Fluid Mechanics in Biophysics

Force and Pressure in a Fluid

Fluids play a crucial role in biophysical systems, transmitting forces and exerting pressure in all directions. Understanding how pressure varies within a fluid is essential for analyzing biological and physical phenomena.

• Uniform Force Transmission: Fluids transmit force uniformly in all directions.

• Isotropic Pressure: At any point in a fluid at rest, the pressure is the same in all directions.

• Pressure Variation with Depth: Pressure increases with depth due to the weight of the fluid above.

Pressure Difference Formula:

• For a fluid of constant density , the pressure difference between two points separated by vertical distance is:

Relationship Between Units of Pressure

Pressure can be measured in various units, and conversion between them is often necessary in scientific calculations.

Pascal's Principle

Pascal's principle states that an increase in pressure at any point in a confined fluid is transmitted undiminished throughout the fluid.

• Pressure Transmission: Any increase in pressure at one point is felt equally at all other points in the fluid.

• Formula for Pressure Increase:

• Force Transmission: The force acting on area is:

Example:

Hydraulic systems, such as brakes and lifts, utilize Pascal's principle to multiply force.

Hydrostatic Skeleton

Many soft-bodied animals use fluid pressure within their bodies to maintain structure and facilitate movement, a concept known as the hydrostatic skeleton.

• Mechanism: Circular and longitudinal muscles contract, changing internal pressure and producing movement.

• Effective Area: The effective area of circular muscle per unit length is .

• Force Generated: where is the force per unit area.

• Surface Area Along Cut:

• Force to Split Cylinder:

• Balance of Forces:

Example:

Worms use hydrostatic skeletons for locomotion by manipulating internal fluid pressure.

Archimedes's Principle

Archimedes's principle explains buoyancy, a key concept in understanding how organisms float or sink in fluids.

• Buoyant Force: A body submerged in a fluid experiences an upward force equal to the weight of the fluid displaced.

Formula:

• where is volume, is fluid density.

Power Required to Remain Float

Animals floating in water must balance gravitational and buoyant forces, and may need to expend energy to remain afloat.

• Fraction Submerged: For an animal of volume and density floating with fraction submerged:

• Difference in Forces:

• Mass of Water Accelerated:

• Force from Momentum Change:

• Velocity Required:

• Kinetic Energy Imparted:

• Power Generated:

Buoyancy of Aquatic Animals

Some aquatic animals adjust their body composition to achieve neutral buoyancy, allowing them to float effortlessly.

• Example: The cuttlefish contains a porous bone of density and body tissue of . To float in sea water (), the percentage of body volume occupied by the bone is calculated as:

Surface Tension

Surface tension arises from molecular forces at the interface between a liquid and another medium, affecting biological and physical phenomena.

• Molecular Attraction: Molecules in a liquid attract each other, resulting in zero net force for interior molecules.

• Surface Molecules: Molecules at the surface experience a net inward force, creating surface tension.

• Biological Example: Water striders exploit surface tension to walk on water.

Formulas:

• Weight of Supported Liquid Column:

• Force Due to Surface Tension:

• Height of Liquid Column:

Example:

Capillary action in plants and the ability of insects to walk on water are direct results of surface tension.

Summary Table: Key Fluid Mechanics Principles

Homework Problems

• Chapter 7, page 93

• Problems 7.6, 7.7

Additional info: These notes expand on the original slides by providing definitions, formulas, and biological examples relevant to college-level physics and biophysics.