Fluid Mechanics Course

Essential Foundation: Critical prerequisite for plasma physics (MHD), astrophysics, and engineering

Viscous Flow

Real fluid behavior - friction, boundary layers, and turbulence

Introduction

Real fluids exhibit viscosity - internal resistance to deformation due to molecular momentum transfer. Viscous effects cause energy dissipation, velocity gradients near surfaces (boundary layers), and fundamentally different flow behavior than ideal fluids.

The Navier-Stokes equations, which include viscous terms, are the complete governing equations for Newtonian fluid flow - one of the most important and challenging equations in physics.

Viscosity

Newton's Law of Viscosity

For a Newtonian fluid, shear stress is proportional to velocity gradient:

$$\tau = \mu \frac{du}{dy}$$

μ = dynamic viscosity [Pa·s = kg/(m·s)]. This linear relationship defines Newtonian fluids.

Dynamic Viscosity (μ)

Absolute measure of fluid's resistance to shear. Water: ~0.001 Pa·s, Air: ~1.8×10⁻⁵ Pa·s

Kinematic Viscosity (ν)

ν = μ/ρ [m²/s]. Ratio of viscous to inertial effects. Appears in Reynolds number.

Non-Newtonian Fluids

Shear-thinning

Viscosity decreases with shear rate (paint, blood, ketchup)

Shear-thickening

Viscosity increases with shear rate (cornstarch in water)

Bingham Plastic

Requires yield stress to flow (toothpaste, mayonnaise)

Navier-Stokes Equations

The complete equations of motion for incompressible Newtonian fluids:

$$\rho\left(\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \rho \vec{g}$$
$$\nabla \cdot \vec{v} = 0 \quad \text{(incompressibility)}$$

Component Form (Cartesian)

$$\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) + \rho g_x$$

Similar equations for v and w components. These coupled, nonlinear PDEs are notoriously difficult to solve - proving existence and smoothness of solutions is a Millennium Prize Problem!

Why So Difficult?

  • • Nonlinearity: The convective term (v·∇)v couples velocity components
  • • Coupling: Pressure and velocity are coupled through continuity
  • • Turbulence: At high Re, solutions become chaotic with wide range of scales

Reynolds Number and Flow Regimes

Reynolds Number

The ratio of inertial to viscous forces - the key dimensionless parameter:

$$Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu} = \frac{\text{Inertial forces}}{\text{Viscous forces}}$$

v = characteristic velocity, L = characteristic length, ν = kinematic viscosity

Flow Regimes

Laminar Flow

Smooth, orderly layers sliding past each other

  • • Pipe flow: Re < 2300
  • • Flat plate: Rex < 5×10⁵
  • • Viscous forces dominate
  • • Predictable, steady (often)

Turbulent Flow

Chaotic, irregular motion with eddies

  • • Pipe flow: Re > 4000
  • • Flat plate: Rex > 5×10⁵
  • • Inertial forces dominate
  • • Enhanced mixing, higher friction

Transition Region

2300 < Re < 4000 for pipes - flow can be either laminar or turbulent depending on disturbances. Hysteresis effects possible.

Pipe Flow

Laminar Pipe Flow (Hagen-Poiseuille)

For fully developed laminar flow in a circular pipe:

$$u(r) = \frac{R^2}{4\mu}\left(-\frac{dp}{dx}\right)\left(1 - \frac{r^2}{R^2}\right) = u_{max}\left(1 - \frac{r^2}{R^2}\right)$$

Parabolic velocity profile. Maximum velocity at centerline is twice the average.

$$Q = \frac{\pi R^4}{8\mu}\left(-\frac{dp}{dx}\right) = \frac{\pi R^4 \Delta p}{8\mu L}$$

Flow rate proportional to R⁴ - halving diameter reduces flow 16×!

Friction Factor

Dimensionless pressure drop (Darcy-Weisbach equation):

$$\Delta p = f \frac{L}{D} \frac{\rho v^2}{2} \quad \text{where} \quad h_f = f\frac{L}{D}\frac{v^2}{2g}$$

Laminar

$$f = \frac{64}{Re}$$

Turbulent (Smooth)

$$f \approx 0.316 Re^{-1/4}$$

Blasius correlation for Re < 10⁵

Moody Diagram

The famous chart relating friction factor f to Reynolds number Re and relative roughness ε/D. Essential for pipe flow calculations. Uses Colebrook-White equation for turbulent flow with roughness effects.

Boundary Layers

Near solid surfaces, viscous effects create a thin boundary layer where velocity varies from zero (at wall, no-slip) to the free-stream value. Outside this layer, the flow can often be treated as inviscid.

Flat Plate Boundary Layer (Blasius)

For laminar flow over a flat plate:

$$\delta = \frac{5x}{\sqrt{Re_x}} \quad \text{where} \quad Re_x = \frac{U_\infty x}{\nu}$$

Boundary layer thickness grows as √x. At higher Rex, the layer is thinner.

Wall Shear Stress

$$\tau_w = 0.332 \rho U_\infty^2 Re_x^{-1/2}$$

Skin Friction Coefficient

$$C_f = \frac{0.664}{\sqrt{Re_x}}$$

Boundary Layer Thickness Definitions

δ (99%)

Distance where u = 0.99U∞

δ* (Displacement)

Effect on outer flow: δ* = ∫(1-u/U)dy

θ (Momentum)

Momentum deficit: θ = ∫(u/U)(1-u/U)dy

Boundary Layer Separation

Adverse pressure gradients (dp/dx > 0) can cause flow reversal near the wall, leading to boundary layer separation. This creates wakes, increases drag dramatically, and can cause stall in airfoils.

Turbulence

Turbulent flows exhibit chaotic, irregular fluctuations with a wide range of length and time scales. Direct simulation (DNS) of all scales is often impractical, leading to modeling approaches.

Reynolds Decomposition

Split velocity into mean and fluctuating parts:

$$u = \bar{u} + u' \quad \text{where} \quad \bar{u'} = 0$$

Reynolds-Averaged Navier-Stokes (RANS)

Time-averaging introduces Reynolds stresses:

$$\rho\left(\frac{\partial \bar{u}_i}{\partial t} + \bar{u}_j\frac{\partial \bar{u}_i}{\partial x_j}\right) = -\frac{\partial \bar{p}}{\partial x_i} + \mu\nabla^2\bar{u}_i - \rho\frac{\partial \overline{u'_i u'_j}}{\partial x_j}$$

The Reynolds stress tensor -ρu'iu'j requires closure models (k-ε, k-ω, etc.)

Kolmogorov Scales

Smallest scales η = (ν³/ε)1/4 where dissipation occurs. Energy cascades from large eddies to small ones.

Turbulent Viscosity

Eddy viscosity μt models enhanced mixing: τturb = μt ∂ū/∂y

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