Fluid Mechanics and Statics Lesson 14 by Owen Borville 12.3.2025
A fluid is a substance that flows freely by the force of gravity and takes the shape of its container. Fluids deform under shear stress by shearing forces. Fluids include liquids, gases, and plasmas. Fluid statics is the physics of stationary fluids.
Density of a substance is the mass per unit volume: ρ = m/V, where ρ is density, m is mass, and V is volume. The SI unit of density is kg/m^3.
Pressure is the force per unit perpendicular area over which the force is applied: P = F/A, P is pressure, F is force, A is area. SI units of pressure are the pascal and 1 Pa = 1 N/m^2. Other types of units are also used to measure pressure, including psi (pounds per square inch), bar, torr, and atmosphere (atm).
Pressure is the weight of the fluid m*g divided by the area A supporting it (the area of the bottom of the container): P = mg/A. Pressure due to the weight of a liquid is: P = h*ρ*g, where P is pressure, h is height of the liquid, ρ is the density of the liquid, and g is the acceleration due to gravity. Pressure is equal to the force per unit area.
Pressure at a depth h in a fluid of constant density is: p = p0 + ρgh
The change in pressure with height in a constant-density fluid: dp/dy = -ρg
Pascal's principle (by Blaise Pascal, 1623-1662) says that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. Pressure at one point in the fluid is applied the same amount everywhere in the fluid.
P1 = P2
F1/A1 = F2/A2
Pascal's principle is the foundation of hydraulic systems, which use this principle to multiply force, as in a hydraulic press or car lift. A hydraulic system is an enclosed fluid system used to exert forces.
Gauge pressure is the pressure relative to atmospheric pressure. Absolute pressure is the sum of gauge pressure and atmospheric pressure [p(abs) = p(g) + p(atm)].
Aneroid gauge measures pressure using a bellows-and-spring arrangement connected to the pointer of a calibrated scale.
Open-tube manometers are used to measure pressure and have U-shaped tubes and one end is always open. A mercury barometer is a device that measures atmospheric pressure.
Buoyant force is the net upward force on any object in any fluid. If the buoyant force is greater than the object's weight, the object will rise to the surface and float. If the buoyant force is less than the object's weight, the object will sink. If the buoyant force equals the object's weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.
Archimedes' principle (287-212 B.C.) states that the buoyant force on an object equals the weight of the fluid it displaces. Specific gravity is the ratio of the density of an object to a fluid (usually water).
Flow rate Q is the volume V flowing past a point in time t, Q = V/t or Q = dV/dt, where V is volume and t is time. SI unit of flow is (m^3/s) but there are other units.
Cohesive forces are attractive forces between molecules of the same type. Adhesive forces are attractive forces between molecules of different type. Surface tension is when cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. Capillary action is the tendency of a fluid to be raised or suppressed in a narrow tube, or capillary tube which is due to the relative strength of cohesive and adhesive forces.
Applications of fluid mechanics are in biological and biomedical systems.
Blood pressure is commonly measured by medical personnel. In addition, the pressures in other parts of the body can be measured and often provide valuable medical information.
The shape of the eye is maintained by fluid pressure (intraocular pressure). Glaucoma is when the circulation in the fluid in the eye is blocked, leading to a buildup in pressure.
Other pressures in the body are pressures in the spinal cord and skull (brain), bladder pressure, and pressure in the skeletal system. Hydrostatic and oncotic pressure exist in the capillaries. Respiratory pressure exists in the lungs during breathing. Pressure exists in the stomach (gastrointestinal pressure). The middle ear has pressure.
Flow rate and velocity are related by Q = Av where A is the cross sectional area of the flow and v is the average velocity.
The equation of continuity states that for an incompressible fluid, the mass flowing into a pipe must equal the mass flowing out of the pipe. Q1 = Q2. An incompressible fluid is a fluid with constant density that does not change when pressure is applied: ρ1A1V1 = ρ2A2V2 or the simplified equation: A1V1 = A2V2 (at constant density)
Bernoulli's (1700-1782) Equation relates fluid speed and pressure and states that the sum on each side of the following equation is constant, or the same at any two points in a incompressible frictionless fluid:
p1 + 1/2ρv1^2 + ρgh1 = p2 + 1/2ρv2^2 + ρgh2
Bernoulli's principle is Bernoulli's equation applied to situations in which the height of the fluid or depth is constant. The terms involving depth (or height h) subtract out from the equation: p1 + 1/2 ρv1^2 = p2 + 1/2 ρv2^2
For an incompressible, frictionless fluid, Bernoulli's equation is: p + 1/2ρv^2 + ρgy = constant
Bernoulli's principle has many applications, such as entrainment (when a moving fluid pulls in and mixes with a surrounding stationary fluid), velocity measurement, airplane aerodynamics and lift, hydraulic systems of pumps and turbines, and fluid flow measurement.
Power in fluid flow is the rate at which work is done or energy in any form is used or supplied.
P + 1/2ρv^2 + ρgh = constant
(P + 1/2 ρv^2 + ρvh)Q = power, where P is power associated with pressure, velocity, and height.
Laminar fluid flow is smooth flow of the fluid in layers that do not mix. Turbulence is described by swirls (or eddies) of mixed layers of fluid. Fluid viscosity (η) is due to friction within a fluid. Viscosity has units of (N/m^2)s or Pa*s.
Fluid viscosity is: η = FL/vA
Fluid flow is proportional to pressure difference and inversely proportional to resistance: Q = (p2-p1)/R
The pressure drop caused by flow and resistance is = p2-p1 = RQ
The Reynolds number N(R) can reveal whether fluid flow is laminar or turbulent. N(R) = 2ρvr/η
For N(R) below about 2000, flow is laminar. For N(R) above about 3000, flow is turbulent. For N(R) between 2000 and 3000, flow could be laminar or turbulent, or both.
Poiseuille's Law for Resistance (of fluid flow) = R = 8ηl/πr^4 for laminar flow in a tube.
Poiseuille's law for flow in a tube is Q (flow rate) = (p2-p1)* πr^4/(8ηl), where l is the length of the pipe or tube.
The pressure drop cause by flow and resistance is: P2-P1 = RQ
When there is an object moving in a fluid: N'R = ρ*v*L/η for an object in fluid, which indicates whether flow is laminar or turbulent.
For N'R less than 1 = flow is laminar. For N'R greater than 10^6, flow is turbulent
Diffusion is the movement of substances due to random thermal molecular motion. The average distance xrms a molecule travels by diffusion in a given amount of time is xrms = √2Dt, where D is the diffusion constant.
Osmosis is a process where molecules of a solvent pass through a semipermeable membrane from a less concentrated solution into a more concentrated one, equalizing the solute concentrations on each side of the membrane. Water molecules of the solvent move toward a higher solute concentration because there are less water molecules on the higher solute concentration side of the membrane. Therefore, each side of the membrane becomes more equally diluted or dissolved in water.
Dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference. Both processes can be reversed by back pressure. Active transport is a process in which a living membrane expends energy to move substances across it.
A fluid is a substance that flows freely by the force of gravity and takes the shape of its container. Fluids deform under shear stress by shearing forces. Fluids include liquids, gases, and plasmas. Fluid statics is the physics of stationary fluids.
Density of a substance is the mass per unit volume: ρ = m/V, where ρ is density, m is mass, and V is volume. The SI unit of density is kg/m^3.
Pressure is the force per unit perpendicular area over which the force is applied: P = F/A, P is pressure, F is force, A is area. SI units of pressure are the pascal and 1 Pa = 1 N/m^2. Other types of units are also used to measure pressure, including psi (pounds per square inch), bar, torr, and atmosphere (atm).
Pressure is the weight of the fluid m*g divided by the area A supporting it (the area of the bottom of the container): P = mg/A. Pressure due to the weight of a liquid is: P = h*ρ*g, where P is pressure, h is height of the liquid, ρ is the density of the liquid, and g is the acceleration due to gravity. Pressure is equal to the force per unit area.
Pressure at a depth h in a fluid of constant density is: p = p0 + ρgh
The change in pressure with height in a constant-density fluid: dp/dy = -ρg
Pascal's principle (by Blaise Pascal, 1623-1662) says that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. Pressure at one point in the fluid is applied the same amount everywhere in the fluid.
P1 = P2
F1/A1 = F2/A2
Pascal's principle is the foundation of hydraulic systems, which use this principle to multiply force, as in a hydraulic press or car lift. A hydraulic system is an enclosed fluid system used to exert forces.
Gauge pressure is the pressure relative to atmospheric pressure. Absolute pressure is the sum of gauge pressure and atmospheric pressure [p(abs) = p(g) + p(atm)].
Aneroid gauge measures pressure using a bellows-and-spring arrangement connected to the pointer of a calibrated scale.
Open-tube manometers are used to measure pressure and have U-shaped tubes and one end is always open. A mercury barometer is a device that measures atmospheric pressure.
Buoyant force is the net upward force on any object in any fluid. If the buoyant force is greater than the object's weight, the object will rise to the surface and float. If the buoyant force is less than the object's weight, the object will sink. If the buoyant force equals the object's weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.
Archimedes' principle (287-212 B.C.) states that the buoyant force on an object equals the weight of the fluid it displaces. Specific gravity is the ratio of the density of an object to a fluid (usually water).
Flow rate Q is the volume V flowing past a point in time t, Q = V/t or Q = dV/dt, where V is volume and t is time. SI unit of flow is (m^3/s) but there are other units.
Cohesive forces are attractive forces between molecules of the same type. Adhesive forces are attractive forces between molecules of different type. Surface tension is when cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. Capillary action is the tendency of a fluid to be raised or suppressed in a narrow tube, or capillary tube which is due to the relative strength of cohesive and adhesive forces.
Applications of fluid mechanics are in biological and biomedical systems.
Blood pressure is commonly measured by medical personnel. In addition, the pressures in other parts of the body can be measured and often provide valuable medical information.
The shape of the eye is maintained by fluid pressure (intraocular pressure). Glaucoma is when the circulation in the fluid in the eye is blocked, leading to a buildup in pressure.
Other pressures in the body are pressures in the spinal cord and skull (brain), bladder pressure, and pressure in the skeletal system. Hydrostatic and oncotic pressure exist in the capillaries. Respiratory pressure exists in the lungs during breathing. Pressure exists in the stomach (gastrointestinal pressure). The middle ear has pressure.
Flow rate and velocity are related by Q = Av where A is the cross sectional area of the flow and v is the average velocity.
The equation of continuity states that for an incompressible fluid, the mass flowing into a pipe must equal the mass flowing out of the pipe. Q1 = Q2. An incompressible fluid is a fluid with constant density that does not change when pressure is applied: ρ1A1V1 = ρ2A2V2 or the simplified equation: A1V1 = A2V2 (at constant density)
Bernoulli's (1700-1782) Equation relates fluid speed and pressure and states that the sum on each side of the following equation is constant, or the same at any two points in a incompressible frictionless fluid:
p1 + 1/2ρv1^2 + ρgh1 = p2 + 1/2ρv2^2 + ρgh2
Bernoulli's principle is Bernoulli's equation applied to situations in which the height of the fluid or depth is constant. The terms involving depth (or height h) subtract out from the equation: p1 + 1/2 ρv1^2 = p2 + 1/2 ρv2^2
For an incompressible, frictionless fluid, Bernoulli's equation is: p + 1/2ρv^2 + ρgy = constant
Bernoulli's principle has many applications, such as entrainment (when a moving fluid pulls in and mixes with a surrounding stationary fluid), velocity measurement, airplane aerodynamics and lift, hydraulic systems of pumps and turbines, and fluid flow measurement.
Power in fluid flow is the rate at which work is done or energy in any form is used or supplied.
P + 1/2ρv^2 + ρgh = constant
(P + 1/2 ρv^2 + ρvh)Q = power, where P is power associated with pressure, velocity, and height.
Laminar fluid flow is smooth flow of the fluid in layers that do not mix. Turbulence is described by swirls (or eddies) of mixed layers of fluid. Fluid viscosity (η) is due to friction within a fluid. Viscosity has units of (N/m^2)s or Pa*s.
Fluid viscosity is: η = FL/vA
Fluid flow is proportional to pressure difference and inversely proportional to resistance: Q = (p2-p1)/R
The pressure drop caused by flow and resistance is = p2-p1 = RQ
The Reynolds number N(R) can reveal whether fluid flow is laminar or turbulent. N(R) = 2ρvr/η
For N(R) below about 2000, flow is laminar. For N(R) above about 3000, flow is turbulent. For N(R) between 2000 and 3000, flow could be laminar or turbulent, or both.
Poiseuille's Law for Resistance (of fluid flow) = R = 8ηl/πr^4 for laminar flow in a tube.
Poiseuille's law for flow in a tube is Q (flow rate) = (p2-p1)* πr^4/(8ηl), where l is the length of the pipe or tube.
The pressure drop cause by flow and resistance is: P2-P1 = RQ
When there is an object moving in a fluid: N'R = ρ*v*L/η for an object in fluid, which indicates whether flow is laminar or turbulent.
For N'R less than 1 = flow is laminar. For N'R greater than 10^6, flow is turbulent
Diffusion is the movement of substances due to random thermal molecular motion. The average distance xrms a molecule travels by diffusion in a given amount of time is xrms = √2Dt, where D is the diffusion constant.
Osmosis is a process where molecules of a solvent pass through a semipermeable membrane from a less concentrated solution into a more concentrated one, equalizing the solute concentrations on each side of the membrane. Water molecules of the solvent move toward a higher solute concentration because there are less water molecules on the higher solute concentration side of the membrane. Therefore, each side of the membrane becomes more equally diluted or dissolved in water.
Dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference. Both processes can be reversed by back pressure. Active transport is a process in which a living membrane expends energy to move substances across it.