Comprehensive Guide to Properties of Fluids and Fluid Dynamics

Density

A measure of how much mass is contained in a given volume.

Density Formula

\rho = \frac{m}{V}, where \rho = density, m = mass, and V = volume.

Pressure

Defined as the magnitude of the force acting perpendicular to a surface divided by the area over which the force is distributed.

Pressure Formula

P = \frac{F}{A}, where P = pressure, F = force, and A = area.

Hydrostatic Pressure

The pressure that increases as you go deeper into a fluid due to the weight of the fluid above.

Hydrostatic Pressure Formula

P = P_0 + \rho g h, where P is pressure at depth h.

Absolute Pressure

The total pressure, including atmospheric pressure.

Gauge Pressure

The pressure relative to atmospheric pressure.

Pascal's Principle

A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid.

Buoyant Force

The upward force exerted by a fluid on an object immersed in it.

Archimedes' Principle

The upward buoyant force on an object is equal to the weight of the fluid displaced by the object.

Hydraulic Lift

A system that uses Pascal's Principle to lift heavy objects.

Continuity Equation

The principle that the mass flow rate must remain constant from one cross-section of a pipe to another.

Flow Rate

The volume of fluid passing a point per unit time, given by Q = A v.

Bernoulli’s Principle

Relates pressure, flow speed, and height in a fluid system.

Bernoulli’s Equation

P1 + \rho g y1 + \frac{1}{2}\rho v1^2 = P2 + \rho g y2 + \frac{1}{2}\rho v2^2 .

Venturi Effect

When velocity increases in a constricted section of a pipe, pressure decreases.

Torricelli’s Theorem

The speed of efflux from a hole in a tank is given by v = \sqrt{2gh}.

Equilibrium in Fluids

States that in static situations, the sum of forces equals zero.

Free Body Diagram (FBD)

A diagram that shows all the forces acting on an object.

Fluid Dynamics

The study of fluids in motion.

Ideal Fluid

A hypothetical fluid that is incompressible and non-viscous.

Incompressible Fluid

Fluid with constant density, primarily applicable to liquids.

Non-viscous Fluid

Fluid with no internal friction or viscosity.

Laminar Flow

Smooth fluid flow with parallel layers, no turbulence.

Irrotational Flow

Fluid particles do not rotate about their own axes.

Fluid Statics

The study of fluids at rest.

Hydrostatic Formula

P = P_0 + \rho g h, used to calculate pressure at a depth in a fluid.

Depth measured in Hydrostatics

In P = P_0 + \rho g h, h is measured down from the surface.

Weight of Fluid Displaced

Used to calculate buoyant force as stated in Archimedes' Principle.

Density of Fluid

The property of the fluid that matters for buoyancy calculations.

Corrected Execution for Buoyant Force

Use the density of the fluid and volume displaced, not the object's mass.

Pressure Conversion

Be aware of atmospheric pressure when calculating absolute pressure.

Constricting Pipe Dynamics

Squeezing a pipe leads to increased fluid velocity and decreased pressure.

Units of Density

Ensure density is in kg/m^3, not g/cm^3.

Standard Density of Water

Approx. \rho_{water} \approx 1000 \, kg/m^3.

Standard Density of Air

Approx. \rho_{air} \approx 1.29 \, kg/m^3 at sea level.

Pascal's Unit

1 Pascal is equal to 1 \, N/m^2.

Atmospheric Pressure

Approx. 1.013 \times 10^5 \, Pa.

Basic Fluid Properties

Density, pressure, and buoyant force define fluid behavior.

Fluid in Motion

Dynamic systems involve the study of how fluids behave during flow.