Equation of Continuity

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2018)
• Priority tier: T4
• Marks (count): 10 (1)
• Average solve time: ~15 min
• Difficulty mix: medium 1
• Section: A | Dominant type: computation

Why This Chapter Matters

The equation of continuity is the mathematical expression of mass conservation in fluid flow — the first equation any fluid mechanics problem must satisfy. UPSC 2018 tested it computationally: verify that a given velocity field satisfies the continuity equation or find a stream function for a given velocity component. The computation is straightforward if you know the two forms (compressible and incompressible) and the stream function relations.

Minimum Theory

Physical Statement

Mass is neither created nor destroyed in fluid flow. If $\rho$ is the fluid density and $\mathbf{v} = (u, v, w)$ is the velocity field, the rate of change of mass in any fixed control volume equals the net mass flux into it.

General (Compressible) Form

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0,$

or in Cartesian components:

$\frac{\partial \rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z} = 0.$

Incompressible Form

For incompressible flow, $\rho = \text{const}$ (density is constant in time and space), so $\partial\rho/\partial t = 0$ and $\rho$ factors out:

$\nabla \cdot \mathbf{v} = 0, \quad \text{i.e.,} \quad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0.$

In 2D incompressible flow (no $z$-dependence, $w = 0$):

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.$

The Stream Function $\psi$

For 2D incompressible flow, the continuity equation is automatically satisfied by introducing the stream function $\psi(x, y)$ defined by:

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

Verification: $\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = \dfrac{\partial^2 \psi}{\partial x\,\partial y} - \dfrac{\partial^2 \psi}{\partial y\,\partial x} = 0$. $\checkmark$

Properties of $\psi$:

• Curves $\psi = \text{const}$ are streamlines of the flow.
• The volume flux per unit width between two streamlines $\psi_1$ and $\psi_2$ is $|\psi_2 - \psi_1|$.
• For irrotational flow ($\nabla \times \mathbf{v} = \mathbf{0}$), both $\phi$ (velocity potential) and $\psi$ satisfy Laplace’s equation: $\nabla^2\phi = 0$ and $\nabla^2\psi = 0$.

Finding $\psi$ from a Given Velocity Component

Given $u(x, y)$ and the incompressibility condition, find $v$ and $\psi$:

1. From $\partial u/\partial x + \partial v/\partial y = 0$, integrate to get $v = -\int (\partial u/\partial x)\,dy + g(x)$.
2. Determine $g(x)$ from any additional condition (e.g., a boundary condition or a given expression for $v$).
3. Find $\psi$ by integrating: $\psi = \int u\,dy$ (treating $x$ as constant), then check $-\partial\psi/\partial x = v$.

Question Archetypes

verify-continuity (1 question; 2018)

Recognition Cues

• An explicit velocity field $(u, v)$ or $(u, v, w)$ is given.
• Asked to “verify”, “check”, or “show” that the continuity equation is satisfied.
• May also ask for the stream function as a follow-up.

Solution Template

1. State the continuity equation for incompressible flow: $\partial u/\partial x + \partial v/\partial y = 0$ (2D) or $\partial u/\partial x + \partial v/\partial y + \partial w/\partial z = 0$ (3D).
2. Compute each partial derivative from the given expressions.
3. Sum the partial derivatives and show the sum is zero.
4. State the conclusion: the velocity field satisfies continuity, hence represents an incompressible flow.
5. If stream function is asked: integrate $u = \partial\psi/\partial y$ to find $\psi$, verify $v = -\partial\psi/\partial x$.

Worked Example

2018 Paper 2, 2018-P2-QMF (10 marks)

For the 2D velocity field $u = x^2 - y^2$, $v = -2xy$, (a) verify that the equation of continuity is satisfied, and (b) find the stream function $\psi$.

Part (a): Verify continuity.

For 2D incompressible flow, the continuity equation is:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.$

Compute:

$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x,$

$\frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(-2xy) = -2x.$

Sum:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 2x + (-2x) = 0. \quad \checkmark$

The velocity field satisfies the equation of continuity. $\blacksquare$

Part (b): Find the stream function.

Use the defining relations $u = \partial\psi/\partial y$ and $v = -\partial\psi/\partial x$.

From $\partial\psi/\partial y = u = x^2 - y^2$, integrate with respect to $y$ (treating $x$ as constant):

$\psi = x^2 y - \frac{y^3}{3} + f(x),$

where $f(x)$ is an arbitrary function of $x$ (the “constant” of integration with respect to $y$).

Now use $v = -\partial\psi/\partial x$:

$-\frac{\partial\psi}{\partial x} = -(2xy + f'(x)) = v = -2xy.$

Therefore:

$-(2xy + f'(x)) = -2xy \implies f'(x) = 0 \implies f(x) = C \text{ (constant)}.$

The stream function is:

$\boxed{\psi = x^2 y - \frac{y^3}{3} + C.}$

Remark. Note that $\psi = x^2 y - y^3/3 = \text{Im}(x+iy)^3 = \text{Im}(z^3)$, confirming that $w = z^3$ is the complex potential — the velocity field corresponds to the cubic flow. $\blacksquare$

Common Traps

• Forgetting to check both partial derivatives: some students check only $\partial u/\partial x$ and forget $\partial v/\partial y$.
• Integration error when finding $\psi$: treating $y$ terms as constants when integrating with respect to $y$.
• Forgetting $f(x)$ after the first integration — it is essential to include the arbitrary function and determine it from the second equation.
• For 3D: forgetting $\partial w/\partial z$ in the continuity equation.

Marks-Aware Writing

This is a 10-mark Section A question, likely split as 5+5 for parts (a) and (b).

For part (a): state the continuity equation, compute partials, show they sum to zero — 3 lines is sufficient.

For part (b): show the integration step clearly, include $f(x)$, and determine it from the second relation — do not just write the answer; the method carries the marks.

Practice Set

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