Potential and Viscous Flows — Irrotational and Laminar Regimes

Scope: rigorous development of potential flow theory and exact viscous solutions of the Navier–Stokes equations. Includes velocity potential formulation, vorticity dynamics, laminar shear and pressure-driven flows, creeping motion, and transition toward turbulence.

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1. Irrotational Flow and the Velocity Potential

For inviscid and irrotational flow: $\nabla\times\mathbf{v} = 0.$ Then $\mathbf{v}$ can be represented as a gradient of a scalar potential: $\mathbf{v} = \nabla\phi.$

For incompressible flow ($\nabla\cdot\mathbf{v}=0$): $\nabla^2 \phi = 0.$ This is Laplace’s equation, governing potential flow.

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2. Streamfunction and Complex Potential

In 2D incompressible flow, define a streamfunction $\psi$: $u = \partial\psi/\partial y, \quad v = -\partial\psi/\partial x.$

Velocity potential $\phi$ and streamfunction $\psi$ satisfy the Cauchy–Riemann relations: $\frac{\partial\phi}{\partial x} = \frac{\partial\psi}{\partial y}, \quad \frac{\partial\phi}{\partial y} = -\frac{\partial\psi}{\partial x}.$

Thus, define a complex potential: $W(z) = \phi + i\psi, \quad z = x + iy.$

The derivative gives complex velocity: $w = u - iv = \frac{dW}{dz}.$ Superposition of elementary potentials yields complex flow fields.

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3. Elementary Potential Flows

Flow Type	Potential Function	Streamfunction	Comments
Uniform flow	$\phi = Ux$	$\psi = Uy$	Constant velocity
Source (strength Q)	$\phi = \frac{Q}{2\pi} \ln r$	$\psi = \frac{Q}{2\pi}\theta$	Radial outflow
Sink	Negative of source	—	Convergent flow
Doublet (limit of source-sink pair)	$\phi = -\frac{\mu \cos \theta}{2\pi r}$	$\psi = -\frac{\mu \sin \theta}{2\pi r}$	Models solid body
Vortex (circulation $\Gamma$)	$\phi = \frac{\Gamma}{2\pi}\theta$	$\psi = -\frac{\Gamma}{2\pi}\ln r$	Tangential velocity

Superposition yields more complex flows: flow past a cylinder, airfoil approximations (Joukowski transform), etc.

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4. Bernoulli’s Equation for Potential Flow

From Euler’s equation for inviscid flow: $\rho\frac{D\mathbf{v}}{Dt} = -\nabla p + \rho\mathbf{g}.$ For steady, irrotational flow, integrate along a streamline: $\frac{p}{\rho} + \frac{v^2}{2} + gz = \text{constant}.$

For compressible flow: $\int_{p_1}^{p_2}\frac{dp}{\rho} + \frac{v_2^2 - v_1^2}{2} + g(z_2 - z_1) = 0.$

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5. Vorticity Dynamics and Circulation

Vorticity $\boldsymbol{\omega} = \nabla\times\mathbf{v}$. For inviscid, barotropic flow: $\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\mathbf{v}.$

5.1 Kelvin’s Circulation Theorem

For inviscid, conservative body forces: $\frac{D\Gamma}{Dt} = 0, \quad \Gamma = \oint_C \mathbf{v}\cdot d\mathbf{s}.$

Circulation is conserved; vortex lines move with the fluid.

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6. Exact Viscous Flow Solutions (Steady, Laminar Regimes)

Consider the incompressible Navier–Stokes equations: $\rho(\mathbf{v}\cdot\nabla)\mathbf{v} = -\nabla p + \mu\nabla^2\mathbf{v}.$

At low Reynolds numbers, inertial terms are small: $0 = -\nabla p + \mu\nabla^2\mathbf{v}.$ These are Stokes flow equations.

6.1 Plane Couette Flow

Between plates separated by distance $h$; top plate moves with velocity $U$. Assume $v_x = u(y), v_y = 0$: $0 = \mu \frac{d^2u}{dy^2}, \quad u(0)=0, \; u(h)=U.$ Solution: $u(y) = U\frac{y}{h}.$

Shear stress: $\tau = \mu U/h$ (uniform across gap).

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6.2 Plane Poiseuille Flow

Pressure-driven flow between stationary plates: $\frac{dp}{dx} = \mu\frac{d^2u}{dy^2}.$ With $u(\pm h)=0$: $u(y) = \frac{1}{2\mu}\frac{dp}{dx}(y^2 - h^2).$

Maximum velocity: $u_{max} = -\frac{1}{2\mu}\frac{dp}{dx}h^2.$

Volumetric flow rate: $Q = -\frac{2h^3}{3\mu}\frac{dp}{dx}.$

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6.3 Hagen–Poiseuille Flow (Circular Pipe)

For fully developed laminar pipe flow: $\frac{dp}{dz} = \mu\left(\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)\right).$ Boundary: $u(R)=0$, symmetry $du/dr=0$ at $r=0$.

Solution: $u(r) = \frac{1}{4\mu}\frac{dp}{dz}(r^2 - R^2).$

Volumetric flow: $Q = -\frac{\pi R^4}{8\mu}\frac{dp}{dz}.$ Average velocity $\bar{u} = Q/(\pi R^2) = -\frac{R^2}{8\mu}\frac{dp}{dz}.$

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6.4 Stokes’ First Problem (Unsteady Plate Motion)

At $t=0$, plate at $y=0$ suddenly moves at speed $U$ in stationary fluid. Governing equation: $\frac{\partial u}{\partial t} = \nu\frac{\partial^2 u}{\partial y^2}, \quad u(0,t)=U, u(\infty,t)=0.$

Solution (error function form): $u(y,t) = U\,\mathrm{erfc}\left(\frac{y}{2\sqrt{\nu t}}\right).$ Velocity decays exponentially with $y$.

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6.5 Stokes’ Second Problem (Oscillating Wall)

Wall oscillates as $u(0,t)=U_0 e^{i\omega t}$. Solution: $u(y,t) = U_0 e^{i\omega t - y/\delta}, \quad \delta = \sqrt{2\nu/\omega}.$ This defines the viscous penetration depth.

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7. Creeping Flow Around a Sphere (Stokes Flow)

Steady low-Re flow around sphere of radius $a$; boundary conditions: $u_r = U\cos\theta\left(1 - \frac{3a}{2r} + \frac{a^3}{2r^3}\right), \quad u_\theta = -U\sin\theta\left(1 - \frac{3a}{4r} - \frac{a^3}{4r^3}\right).$

Pressure field: $p - p_\infty = -\frac{3\mu U a}{2r^2}\cos\theta.$

Total drag force: $F_D = 6\pi\mu aU.$

This is Stokes’ law, valid for $Re = \rho U a/\mu \ll 1.$

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8. Energy Dissipation in Laminar Flow

Viscous dissipation per unit volume: $\Phi = 2\mu e_{ij}e_{ij}.$ Total mechanical energy balance: $\frac{\partial}{\partial t}\left(\frac{\rho v^2}{2}\right) + \nabla\cdot(p\mathbf{v}) = -\Phi - \nabla\cdot(\mathbf{q}_{mech}).$

For steady Poiseuille flow, total head loss equals energy dissipated by viscous forces.

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9. Transition from Laminar to Turbulent Flow

Laminar flow becomes unstable when inertial forces overcome viscous damping.

• Characterized by Reynolds number: $Re = \rho UL/\mu.$
• Pipe flow transition at $Re_{cr} \approx 2300.$

Perturbation amplification governed by linear stability analysis: $\frac{d}{dt}(\delta\mathbf{v}) = \mathcal{L}(\delta\mathbf{v}).$ When eigenvalues of operator $\mathcal{L}$ have positive real parts $\to$ instability.

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10. Potential vs. Viscous Flow Summary

Feature	Potential Flow	Viscous Flow
Governing eq.	Laplace ($\nabla^2\phi=0$)	Navier–Stokes
Assumptions	Inviscid, irrotational	Newtonian viscosity
Dissipation	None	Finite ($\mu>0$)
Flow type	Ideal (no drag)	Real (with drag)
Examples	Flow over airfoil (lift)	Poiseuille, Couette

D’Alembert’s paradox: potential flow predicts zero drag—resolved only when viscosity (boundary layer separation) is included.

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11. Summary Equations

Equation	Expression	Description
Laplace equation	$\nabla^2\phi = 0$	Incompressible potential flow
Bernoulli	$p/\rho + v^2/2 + gz = \text{const}$	Energy balance along streamline
Navier–Stokes	$\rho Dv/Dt = -\nabla p + \mu\nabla^2 v$	Viscous flow dynamics
Stokes flow	$0 = -\nabla p + \mu\nabla^2 v$	Creeping flow ($Re \ll 1$)
Stokes’ drag	$F_D = 6\pi\mu aU$	Low-Re sphere drag
Poiseuille	$Q = \pi R^4\Delta p/(8\mu L)$	Laminar pipe flow rate

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12. Cross-Links

• fundamentals.md — conservation laws and Navier–Stokes derivation.
• 02_Boundary_Layers_and_Separation.md — viscous–inviscid interaction and drag mechanisms.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy generation due to viscous dissipation.