Fluid Mechanics

Fluid

Any substance that cannot resist a shear force without motion and does not return to its initial state when the stress is removed (e.g., liquids and gases).

Stress (τ)

Force per unit area. For shear, the force is tangential to the area. Equation: τ=F/A.

Strain (e)

Measures deformation in a fluid, defined as the displacement of the upper surface relative to the height. Equation:

e=\frac{\partial X}{\partial y}

Solid vs. Fluid Response to Stress

Solid: stress proportional to strain (τsolid​=Ge), returns to shape. Fluid: no restoring force, stress proportional to rate of strain (τfluid​=μ de/dt​).

Viscosity (μ)

A constant representing the relationship between stress and the time-dependent strain in a fluid. Equation:

\tau=\mu de/dt

Newtonian Fluid

Fluid where viscosity (μ) is not a function of fluid velocity

Non-Newtonian Fluid

Fluid where viscosity is a function of fluid velocity

Lagrangian Fluid Element

A description of fluid motion where the fluid element moves with the fluid; no flow into or out of the element, so mass is constant.

Eulerian Fluid Element

A description of fluid motion where the fluid element is fixed in space; mass may vary with time, and fluid velocity is a function of position and time (u=u(x,y,z,t)).

Streamline

A 3D curve tangential to the local fluid velocity vector at a fixed time (dr∝u). Streamlines cannot intersect. Equation: ​

dx/u_{x}=dy/u_{y}=dz/u_{z}

Laminar Flow

Smooth flow profile, usually matching material surfaces, where fluid layers slide past each other without significant mixing.

Viscous Flow

Fluid flow where viscosity is an important factor influencing the motion.

Steady Flow

Fluid flow where the fluid velocity vector field is not a function of time

Poiseuille Flow

Cylindrically symmetric, viscous, steady, laminar flow through a cylindrical pipe, characterized by a parabolic velocity profile. Velocity equation:

u_z​(r)=(P_1​−P_2​)​(a^2−r^2)/4μL

Poiseuille Flow Rate (Mass)

The total mass flowing through a cylindrical pipe per second for constant density. Equation:

M=\pi\rho(P_1-P_2)a^4/8\mu L

Kinematic Viscosity (ν)

Viscosity divided by fluid density. Equation:

ν=μ/ρ

Reynolds Number (Re)

A dimensionless parameter quantifying the ratio of inertial forces to viscous forces in a fluid flow. Equation:

Re=\rho_0L_0u_0/\mu

Small Re means viscous-dominated flow.

Continuum Hypothesis

The assumption that a fluid can be treated as a continuous medium, ignoring molecular-level details, valid for length scales much larger than the mean-free-path.

Conservation of Mass (Continuity Equation)

The principle that mass is conserved in a fluid. General form:

\frac{\partial\rho}{\partial t}+\nabla\cdot\underline{u}\rho=0

For incompressible flow: ∇⋅u=0.

Advective Derivative (Total Derivative)

Represents the rate of change of a fluid property at a fixed point in space plus the additional rate of change due to the fluid element moving. Equation:

\frac{D}{Dt}=\frac{\partial}{\partial t}+\underline{u}\cdot\nabla

Navier-Stokes (N-S) Equation

\rho\partial\underline{u}/\partial t+\rho(\underline{u}\cdot\nabla)\underline{u}=-\nabla P+\mu\nabla^2\underline{u}-\rho g\underline{k̂}

Euler's Equations

The Navier-Stokes equations with the viscosity term set to zero (inviscid N-S equations).

Bernoulli's Principle

P+\rho gz+\frac{\rho u^2}{2}= constant along streamline

Vorticity (ω)

A measure of the local rotation of a fluid element, defined as the curl of the velocity vector field. Equation:

\underline{\omega}=\nabla\times\underline{u}

Irrotational Flow

A fluid flow where the vorticity is zero (∇×u=0).

Circulation (K)

The line integral of the fluid velocity around an arbitrary closed path (Γ). Equation:

K=\oint_{\Gamma}\underline{u}\cdot\underline{dl}

Kelvin's Circulation Theorem

States that circulation is conserved for an inviscid fluid if it is either incompressible or barotropic.

Barotropic Fluid

A fluid where density is a function of pressure (ρ=ρ(P)).

Potential Flow

A flow that is irrotational (∇×u=0). If also incompressible, it satisfies Laplace's equation for a scalar potential\nabla^2\phi=0

Boundary Layer

A region near a material object where viscous forces are significant, typically where the flow transitions from zero at the surface to the free stream velocity.

Boundary Layer Thickness (δ)

The thickness of the layer where viscosity cannot be neglected, scaling as

\frac{\delta}{d}=\frac{1}{\sqrt{\operatorname{Re}}}

Starting/Trailing Vortex

A detached vortex formed in the boundary layer of an aerofoil, having circulation opposite to that generated on the aerofoil, and remaining where it was formed.

Magnus Effect

The phenomenon where a rotating object or an object moving through a fluid with circulation experiences a force perpendicular to the direction of motion, contributing to lift on an aerofoil. Lift equation:

\underline{L}=\rho\underline{u}\times\underline{K}

Strouhal Number (St)

A dimensionless parameter describing oscillating flow mechanisms, such as vortex shedding. Equation:

St=\frac{D}{Pv}

Vortex Shedding

The oscillating flow that occurs when a fluid flows past a blunt body at certain Reynolds numbers, causing vortices to be shed from alternating sides of the body.

Turbulence

A fluid flow regime characterized by chaotic changes in pressure and flow velocity, occurring at high Reynolds numbers due to complex vortex interactions.

Energy Cascade

In turbulence, the process where energy is transferred from large-scale motions to smaller-scale motions, eventually dissipating into heat at the smallest scales due to viscosity.

Drag

The resistance force experienced by an object moving through a fluid, often due to pressure differences and viscous effects, particularly in turbulent regions.