Fluid Dynamics, Viscosity, Poiseuille’s Equation, and Oscillations: Physics for Life Sciences I (Lecture 23 Study Notes)

Fluid Properties and Pressure

Fluids: Definition and Types

Fluids are substances that can flow, including both liquids and gases. Their molecular arrangement and ability to change shape distinguish them from solids.

• Liquids are nearly incompressible; their molecules are closely packed but can move freely.

• Gases are compressible; their volume can be easily increased or decreased due to the greater spacing between molecules.

Pressure in Liquids

Pressure in a fluid at rest depends on depth and is described by the hydrostatic pressure equation:

• Hydrostatic Pressure: $p = p_0 + ho g d$ where $p_0$ is atmospheric pressure, $ho$ is fluid density, $g$ is acceleration due to gravity, and $d$ is depth.

• Pascal's Principle: A change in pressure at one point in an incompressible fluid is transmitted equally throughout the fluid: $p' = p + riangle p$.

Barometers

Barometers measure atmospheric pressure using a column of liquid. The height of the column is related to atmospheric pressure:

• $p_{atm} = ho g h$

• Mercury is commonly used due to its high density, resulting in a shorter column compared to water.

Archimedes’ Principle and Buoyancy

Buoyancy is the upward force exerted by a fluid on an object immersed in it. Archimedes’ principle states:

• Buoyant Force: $F_B = ho_f V_f g$ where $ho_f$ is fluid density, $V_f$ is volume of fluid displaced, and $g$ is gravity.

• Sink: $ho_{avg} > ho_f$ → $F_B < w_o$

• Float: $ho_{avg} < ho_f$ → $F_B > w_o$

• Neutral Buoyancy: $ho_{avg} = ho_f$ → $F_B = w_o$

Applications

• Boats float when the buoyant force equals their weight.

• Balloons rise when the buoyant force exceeds their weight.

Fluid Dynamics

Streamlines and Fluid Elements

Fluid flow can be visualized using streamlines, which trace the paths of individual fluid particles. Fluid elements contain a fixed volume, but their shape may change as they move.

Equation of Continuity

For incompressible, laminar flow, the equation of continuity relates the speed and cross-sectional area at different points in a tube:

• $v_1 A_1 = v_2 A_2$

• Volume Flow Rate: $Q = rac{ riangle V}{ riangle t}$

Pressure Gradients and Acceleration

Fluid elements accelerate when moving from wider to narrower sections of a tube due to pressure differences:

• A pressure gradient is a region with changing pressure from one point to another.

• An ideal fluid accelerates whenever there is a pressure gradient.

Bernoulli Effect

Along a streamline, pressure is higher where the fluid moves slower and lower where it moves faster. This is known as the Bernoulli effect, discovered by Daniel Bernoulli.

• The speed of a fluid can be measured by a Venturi tube.

Bernoulli’s Equation

Bernoulli’s equation is derived from conservation of energy and relates pressure, speed, and height at two points along a streamline:

• $p_1 + rac{1}{2} ho v_1^2 + ho g y_1 = p_2 + rac{1}{2} ho v_2^2 + ho g y_2$

Applications

• Lift on Airplane Wings: Faster airflow above the wing creates lower pressure, while slower airflow below creates higher pressure, resulting in lift.

• Magnus Effect: A spinning object in a fluid experiences a force perpendicular to its motion due to pressure differences caused by varying fluid velocities around the object.

Worked Example: Pressure in an Irrigation System

Bernoulli’s equation and the equation of continuity are used to solve for pressure differences in pipes with varying diameters and elevations.

• Given: $v_1 = 5.0$ m/s, $p_1 = 75$ kPa, $ho = 1000$ kg/m3

• Use: $p_2 = p_1 + rac{1}{2} ho (v_1^2 - v_2^2) + ho g (y_1 - y_2)$

• Find $v_2$ using $v_1 A_1 = v_2 A_2$

Viscosity and Poiseuille’s Equation

Viscosity

Viscosity is a measure of a fluid’s resistance to flow. Real fluids require a pressure difference to maintain constant flow speed.

• Coefficient of Viscosity: $ext{Units: N} ext{ s/m}^2$ or $ext{Pa} ext{ s}$

• Viscosity decreases rapidly with increasing temperature.

Table: Viscosities of Fluids

Fluid	${{η}}$ (Pa·s)
Air 20°C	$1.8 imes 10^{-5}$
Water 20°C	$1.0 imes 10^{-3}$
Water 40°C	$0.65 imes 10^{-3}$
Water 60°C	$0.47 imes 10^{-3}$
Whole blood 37°C	$3.5 imes 10^{-3}$
Motor oil -35°C	$3 imes 10^1$
Motor oil 100°C	$0.01$
Molasses 20°C	$5$

Poiseuille’s Equation

Describes viscous flow through a tube:

• Average Speed: $v_{avg} = rac{R^2}{8 ext{η}L} riangle p$

• Volume Flow Rate: $Q = v_{avg} A = rac{ ext{π} R^4 riangle p}{8 ext{η}L}$

• Flow rate increases dramatically with tube radius (to the fourth power).

The Circulatory System

Blood Flow and Pressure

Principles of fluid dynamics and viscosity apply to blood flow in the circulatory system.

• In large vessels, viscosity effects are negligible; in small vessels, viscosity is significant.

• Most pressure drop occurs in smaller vessels (arterioles and capillaries).

Blood Pressure

• Measured in mm Hg (millimeters of mercury).

• Systolic pressure: Peak pressure, typically 120 mm Hg.

• Diastolic pressure: Base pressure, typically 80 mm Hg.

• $1 ext{ mm Hg} = 133 ext{ Pa}$

Example: Change in Blood Pressure with Height

Raising a blood pressure cuff above the heart introduces a pressure difference due to gravity:

• $riangle p = ho g d$

• For $d = 0.15$ m, $ho = 1050$ kg/m3, $g = 9.8$ m/s2: $riangle p = 1540$ Pa $= 12$ mm Hg

Oscillations and Simple Harmonic Motion (SHM)

Equilibrium and Restoring Forces

Oscillatory systems have an equilibrium position and a restoring force that acts to return the system to equilibrium.

• Example: A marble in a bowl returns to the bottom when displaced.

• Restoring force: The net force directed toward equilibrium.

Oscillation: Period and Frequency

• Period ($T$): Time to complete one cycle.

• Frequency ($f$): Number of cycles per second ($f = 1/T$), measured in hertz (Hz).

Table: Common Units of Frequency

Frequency	Period
$10^3$ Hz = 1 kHz	1 ms
$10^6$ Hz = 1 MHz	1 μs
$10^9$ Hz = 1 GHz	1 ns

Simple Harmonic Motion (SHM)

SHM is a sinusoidal oscillation, typically modeled by a mass on a spring or a pendulum.

• Sinusoidal form: $x(t) = A ext{cos}( ext{ω} t + ext{φ})$

• Amplitude ($A$): Maximum displacement from equilibrium.

Linear Restoring Forces and SHM

• For a spring: $F_{net} = -k x$ (Hooke’s Law)

• Oscillation about equilibrium with a linear restoring force is always SHM.

Vertical Mass on a Spring

• Equilibrium position determined by $k riangle L = mg$

• Restoring force for displacement: $F_{net} = -k y$

Pendulum

• Restoring force for small angles: $F_{net} = -mg heta$ (using small-angle approximation $ext{sin} heta ightarrow heta$)

• Pendulum undergoes SHM for small displacements.

Summary Table: Frequency and Period

Oscillator	Frequency	Period
Mass on Spring	$f = rac{1}{2 ext{π}} rac{k}{m}$	$T = 2 ext{π} rac{m}{k}$
Pendulum	$f = rac{1}{2 ext{π}} rac{g}{L}$	$T = 2 ext{π} rac{L}{g}$

Key Concepts

• Fluid dynamics describes the motion of fluids and the forces involved.

• Bernoulli’s equation is a statement of energy conservation for ideal fluids.

• Poiseuille’s equation governs viscous flow through tubes.

• Oscillation is repetitive motion about equilibrium, characterized by amplitude, period, and frequency.

• Simple harmonic motion occurs when a linear restoring force acts to return a system to equilibrium.

Additional info: These notes expand on the provided slides with definitions, equations, and context for each topic, including worked examples and tables for viscosity and oscillation parameters.