Fluid Mechanics - Viscosity and Density

Definition: A measure of a fluid's resistance to deformation or flow.
Dynamic Viscosity (( \mu )): The ratio of shear stress to shear rate; units are Pascal-seconds (Pa·s) or poise.
Kinematic Viscosity (( \nu )): The ratio of dynamic viscosity to fluid density; units are square meters per second (m²/s).
Factors Affecting Viscosity:
- Temperature: Generally, viscosity decreases with increasing temperature for liquids and increases for gases.
- Pressure: For most liquids, viscosity increases with pressure.
Types of Fluids:
- Newtonian: Viscosity remains constant regardless of shear rate.
- Non-Newtonian: Viscosity changes with shear rate (e.g., shear-thinning, shear-thickening).

Definition: The mass per unit volume of a fluid; usually denoted by ( \rho ).
Units: Kilograms per cubic meter (kg/m³).
Factors Influencing Density:
- Temperature: Density generally decreases with increasing temperature for liquids and gases.
- Pressure: Density of liquids is relatively incompressible, but gases can have significant changes with pressure.
Specific Gravity: Ratio of a fluid's density to the density of water (at 4°C).

Definition: The force exerted per unit area within a fluid; denoted as ( P ).
Units: Pascals (Pa), bar, or atmospheres (atm).
Hydrostatic Pressure: Due to the weight of the fluid column above; given by ( P = \rho gh ), where:
- ( \rho ) = fluid density
- ( g ) = acceleration due to gravity
- ( h ) = height of the fluid column.
Pressure Variation:
- In a static fluid, pressure increases with depth.
- In dynamic fluids, pressure can change due to flow conditions.

Types of Flow:
- Laminar Flow: Smooth, orderly flow; characterized by low Reynolds numbers (Re < 2000).
- Turbulent Flow: Chaotic, irregular flow; characterized by high Reynolds numbers (Re > 4000).
Reynolds Number (Re): A dimensionless number that predicts flow regime; calculated as ( Re = \frac{\rho vD}{\mu} ), where:
- ( v ) = fluid velocity
- ( D ) = characteristic length (e.g., diameter of a pipe).
Continuity Equation: For incompressible fluids, ( A_1v_1 = A_2v_2 ), where ( A ) is the cross-sectional area and ( v ) is the velocity of flow.

Definition: The upward force exerted by a fluid on an object submerged in it.
Archimedes' Principle: An object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object.
Factors Affecting Buoyancy:
- Density of the fluid: Higher density fluids exert greater buoyant forces.
- Volume of the displaced fluid: Larger displaced volume leads to greater buoyant force.
Conditions for Floating and Sinking:
- An object floats if its density is less than the fluid's density.
- An object sinks if its density is greater than the fluid's density.
- An object is neutrally buoyant if its density equals that of the fluid.

Viscosity: Measures a fluid's resistance to flow and deformation.
Dynamic Viscosity (( \mu )): Ratio of shear stress to shear rate, measured in Pascal-seconds (Pa·s) or poise.
Kinematic Viscosity (( \nu )): Dynamic viscosity divided by fluid density, measured in square meters per second (m²/s).
Temperature Influence: Generally, liquid viscosity decreases with rising temperature, while gas viscosity increases.
Pressure Influence: Liquid viscosity typically increases with pressure, whereas gases show more variability.
Newtonian Fluids: Maintain constant viscosity regardless of shear rate.
Non-Newtonian Fluids: Exhibit variable viscosity in response to shear rates, such as shear-thinning and shear-thickening behaviors.

Density (( \rho )): Mass per unit volume of a fluid, usually expressed in kilograms per cubic meter (kg/m³).
Temperature Effect: With rising temperature, the density of liquids and gases generally decreases.
Pressure Effect: While liquids are mostly incompressible, gases can show significant density changes under varying pressures.
Specific Gravity: The ratio of a fluid's density to that of water at 4°C.

Pressure (( P )): Force per unit area within a fluid, commonly in Pascals (Pa), bars, or atmospheres (atm).
Hydrostatic Pressure: Calculated as ( P = \rho gh ), depends on the height of the fluid column, where:
- ( \rho ): Fluid density
- ( g ): Acceleration due to gravity
- ( h ): Height of the fluid column
Pressure Variation: In static fluids, pressure rises with depth; in dynamic fluids, pressure may vary based on flow dynamics.

Laminar Flow: Characterized by smooth and orderly motion, typically with Reynolds numbers (Re) less than 2000.
Turbulent Flow: Marked by chaotic and irregular flow patterns, with Reynolds numbers exceeding 4000.
Reynolds Number (Re): A dimensionless value that indicates flow regime, calculated by ( Re = \frac{\rho vD}{\mu} ).
Continuity Equation: Represents the conservation of mass in incompressible fluids, expressed as ( A_1v_1 = A_2v_2 ), where ( A ) is cross-sectional area and ( v ) is flow velocity.

Buoyancy: The upward force from a fluid on a submerged object.
Archimedes' Principle: States that a submerged object experiences a buoyant force equivalent to the weight of the fluid displaced.
Density Impact on Buoyancy: Higher density fluids generate greater buoyant forces.
Displaced Volume: A larger volume of displaced fluid results in an increased buoyant force.
Floating and Sinking Conditions:
- An object floats if it is less dense than the fluid.
- An object sinks if it is denser than the fluid.
- Neutral buoyancy occurs when the object's density matches that of the fluid.