Untitled

• Fluid mechanics studies the fundamentals of flow

Movement of Flow

• Described by two methods: Lagrangian and Eulerian

Lagrangian Method

• Tracks an arbitrary particle and its kaleidoscopic changes in velocity and acceleration

Eulerian Method

• Studies changes in velocity and pressure at fixed positions (x, y, z) in space and at time (t)

Streamline

• A curve formed by the velocity vectors of each fluid particle at a specific time
• Tangent at each point indicates the direction of the fluid at that point
• Can be obtained by drawing a curve following the flow trace of floating powder on flowing water

Streamline Properties

• Velocity vector has no normal component, so no flow crosses the streamline
• For two-dimensional flow, with streamline gradient dy/dx, the streamline equation is dx/u = dy/v
• Where u is the velocity in the x-direction and v is the velocity in the y-direction
• Streamlines around a body vary based on the relative relationship between the observer and the body

Streak Line

• Line formed by a series of fluid particles passing a certain point in the stream one after another

Path Line

• Path of one particular particle starting from one particular point in the stream

Steady Flow

• Streak lines, path lines, and streamlines all occur together

Stream Tube

• Formed by drawing streamlines passing through all points on a given closed curve in a flow
• Fluid is regarded as flowing in a solid tube, and is useful for studying fluid in steady motion as no fluid enters or leaves through its walls

Steady Flow

• Flow state does not change with time, as expressed by velocity, pressure, or density at a position
• A tap with a stationary handle and constant opening leaving is an example resulting in steady flow

Unsteady Flow

• Flow state changes with time
• A tap with a handle being turned is an example of unsteady flow

Three-Dimensional Flow

• Flows have velocity components in x, y, and z directions
• Velocity components in x, y, and z axial directions as u, v, and w with corresponding equations

Two-Dimensional Flow

• Flow state is the same on all planes parallel to a cut plane and is described by two coordinates (x and y)
• Velocity components in x and y directions are expressed as u and v, with corresponding equations

One-Dimensional Flow

• Flow state is determined by one coordinate (x only)
• Velocity (u) depends only on coordinates (x) and (t) with corresponding equations

Laminar Flow

• Characterized by colored liquid flowing without mixing with peripheral water

Turbulent Flow

• Characterized by colored liquid becoming turbulent and mingling with peripheral water

Critical Velocity

• The flow velocity at which laminar flow transitions to turbulent flow

Reynolds Number (Re)

• Used to predict weather a flow is laminar or turbulent
• Reynolds discovered laminar flow turns turbulent when the non-dimensional quantity ρvd/μ reaches a certain amount
• Regardless of average velocity (v), glass tube diameter (d), water density (ρ), and water viscosity (μ) values
• Defined as Re = ρvd/μ = vd/ν, where ν is kinematic viscosity
• Critical Reynolds number obtained when the velocity is the critical velocity (v_c) with equation Re_c = v_cd/ν

Factors Affecting Reynolds Number

• Turbulence in the fluid entering the tube
• Lower critical Reynolds number. Flow remains laminar; estimated at 2320.

Incompressible Flow

• Liquids are considered incompressible fluids

Compressible Flow

• Gases are considered compressible fluids

Compressibility

• Compressibility is considered for liquids under high pressure, like oil in hydraulic machines
• Compressibility of gasses are ignored with small changes in pressure

Mach Number (M)

• Ratio is used as standard to judge compressibility, alongside the compressibility of the fluid can dominate so that the ratio of the inertia force to the elasticity
• M = v / a where a is the velocity of sound.

Subsonic Flow

• M < 1

Sonic Flow

• M = 1

Supersonic Flow

• M > 1

Transonic Flow

• When M = 1, M < 1, and M > 1 zones occur simultaneously

Fluid Particle Deformation and Rotation

• Fluid particles flowing through a narrow channel undergo deformation and rotation

Angular Velocities

• AB in the x direction moves to A'B' while rotating by dε₁
• AD in the y direction rotates by dε₂
• Equations are provided for dε₁, dθ₁, dε₂, and dθ₂
• Angular velocities of AB and AD are ω₁ and ω₂ with corresponding equations

Average Angular Velocity (ω)

• Equation presented for the calculation of average angular velocity (ω)
• Defined as (∂v/∂x - ∂u/∂y).

Vorticity (ξ)

• Expressed with corresponding equation as:
• ξ = ∂v/∂x - ∂u/∂y
• Ir-rotational flow occurs when vorticity is zero.

Cylinder Vessel Spin

• Cylindrical vessel filled with liquid spinning about the vertical axis at a certain angular velocity
• Liquid exhibiting a rotary movement along the flow line along with the element rotating at the same time.

Free Vortex Flow

• Occurs when liquid flows through a small hole at the bottom of a vessel
• It is a type of ir-rotational flow where microelements face the same direction without performing rotation.

Assuming a Closed Curve (s)

• The integrated velocity (ν_s) along this curve is defined as the circulation (Γ)
• Counter-clockwise rotation is taken as positive
• Equation for the circulation is: Γ = ∫ ν_s ds = ∫ ν_s cosθ ds

Circulation Calculation

• Area enclosed by closed curve (s) is divided into x and y parallel lines for calculation
• Micro-Rectangle ABCD formula shown
• General formula shown dΓ = ζdx dy = ζdA

Stoke's Theorem

• Vorticity (ζ) is two times the angular velocity (ω) of rotational flow
• Circulation is equal to the product of vorticity by area
• After integrating and cancelling, integration is completed on closed curves

Outcomes of No Vorticity Inside Closed Curve

• Circulation around it is zero

Use of Stokes' Theorem in Fluid Dynamics

• Studying flow inside impellers of pumps and blowers
• Studying flow around an aircraft wing