Fluid mechanics Study Guide

📖 Core Concepts Fluid Mechanics – study of liquids, gases, plasmas and the forces acting on them; split into fluid statics (at rest) and fluid dynamics (in motion). Continuum Assumption – treat fluid as continuous matter; properties (ρ, p, T, u) defined at infinitesimal volumes large compared with molecular scales. Conservation Laws – mass, momentum, and energy are conserved inside any control volume. Knudsen Number $Kn=\lambda/L$ – measures molecular mean‑free‑path ($\lambda$) vs characteristic length ($L$); $Kn<0.1$ ⇒ continuum valid. Reynolds Number $Re=\rho UL/\mu$ – ratio of inertial to viscous forces; determines laminar vs turbulent regime. Viscosity – internal friction; zero for an inviscid fluid, linear with shear rate for a Newtonian fluid ($\tau=\mu\dot\gamma$). Navier–Stokes Equations – momentum balance for a viscous fluid; reduce to Euler equations when $\mu=0$. --- 📌 Must Remember Archimedes’ principle: upward force = weight of displaced fluid. Pascal’s law: pressure change applied to an incompressible fluid is transmitted undiminished throughout. Continuum validity: $Kn<0.1$. Reynolds number thresholds (order of magnitude): $Re\lesssim 2000$ → laminar (steady, smooth). $Re\gtrsim 4000$ → turbulent (chaotic). Newtonian fluid stress: $\tau = \mu \dot\gamma$. Navier–Stokes (incompressible): $$\rho\left(\frac{\partial \mathbf{u}}{\partial t}+ \mathbf{u}\cdot\nabla\mathbf{u}\right)= -\nabla p + \mu\nabla^{2}\mathbf{u}+ \rho\mathbf{g}$$ Euler equations: same as above with $\mu=0$. Boundary‑layer no‑slip: fluid velocity equals solid surface velocity at the wall. --- 🔄 Key Processes Applying Conservation of Mass (Continuity): For incompressible flow: $\nabla\cdot\mathbf{u}=0$. For control volume: $\displaystyle \frac{d}{dt}\int{CV}\rho\,dV = -\int{CS}\rho\mathbf{u}\cdot d\mathbf{A}$. Evaluating Flow Regime: Compute $Re=\rho UL/\mu$. Compare to laminar/turbulent limits → choose analytical (laminar) or CFD/turbulence model (turbulent). Deriving Pressure Variation with Depth (Hydrostatics): $dp = -\rho g\,dz$ → $p(z)=p0+\rho g h$. Setting Up Navier–Stokes for a Problem: Identify control volume, assume incompressible/steady if justified. Write momentum equations, apply boundary conditions (no‑slip, pressure inlet/outlet). Simplify (e.g., neglect $\mu\nabla^2\mathbf{u}$ for high‑Re inviscid flow). CFD Workflow (high‑level): Geometry → Mesh → Governing equations (NS) → Solver settings (turbulence model) → Convergence check → Post‑process results. --- 🔍 Key Comparisons Inviscid vs Viscous Inviscid: $\mu=0$, no shear stress, Euler equations, only good away from walls. Viscous: $\mu>0$, shear stress present, Navier–Stokes, essential in boundary layers. Newtonian vs Non‑Newtonian Newtonian: $\tau \propto \dot\gamma$ (linear). Non‑Newtonian: $\tau$–$\dot\gamma$ relation non‑linear (shear‑thinning, shear‑thickening, yield stress). Ideal vs Real Fluid Ideal: non‑viscous and incompressible (theoretical tool). Real: has viscosity, may be compressible; only approximated as ideal when viscous effects negligible. Laminar vs Turbulent Flow Laminar: smooth streamlines, $Re$ low, analytical solutions often possible. Turbulent: chaotic eddies, $Re$ high, requires statistical models or CFD. --- ⚠️ Common Misunderstandings “Zero viscosity = no drag.” Real fluids always have some viscosity; only superfluids approach $\mu=0$. “High Reynolds number always means turbulent.” Transitional ranges exist (≈2000–4000); geometry and disturbances also matter. “Euler equations work near walls.” They ignore the no‑slip condition; boundary layers must be treated with Navier–Stokes or boundary‑layer theory. “All fluids are Newtonian.” Many biological or industrial fluids (blood, polymer melts) are non‑Newtonian. --- 🧠 Mental Models / Intuition Fluid parcels as “tiny ships”: they carry mass, momentum, and energy; forces act on them just like on a small boat. Reynolds number as “inertia vs friction”: imagine pushing a sled (low $Re$ – friction dominates) vs a race car (high $Re$ – inertia dominates). Boundary layer as “traffic jam”: fluid right at the wall slows to zero (no‑slip), creating a thin region where velocity changes rapidly. Continuum ≈ “smooth road”: if your car’s wheels (molecules) are much smaller than the road’s bumps (system length), you can treat the road as smooth. --- 🚩 Exceptions & Edge Cases $Kn \ge 0.1$ (rarefied gases): continuum equations break down; need Boltzmann equation or DSMC methods. Superfluid helium: behaves as an inviscid fluid but exhibits quantum effects; not covered by classical NS. Highly compressible flows (Mach > 0.3): density variations become significant; incompressible assumption fails. Non‑Newtonian shear‑thickening fluids: apparent viscosity increases with shear rate, opposite of many polymer solutions. --- 📍 When to Use Which Inviscid (Euler) analysis: Flow away from solid surfaces, $Re \gg 1$, and viscous forces negligible. Viscous (Navier–Stokes) analysis: Near walls, low‑$Re$ devices, or any situation where shear stress matters. Laminar analytical solution: Simple geometry, $Re$ low, steady state (e.g., Hagen–Poiseuille flow). Turbulent CFD: $Re$ high, complex geometry, or unsteady flow where analytical methods fail. Newtonian model: Water, air, most gases under standard conditions. Non‑Newtonian model: Blood, polymer melts, ketchup, paints – use power‑law or Bingham models. --- 👀 Patterns to Recognize Linear pressure gradient → Poiseuille (parabolic) velocity profile in straight pipes. $p$ decreasing linearly with depth → hydrostatic balance. Separation of variables in NS → possible only for low‑$Re$, simple geometries. Sharp rise in drag coefficient near critical $Re$ → transition to turbulence. Boundary‑layer thickness $\delta \sim \sqrt{\nu x/U}$ for laminar flat‑plate flow. --- 🗂️ Exam Traps Choosing Euler instead of Navier–Stokes for a problem that mentions a solid wall → you’ll miss the no‑slip condition. Using $Re$ formula with wrong characteristic length (e.g., pipe diameter vs hydraulic radius) → answer off by a factor. Assuming incompressibility for high‑speed gas flows (Mach > 0.3) → density changes invalidate the simplification. Mixing up pressure drop signs: $dp = -\rho g\,dz$; forgetting the negative leads to pressure increasing with depth incorrectly. Treating a non‑Newtonian fluid as Newtonian when the problem states “shear‑thickening” or “yield stress.” Neglecting the Knudsen number for micro‑scale flows (MEMS); continuum equations give non‑physical results. ---