M. (1968). Theoretical Hydrodynamics. Macmillan.
- Munson, B. R., Young, D. F., Okiishi, T. H., & Huebsch, W. W. (2013). Fundamentals of Fluid Mechanics. Wiley.
- Schlichting, H., & Gersten, K. (2017). Boundary-Layer Theory. Springer.
- Vogel, S. (1994). Life in Moving Fluids: The Physical Biology of Flow. Princeton University Press.
- White, F. M. (2011). Fluid Mechanics. McGraw-Hill Education.