Fluid Mechanics Concepts

Basic Fluid Mechanics and Hydraulic Machines

• Fluid mechanics deals with the behavior of fluids and gases when they are at rest and in motion.
• Understanding fluid mechanics is important in many branches of engineering, including biomechanics, oceanography, chemical processing, aeronautics, mechanical engineering, civil engineering, and environmental science.

Dimensions and Systems of Units

• There are four fundamental dimensions: length (L), mass(M), time (T) and temperature (Θ).
• All other dimensions can be expressed in terms of these fundamental dimensions.
• Force is expressed using Newton's second law with this formula F=ma.
• F dimensionally = ML/T^2
• The system to be used is the SI (System International).
• Derived units are Area (L^2), Volume (L^3), Velocity (L/T), Density (M/L^3), Pressure (M/LT^2), Work (ML^2/T^2), Power(ML^2/T^3), Viscosity (M/LT) and flow rate (L^3/T).
• Weight is related to mass using the formula W = mg where g is the local gravity.
• The standard value for g is 9.80 m/s^2, and the SI unit for weight is expressed in Newtons (N).

Non-Dimensional Quantity

• A non-dimensional quantity has no unit, only a number.
• Common dimensionless parameters include:
  • Euler's number (Eu)
  • Reynold's number (Re)
  • Froude number (Fr)
  • Mach number (M)
  • Weber number (We)
  • Strouhal (S)
• Fp = pressure force which can be expressed as Δ pA ~ Δ pr^2.
• Fi = inertial force which can be expressed as m(dv/ds) ~ ρl^2v(v/l) = ρl^2v^2.
• Fv: Viscous force, expressed by τA = μ(du/dy)A ~ μlv
• Fg: Gravity force, expressed by mg ~ ρl^3g.
• FB: Compressibility force, expressed by BA ~ (dp/ρ)p = ρc^2l^2.
• FS: Surface tension force, expressed by σl.
• Fω Centrifugal force, expressed by m r ω² ~ρ l^3ω² =ρl^3ω².

Reynold's Number

• The ratio of inertial force to viscous force is a significant parameter.
• Reynolds number Re = (ρl^2v)/(μlv) = (ρlv)/μ
• It is significant for pipe flow, it characterizes the flow regime with viscous forces.

Euler Number

• Expressed as Eu = (Δp)/(ρv^2)
• Useful in fluid flow in pumps, indicates the pressure drop.

Mach number

• The Mach number, expressed as M = v/c, is significant when compressibility matters, as in airfoils on aircraft.

Pressure Scales

• In fluid mechanics, pressure comes from a normal compressive force acting on an area.
• Defined as force per unit area (P = F/A).
• The SI unit is Newtons per square meter (N/m^2) or Pascal (Pa).
• Pascal is small unit, so kilo Pascal (kPa) or Mega Pascal (MPa) are typically used.
• Standard atmospheric pressure at sea level is 101.3 kPa.
• Gauge pressure is recorded by the gauge or manometer
• Absolute pressure = gauge pressure + atmospheric pressure (Pa = Pg + Patm).

Gauge Pressure

• Negative gauge pressure occurs when the absolute pressure is less than atmospheric pressure, also called vacuum.
• For negative pressures: P(absolute) = P(atm) - P(gauge).

Fluid Properties

• A liquid is a state of matter where molecules can change positions but are restricted by cohesive forces to maintain a fixed volume.
• Liquids take the shape of their container.

Density and Specific Weight

• Fluid density is mass per unit volume (ρ = mass/volume = m/V).
• Specific weight is weight per unit volume (γ = weight/volume = mg/V).
• γ = ρg
• Specific gravity (S) is the ratio of the density of a liquid to the density of water (S = density of liquid / density of water).
• Hydrometer measures specific gravity of liquid.
• Density, specific weight, and specific gravity of water at standard conditions table.
  • Water has a density of 1000 Kg/m^3
  • Water’s specific weight is 9800 N/m^3
  • Water has a specific gravity of 1

Viscosity

• Shear stresses arise when a fluid is in motion. Particles move differently, deforming the fluid.
• A fluid at rest has no shear forces.
• Fluid sticks to solid boundaries (no slip condition).
• The velocity of fluid varies from layer to layer:
• Shear stress (τ) is given by τ = μ (du/dy), where μ is dynamic viscosity and du/dy is the velocity gradient.
• Units of τ are N/m^2 or Pa, and units of μ are Ns/m^2.
• Kinematic viscosity (ν) is the ratio of dynamic viscosity to density:
• ν = μ/ρ
• Unit of ν is m^2/s.
• Viscosity affects fluid flow; higher viscosity (like honey) implies slower flow.
• Accounted for pressure and energy losses in pipes.
• Newtonian fluids:
  • Shear stress = proportional to velocity gradient such as water, air, and oil.
• Non-Newtonian fluids:
  • Do not obey the above viscosity law such as milk, plastic, and paint.

Surface Tension

• Force that appears only in liquids at an interface, typically liquid-gas.
• Has units of force per unit length (N / m).
• The force due to surface tension is the surface tension, multiplied by the length or the circumference For droplet of water, the force exerted is F = p ρ πR^2 Pressure force and tension force must balance each other: πr² = 2πRσ p ρπΓ , so p = (2σ)/R.

Capillary Action

• Rise of liquid in a clean glass capillary tube due to surface tension.
• Liquid forms a contact angle β with the tube; β is zero for water and greater than and = 90°
• Water's beta = 0: a liquid is defined as having capillary drop instead of rise.
• Equate vertical component of surface tension force and weight of liquid column to find capillary rise (h)

Compressibility and Mach Number

• Fluids deform due to pressure changes; greater pressures lead to increased density.
• Compressibility is the change in pressure divided by the relative change in density at constant temperature.
• 21 MPa it is needed to cause a 1% change in density for water, so water is incompressible. Gases will however change.
• Speed of sound (C) is related to changes in pressure and density by C = √(dp/dρ).
• For gases undergoing isentropic process C √(KP/ρ), making use of ideal gas: p = RT
• C = √(kRT)
• Mach number (M) is the ratio of fluid velocity to local sound velocity expressed as M = V/C.
• Compressible flow is subsonic (M < 1).
• Supersonic flow is (M > 1)
• Transonic flow is approximately (M = 1)
• Hypersonic flow is generally above (M > 5).
• Air is considered incompressible if (M < 0.3) or air velocity is around 100 m/s.

Fluid Flow

• Liquid is fluid statics (water container).
• Moving rivers are characterized as fluid dynamics (rivers, pipes, turbines, air, water).
• External flows include studying things aeroplanes aerofoils and ships & rockets.
• Internal flows like flow in a pipe refers to pumps.
• Compressible flow sees density varies.
• Incompressible flow sees density constant.
• Mach number criterion of density of water vs air.
• Equations derived are Bernoullis and Eulers from second or third law of motion, with continuity equations and momentum Equations for the help of this
• Jet has impact on hydro turbines.

Scope of fluid mechanics

• Dimensional analyses deals with:

  • measurement of SI units and non dimensional qualities
  • measurement of mass and capillary.

• Fluid statics:

  • fluid pressure
  • hydro static forces and stability.

• Fluid kinematics:

  • one, or two or three dimensions steady flows
  • unsteady flows or streamlines.

• Internal flows

  • single and multi pipe and turbulent flow, and losses

• External flows

  • friction and pressure drag, drag coefficients and flow concepts.
  • Also studies how turbomachines interact with energy along with angular momentum.

• Pump

  • centrifugal pump
  • laws, impulse and reaction.

Laminar and Turbulent Flow

• Fluid move around closed conduits.
• Osbourne did experiments on laminar or turbulent flow in a pipe.
• Laminar flow is where there is a parallel streaming on one line whereas turbulent flow is where lines diffuse.
• Reynolds no distinguishes flow types:
  • Re less tha 2100 equals laminar vs greater than 4000.
  • Reynolds is found using the fluid velocity and viscosity and the diameter for pipe flow.

Momentum Equation for One-Dimensional and Two-Dimensional Flow

• Momentum is the product of the mass of a particle and the velocity. Momentum = mV

• Momentum changes velocity corresponding via Newton's second law of relation w applied force

• In a straight line the incoming outgoing and outgoing velocity flows are in the same direction

• When there is no storage within the control volume due to continuity equation, ṁ = ρ2 A2 V2 = ρ𝐴𝑉.

• The Rate of momentum acorss the AB is equal to mass and times the velocity

• Newton's laws are equated as: F = ṁ (V2 - V₁)

• Forces Fx and Fy when combined results in: F = √Fx^+ Fy^=

• Here is the force acting on the fluid equal to opposting forces on the X Y respectively

Jet Striking a Plate

• Can hit one of these three ways:
  • jet is striking a plate which is perpendicular
  • jet is inclined to plate
  • palte is moving in the direction of the jet.
• Normal jet velocity Vnormal = (V-u) cosΘ
• Mass flow rate m = pA (V-U cosΘ
• Newton's law of motion; Force equal to given is F = pA(V-U cosΘ.
• When this moving plane, reduction with a position will need the Jet reduces it down

Force Exerted on Jet Deflected by Moving Vane

• The fluid leaves the vane at relative velocity V₁₂ making an angle B₂ with the direction of the vane.
• Tangential velocity is exit velocities: Fx = m (VIx - V2x) F₁ = m (Vly - V2y)

Euler's equations

• A relationship amongst velocity pressure elevation and density is a streamline.
• A stream surrounding the stream tube has a cross section as an area of delta s shown in Fig.
• Area and velocity is expressed by: => PAV δV => P V^ g/dz- pg A dz =>P + V^/ρ gz = constants

===Hydro Power Plants

• Hydropower plants the source of energy is water.
• Dams construction creates a cement wall storing reservoir
• the turbine and tail is levels between which is work .
• Forces: Wind pressure and forces on the upstream level of the dam. The force is zero at the level at the top and increases and the slope is on the higher range.
• The forces resist the pressure

Questions cover non-dimensional quantities, viscosity, surface tension, Euler's theory, and Pelton turbines. Explore fluid properties, dimensionless numbers, and turbine applications. Understand the relationship between inertia, elastic forces and energy transfer.