Two-Dimensional and Axisymmetric Flow

At a Glance

Frequency: 1 sub-part across 1 of 13 years (2015)
Priority tier: T4
Marks (count): 20 (1)
Average solve time: ~30 min
Difficulty mix: medium 1
Section: B | Dominant type: derivation

Why This Chapter Matters

The connection between 2D irrotational flow and complex analysis is one of the most elegant results in applied mathematics: the velocity potential and stream function satisfy the Cauchy-Riemann equations, making the complex potential $w = \phi + i\psi$ an analytic function of $z = x + iy$ . UPSC 2015 tested this with a derivation of the complex potential for a standard flow — a Section B question requiring both theoretical justification and explicit computation.

Minimum Theory

2D Irrotational Incompressible Flow

Consider a 2D flow in the $xy$ -plane with velocity $(u, v)$ :

Incompressibility: $\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = 0$ (continuity equation).
Irrotationality: $\dfrac{\partial v}{\partial x} - \dfrac{\partial u}{\partial y} = 0$ (no vorticity).

Velocity Potential and Stream Function

Velocity potential $\phi$ (exists for irrotational flow):

$u = \frac{\partial \phi}{\partial x}, \qquad v = \frac{\partial \phi}{\partial y}.$

Irrotationality is automatic: $\partial v/\partial x - \partial u/\partial y = \phi_{xy} - \phi_{yx} = 0$ . Incompressibility gives $\nabla^2\phi = 0$ (Laplace’s equation for $\phi$ ).

Stream function $\psi$ (exists for incompressible flow):

$u = \frac{\partial \psi}{\partial y}, \qquad v = -\frac{\partial \psi}{\partial x}.$

Continuity is automatic. Irrotationality gives $\nabla^2\psi = 0$ (Laplace’s equation for $\psi$ ).

The Complex Potential — Key Result

Define the complex potential:

$w = \phi(x,y) + i\,\psi(x,y).$

Claim. $w$ is an analytic function of $z = x + iy$ .

Proof. Check the Cauchy-Riemann equations for $w = \phi + i\psi$ :

$\frac{\partial \phi}{\partial x} = u = \frac{\partial \psi}{\partial y} \quad \checkmark$

$\frac{\partial \phi}{\partial y} = v = -\left(-\frac{\partial \psi}{\partial x}\right) \implies \frac{\partial \phi}{\partial y} = -\left(-v\right)$

More carefully: $\phi_x = u = \psi_y$ and $\phi_y = v = -\psi_x$ . Thus:

$\phi_x = \psi_y \quad \text{and} \quad \phi_y = -\psi_x.$

These are exactly the Cauchy-Riemann equations for $w = \phi + i\psi$ . Hence $w$ is analytic. $\blacksquare$

Complex Velocity

The derivative of the complex potential gives the complex velocity:

$\frac{dw}{dz} = \frac{\partial \phi}{\partial x} + i\frac{\partial \psi}{\partial x} = u + i(-v) = u - iv.$

Thus if you know $w(z)$ , the velocity components are recovered by:

$u - iv = \frac{dw}{dz}.$

The speed is $|dw/dz| = \sqrt{u^2 + v^2}$ .

Standard Complex Potentials

Flow type	Complex potential $w(z)$	Complex velocity $dw/dz$
Uniform flow at speed $U$ (along $x$ -axis)	$Uz$	$U$
Uniform flow at angle $\alpha$	$Ue^{-i\alpha}z$	$Ue^{-i\alpha}$
Line source/sink of strength $m$ at origin	$\dfrac{m}{2\pi}\log z$	$\dfrac{m}{2\pi z}$
Line vortex of strength $\kappa$ at origin	$\dfrac{-i\kappa}{2\pi}\log z$	$\dfrac{-i\kappa}{2\pi z}$
Doublet of strength $\mu$ at origin	$\dfrac{\mu}{2\pi z}$	$\dfrac{-\mu}{2\pi z^2}$
Flow past cylinder of radius $a$ , speed $U$	$U\!\left(z + \dfrac{a^2}{z}\right)$	$U\!\left(1 - \dfrac{a^2}{z^2}\right)$

Axisymmetric Flow (Stokes Stream Function)

For flow with symmetry about the $z$ -axis (cylindrical coordinates $r, \theta, z$ with no $\theta$ -dependence), a Stokes stream function $\Psi(r,z)$ is defined so that:

Streamlines are curves $\Psi = \text{const}$ .
The volume flux through any circle of radius $r$ centred on the axis, in the plane $z = \text{const}$ , equals $2\pi\Psi$ .
Velocity components: $u_r = -\dfrac{1}{r}\dfrac{\partial \Psi}{\partial z}$ , $u_z = \dfrac{1}{r}\dfrac{\partial \Psi}{\partial r}$ .

Unlike the 2D case, $\Psi$ does not generally satisfy Laplace’s equation; instead it satisfies the Stokes stream function equation.

Question Archetypes

Archetype	Recognition
find-complex-potential	”Find the complex potential for the given 2D irrotational flow”
derive-cr-structure	”Show that $\phi$ and $\psi$ satisfy the C-R equations / $w$ is analytic”
combine-flows	”Superpose source, vortex, uniform stream; find the combined $w$ “
axisymmetric-properties	”State properties of the Stokes stream function; find flux”

find-complex-potential (1 question; 2015)

Recognition Cues

A 2D irrotational flow is described (uniform stream + source, doublet, vortex, etc.).
Asked to find $w(z)$ , or the velocity potential $\phi$ , or the stream function $\psi$ .
May ask to verify that $w$ is analytic or to find the velocity at a point.

Solution Template

Identify the flow components (uniform stream, source, vortex, doublet, etc.) from the problem description.
Write the complex potential for each component using the standard table.
Superpose: $w = w_1 + w_2 + \ldots$ (complex potentials add linearly).
Extract $\phi = \text{Re}(w)$ and $\psi = \text{Im}(w)$ if asked.
Compute $dw/dz = u - iv$ to find velocity components.
If asked to prove $w$ is analytic: verify C-R equations $\phi_x = \psi_y$ , $\phi_y = -\psi_x$ using the flow relations $u = \phi_x = \psi_y$ , $v = \phi_y = -\psi_x$ .

Worked Example

2015 Paper 2, 2015-P2-QMF (20 marks)

Find the complex potential for the 2D irrotational flow consisting of a uniform stream of speed $U$ parallel to the $x$ -axis superposed with a line source of strength $m$ at the origin. Find the velocity at the stagnation point and the equation of the streamline through the stagnation point.

Step 1: Write the complex potential.

Uniform stream: $w_1 = Uz$ .

Line source of strength $m$ at origin: $w_2 = \dfrac{m}{2\pi}\log z$ .

Superposed complex potential:

$w(z) = Uz + \frac{m}{2\pi}\log z.$

Step 2: Find the complex velocity.

$\frac{dw}{dz} = U + \frac{m}{2\pi z} = u - iv.$

Step 3: Find the stagnation point.

At a stagnation point, $u = v = 0$ , i.e., $dw/dz = 0$ :

$U + \frac{m}{2\pi z} = 0 \implies z = -\frac{m}{2\pi U}.$

This is the point $\left(-\dfrac{m}{2\pi U},\, 0\right)$ on the negative $x$ -axis. The velocity there is zero (stagnation).

Step 4: Stream function and stagnation streamline.

Write $z = re^{i\theta}$ (polar form). Then $\log z = \ln r + i\theta$ , so:

$w = U(x + iy) + \frac{m}{2\pi}(\ln r + i\theta),$

$\psi = \text{Im}(w) = Uy + \frac{m\theta}{2\pi} = Ur\sin\theta + \frac{m\theta}{2\pi}.$

At the stagnation point $z = -m/(2\pi U)$ : here $r = m/(2\pi U)$ , $\theta = \pi$ , so:

$\psi_{\text{stag}} = U \cdot \frac{m}{2\pi U} \cdot \sin\pi + \frac{m\cdot\pi}{2\pi} = 0 + \frac{m}{2} = \frac{m}{2}.$

The stagnation streamline is $\psi = m/2$ :

$\boxed{Ur\sin\theta + \frac{m\theta}{2\pi} = \frac{m}{2}.}$

This streamline divides the flow: above and below it the stream passes around the “body” formed by the stagnation dividing streamline.

Step 5: Verify $w$ is analytic.

$w(z) = Uz + \dfrac{m}{2\pi}\log z$ is a sum of analytic functions (linear function, which is entire, plus $\log z$ , which is analytic on $\mathbb{C}\setminus(-\infty,0]$ ). Hence $w$ is analytic on $\mathbb{C}\setminus\{0\} \cap \mathbb{C}\setminus(-\infty,0]$ , i.e., on $\mathbb{C}$ cut along the negative real axis. The corresponding $\phi$ and $\psi$ automatically satisfy the C-R equations:

$\phi_x = \psi_y = u, \quad \phi_y = -\psi_x = v. \quad \blacksquare$

$\boxed{w(z) = Uz + \frac{m}{2\pi}\log z}$

Common Traps

Sign error in the source complex potential: $w = (m/2\pi)\log z$ for a source (strength $m > 0$ ); a sink has $m < 0$ . Do not negate $\log z$ unless the problem specifies a sink.
Confusing complex velocity: $dw/dz = u - iv$ , not $u + iv$ . The imaginary part has a minus sign.
Stagnation point: set $dw/dz = 0$ , not $|dw/dz|= 0$ computed separately for $u$ and $v$ .
Forgetting the branch cut of $\log z$ when stating where $w$ is analytic.
For axisymmetric flow: confusing the 2D stream function (where flux = $\psi_2 - \psi_1$ ) with the Stokes stream function (flux = $2\pi(\Psi_2 - \Psi_1)$ ).

Marks-Aware Writing

This is a 20-mark Section B derivation question. UPSC expects:

Explicit statement of each component’s complex potential — do not just write the answer.
Superposition step clearly shown.
$dw/dz$ computed and interpreted as $u - iv$ .
Stagnation point found by setting $dw/dz = 0$ , solving for $z$ .
Stream function extracted as $\psi = \text{Im}(w)$ — shown in polar coordinates.
Stagnation streamline: evaluate $\psi$ at the stagnation point, then write the streamline equation.
Analyticity of $w$ : mention C-R, even if briefly.

Marks are distributed across all these steps. A bare final answer without working receives minimal credit.

Practice Set

Only one historical question on this atom (shown above).