Euler’s equation of motion for inviscid flow

At a Glance

• Frequency: 2 sub-parts across 2 of 14 years (2017, 2024)
• Priority tier: T3
• Marks (count): 15 (1), 20 (1)
• Average solve time: ~15 min
• Difficulty mix: hard 1, medium 1
• Section: B | Dominant type: computation

Why This Chapter Matters

Euler’s equation is the foundation of inviscid fluid mechanics: it appears in Section B (optional), carries up to 20 marks, and connects directly to Bernoulli’s theorem — the primary computational tool in both questions. The 2024 question (20 marks) asks for pressure in a 2D doublet-flow field after verifying incompressibility and irrotationality, while the 2017 question (15 marks) requires the compressible-flow version of Bernoulli to prove a velocity ratio formula. Both question types are entirely mechanical once you know the two Bernoulli forms and one algebraic identity: $(x^2-y^2)^2 + 4x^2y^2 = (x^2+y^2)^2$.

Minimum Theory

Euler’s Equation. For an inviscid fluid (viscosity $\mu = 0$) the Navier-Stokes equation reduces to Euler’s equation:

$\rho\!\left[\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v}\right] = -\nabla p + \rho\mathbf{F}.$

There is no viscous term. For steady flow ($\partial_t = 0$) this becomes $\rho(\mathbf{v}\cdot\nabla)\mathbf{v} = -\nabla p + \rho\mathbf{F}$.

Bernoulli’s Theorem (incompressible, irrotational). For steady, inviscid, incompressible flow that is also irrotational ($\nabla \times \mathbf{v} = 0$), Bernoulli’s equation holds throughout the fluid (not just along a streamline):

$p + \frac{1}{2}\rho|\mathbf{v}|^2 + \rho gz = p_\infty = \text{const.}$

Equivalently $p = p_\infty - \tfrac{1}{2}\rho|\mathbf{v}|^2$ (neglecting gravity or setting $z = \text{const}$).

Bernoulli’s Theorem (compressible, isothermal). For steady compressible flow with $p = K\rho$ (isothermal, $K$ constant), the compressible Bernoulli integral along a streamline is $\tfrac{|\mathbf{v}|^2}{2} + \int dp/\rho = \text{const}$. With $p = K\rho$, $\int dp/\rho = K\ln\rho$, giving:

$\frac{|\mathbf{v}|^2}{2} + K\ln\rho = \text{const.}$

Continuity for steady compressible flow: $\rho A q = \text{const}$ (mass flux), where $A$ is the cross-sectional area.

Question Archetypes

bernoulli-pressure (1 question(s); 2024)

Recognition Cues — The question hands you $u(x,y)$ and $v(x,y)$ with a rational form involving $(x^2+y^2)^n$ in the denominator. You are asked to “verify” the flow satisfies the equations of motion and to “determine the pressure”. The phrase “inviscid incompressible” tells you Euler applies; you must first check $\nabla\cdot\mathbf{v} = 0$ (incompressibility) and $\omega_z = 0$ (irrotationality) before invoking Bernoulli.

Solution Template

1. Compute $\partial u/\partial x + \partial v/\partial y$ using the quotient rule. Show the sum is zero (incompressibility).
2. Compute $\omega_z = \partial v/\partial x - \partial u/\partial y$. Show it is zero (irrotationality).
3. Compute $|\mathbf{v}|^2 = u^2 + v^2$. Simplify using the identity $(x^2-y^2)^2 + 4x^2y^2 = (x^2+y^2)^2$ if applicable.
4. Apply Bernoulli: $p = p_\infty - \tfrac{1}{2}\rho|\mathbf{v}|^2$.

Worked Example(s)

2024 Paper 2, 2024-P2-Q7c (20 marks)

Let $u = B(x^2-y^2)/(x^2+y^2)^2$, $v = 2Bxy/(x^2+y^2)^2$, $w = 0$ satisfy the equations of motion for inviscid incompressible flow. Determine the pressure.

Step 1 — Incompressibility. Compute $\partial u/\partial x$ by quotient rule:

$\frac{\partial u}{\partial x} = \frac{B\cdot 2x(x^2+y^2)^2 - B(x^2-y^2)\cdot 4x(x^2+y^2)}{(x^2+y^2)^4} = \frac{2Bx(3y^2 - x^2)}{(x^2+y^2)^3}.$

Compute $\partial v/\partial y$:

$\frac{\partial v}{\partial y} = \frac{2Bx(x^2+y^2)^2 - 2Bxy\cdot 4y(x^2+y^2)}{(x^2+y^2)^4} = \frac{2Bx(x^2 - 3y^2)}{(x^2+y^2)^3}.$

Sum: $\dfrac{2Bx[(3y^2-x^2)+(x^2-3y^2)]}{(x^2+y^2)^3} = 0.$ Incompressible. ✓

Step 2 — Irrotationality. Compute $\partial v/\partial x$:

$\frac{\partial v}{\partial x} = \frac{2By(y^2 - 3x^2)}{(x^2+y^2)^3}.$

Compute $\partial u/\partial y$:

$\frac{\partial u}{\partial y} = \frac{-2By(x^2+y^2)^2 - B(x^2-y^2)\cdot 4y(x^2+y^2)}{(x^2+y^2)^4} = \frac{-2By[x^2+y^2+2(x^2-y^2)]}{(x^2+y^2)^3} = \frac{2By(y^2-3x^2)}{(x^2+y^2)^3}.$

So $\partial v/\partial x = \partial u/\partial y$, hence $\omega_z = 0$. Irrotational. ✓

Step 3 — Compute $|\mathbf{v}|^2$.

$u^2 + v^2 = \frac{B^2(x^2-y^2)^2 + 4B^2x^2y^2}{(x^2+y^2)^4} = \frac{B^2[(x^2-y^2)^2 + 4x^2y^2]}{(x^2+y^2)^4}.$

Apply the identity $(x^2-y^2)^2 + 4x^2y^2 = x^4 + 2x^2y^2 + y^4 = (x^2+y^2)^2$:

$|\mathbf{v}|^2 = \frac{B^2}{(x^2+y^2)^2}.$

Step 4 — Pressure via Bernoulli.

$\boxed{p = p_\infty - \frac{\rho B^2}{2(x^2+y^2)^2}.}$

As $|z| \to \infty$, $|\mathbf{v}| \to 0$ and $p \to p_\infty$ (far-field pressure). ✓

Common Traps

• Verify both conditions before invoking Bernoulli. Incompressibility alone is not enough; irrotationality is what allows Bernoulli to hold everywhere (not just along a streamline). Check both $\nabla\cdot\mathbf{v} = 0$ and $\omega_z = 0$.
• The key identity. $(x^2-y^2)^2 + 4x^2y^2 = (x^2+y^2)^2$ makes $|\mathbf{v}|^2$ collapse to $B^2/(x^2+y^2)^2$. Without this simplification the pressure formula looks intractable.
• Quotient-rule bookkeeping. Each partial derivative requires careful application of the quotient rule. Setting up the numerator systematically avoids sign errors.

compressible-bernoulli (1 question(s); 2017)

Recognition Cues — The problem involves “steady” flow through a pipe or nozzle of varying cross-section. A density–pressure relation is given, typically $K = p/\rho$ (isothermal) stated as a constant. You are asked to prove a ratio formula relating velocities at the two ends. The key is that density is not constant (compressible), so mass continuity keeps $\rho A q$ rather than just $Aq$.

Solution Template

1. Write the mass continuity equation $\rho_1 A_1 q_1 = \rho_2 A_2 q_2$. Note $A \propto D^2$ (diameter squared). Express $v/V = (\rho_1/\rho_2)(D^2/d^2)$.
2. Write the compressible Bernoulli integral: $\tfrac{q^2}{2} + \int dp/\rho = \text{const}$.
3. Use the given thermodynamic relation to evaluate $\int dp/\rho$. For $p = K\rho$ (isothermal): $\int dp/\rho = K\ln\rho$.
4. Apply Bernoulli at both ends; solve for $\rho_1/\rho_2$ in terms of the velocities.
5. Substitute into the continuity result to obtain the required formula.

Worked Example(s)

2017 Paper 2, 2017-P2-Q7c (15 marks)

A stream rushes through a conical pipe with end diameters $D$ and $d$. If $V$ and $v$ are the corresponding velocities and $K = p/\rho$ is constant, prove $v/V = (D^2/d^2)\,e^{(v^2-V^2)/2K}$.

Step 1 — Continuity. For steady flow, mass flux is constant: $\rho_1 A_1 V = \rho_2 A_2 v$. With $A \propto D^2$:

$\rho_1 D^2 V = \rho_2 d^2 v \quad\Rightarrow\quad \frac{v}{V} = \frac{\rho_1}{\rho_2}\cdot\frac{D^2}{d^2}. \qquad(1)$

Step 2 — Compressible Bernoulli. Along a streamline with $p = K\rho$:

$\frac{q^2}{2} + \int\frac{dp}{\rho} = \text{const}, \quad \int\frac{dp}{\rho} = \int\frac{K\,d\rho}{\rho} = K\ln\rho.$

Applying at both ends:

$\frac{V^2}{2} + K\ln\rho_1 = \frac{v^2}{2} + K\ln\rho_2 \quad\Rightarrow\quad K\ln\frac{\rho_1}{\rho_2} = \frac{v^2 - V^2}{2}.$

Hence:

$\frac{\rho_1}{\rho_2} = e^{(v^2-V^2)/2K}. \qquad(2)$

Step 3 — Combine (1) and (2):

$\boxed{\frac{v}{V} = \frac{D^2}{d^2}\,e^{(v^2-V^2)/2K}.}$

Common Traps

• Do not cancel density in continuity. The fluid is compressible, so $\rho_1 \ne \rho_2$. Cancelling $\rho$ as in incompressible flow eliminates the exponential entirely and gives only $v/V = D^2/d^2$.
• Pressure integral is $K\ln\rho$, not $K/\rho$. The isothermal relation $p = K\rho$ gives $dp = K\,d\rho$, so $\int dp/\rho = K\int d\rho/\rho = K\ln\rho$. This logarithm is what produces the exponential in the final formula.
• Area scales as diameter squared. The circular cross-section gives $A = \pi D^2/4 \propto D^2$, so the continuity ratio picks up $D^2/d^2$, not $D/d$.

Marks-Aware Writing

15-mark answer (compressible-bernoulli). Three clear steps: (1) continuity with the $D^2/d^2$ ratio explicitly derived; (2) compressible Bernoulli integral with the isothermal substitution shown step by step; (3) algebraic combination. Each step should display one key equation. The result must match the boxed formula exactly — leave nothing implicit.

20-mark answer (bernoulli-pressure). Four steps, each needing workings: (1) full quotient-rule computation of $\partial u/\partial x$ and $\partial v/\partial y$; (2) full computation of $\partial v/\partial x$ and $\partial u/\partial y$; (3) algebra showing $|\mathbf{v}|^2 = B^2/(x^2+y^2)^2$ via the $(x^2\pm y^2)$ identity; (4) Bernoulli giving the pressure formula. The derivations in steps 1 and 2 are the main marks-earners — a final answer without the algebra will lose most of the credit.

Practice Set

• 2013-P2-Q8b (15 m) — — Hint: irrotational flow and Bernoulli; check the vorticity condition before applying the theorem.
• 2019-P2-Q8b (15 m) — — Hint: verify incompressibility and find the pressure via Bernoulli for a given 2D velocity field.