Question

For a 2D ideal flow, let \(\varphi\) be the velocity potential and \(\psi\) be the stream function. Which one of the following is TRUE?

• A. \(\nabla^2 \varphi = 0\) and \(|\nabla \psi|^2 = |\nabla \varphi|^2\)
• B. \(\nabla^2 \varphi = 0\) and \(\nabla \psi . \nabla \varphi \neq 0\)
• C. \(\nabla^2 \psi = 0\) and \(|\nabla \psi|^2 \neq |\nabla \varphi|^2\)
• D. \(\nabla^2 \psi = 0\) and \(\nabla \psi \times \nabla \varphi = 0\)

Hint:

In potential flow theory, the conditions \(\nabla^2 \varphi = 0\) and \(\nabla^2 \psi = 0\) are fundamental. The orthogonality condition \(\nabla \psi . \nabla \varphi = 0\) means that streamlines (lines of constant \(\psi\)) and equipotential lines (lines of constant \(\varphi\)) intersect at right angles.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An ideal flow is a theoretical fluid flow that is both incompressible and irrotational.
- Incompressibility leads to the continuity equation \(\nabla . \vec{V} = 0\).
- Irrotationality means the vorticity is zero, \(\nabla \times \vec{V} = 0\).
The velocity potential (\(\varphi\)) exists for irrotational flows (\(\vec{V} = \nabla \varphi\)).
The stream function (\(\psi\)) exists for 2D incompressible flows (\(u = \partial\psi/\partial y, v = -\partial\psi/\partial x\)).
Step 2: Key Formula or Approach:
We need to check the validity of the mathematical statements in each option based on the properties of ideal flow.
1. Check if \(\varphi\) and \(\psi\) satisfy the Laplace equation (\(\nabla^2 f = 0\)).
2. Check the relationship between the magnitudes of the gradients, \(|\nabla \varphi|\) and \(|\nabla \psi|\).
3. Check the relationship between the gradient vectors themselves (dot product, cross product).
Step 3: Detailed Explanation or Calculation:
Let the velocity components be \(u\) and \(v\).
From the definition of velocity potential: \(u = \frac{\partial \varphi}{\partial x}\) and \(v = \frac{\partial \varphi}{\partial y}\).
From the definition of stream function: \(u = \frac{\partial \psi}{\partial y}\) and \(v = -\frac{\partial \psi}{\partial x}\).
Laplace Equations:
- For an incompressible flow, \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\). Substituting the definitions from \(\varphi\): \[ \frac{\partial}{\partial x}\left(\frac{\partial \varphi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{\partial \varphi}{\partial y}\right) = \frac{\partial^2 \varphi}{\partial x^2} + \frac{\partial^2 \varphi}{\partial y^2} = \nabla^2 \varphi = 0 \] So, \(\nabla^2 \varphi = 0\) is TRUE.
- For an irrotational flow, vorticity is zero: \(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0\). Substituting the definitions from \(\psi\): \[ \frac{\partial}{\partial x}\left(-\frac{\partial \psi}{\partial x}\right) - \frac{\partial}{\partial y}\left(\frac{\partial \psi}{\partial y}\right) = -\left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right) = -\nabla^2 \psi = 0 \] So, \(\nabla^2 \psi = 0\) is also TRUE.
Magnitudes of Gradients:
The magnitude of the velocity vector is \(|\vec{V}| = \sqrt{u^2 + v^2}\).
\(|\nabla \varphi|^2 = \left(\frac{\partial \varphi}{\partial x}\right)^2 + \left(\frac{\partial \varphi}{\partial y}\right)^2 = u^2 + v^2 = |\vec{V}|^2\).
\(|\nabla \psi|^2 = \left(\frac{\partial \psi}{\partial x}\right)^2 + \left(\frac{\partial \psi}{\partial y}\right)^2 = (-v)^2 + (u)^2 = u^2 + v^2 = |\vec{V}|^2\).
Therefore, \(|\nabla \psi|^2 = |\nabla \varphi|^2 = |\vec{V}|^2\). This statement is TRUE.
Dot and Cross Products of Gradients:
- Dot Product: \(\nabla \psi . \nabla \varphi = \frac{\partial \psi}{\partial x}\frac{\partial \varphi}{\partial x} + \frac{\partial \psi}{\partial y}\frac{\partial \varphi}{\partial y} = (-v)(u) + (u)(v) = 0\). This means the gradients are orthogonal, so the statement \(\nabla \psi . \nabla \varphi \neq 0\) is FALSE.
- Cross Product (in 2D): \(\nabla \psi \times \nabla \varphi = (\psi_x \hat{i} + \psi_y \hat{j}) \times (\varphi_x \hat{i} + \varphi_y \hat{j}) = (\psi_x \varphi_y - \psi_y \varphi_x)\hat{k}\). Substituting \( \psi_x = -v, \psi_y = u, \varphi_x = u, \varphi_y = v \), we get \( ((-v)(v) - (u)(u))\hat{k} = -(u^2+v^2)\hat{k} = -|\vec{V}|^2 \hat{k} \). This is only zero if the velocity is zero. So, \(\nabla \psi \times \nabla \varphi = 0\) is FALSE in general.
Evaluating the Options:
(A) \(\nabla^2 \varphi = 0\) (True) and \(|\nabla \psi|^2 = |\nabla \varphi|^2\) (True). This option is TRUE.
(B) \(\nabla^2 \varphi = 0\) (True) and \(\nabla \psi . \nabla \varphi \neq 0\) (False). This option is FALSE.
(C) \(\nabla^2 \psi = 0\) (True) and \(|\nabla \psi|^2 \neq |\nabla \varphi|^2\) (False). This option is FALSE.
(D) \(\nabla^2 \psi = 0\) (True) and \(\nabla \psi \times \nabla \varphi = 0\) (False). This option is FALSE.
Step 4: Final Answer:
The only true statement is (A).
Step 5: Why This is Correct:
For a 2D ideal (incompressible and irrotational) flow, both the velocity potential \(\varphi\) and the stream function \(\psi\) satisfy the Laplace equation. Additionally, the squared magnitude of the gradient of both functions is equal to the squared magnitude of the fluid velocity.