Understanding Irrotational Flow

In fluid dynamics, a flow is described as irrotational if the curl of the velocity field is zero. The velocity field $\mathbf{v}$ represents the velocity of the fluid particles at any point in space.

For a two-dimensional velocity field given by $\mathbf{v} = u(x,y)\mathbf{i} + v(x,y)\mathbf{j}$, the condition for irrotational flow is that the component of the curl in the $\mathbf{k}$ direction is zero. Mathematically, this is expressed as:

\(\text{curl } \mathbf{v} \cdot \mathbf{k} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0\)

This condition simplifies to:

\(\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}\)

Analyzing the Given Velocity Field

The given velocity field is $\mathbf{v} = (2 + 6x - 6y)\mathbf{i} + (3x + cx - y)\mathbf{j}$.

From this equation, we can identify the components of the velocity field:

• The x-component of velocity, $u$, is \(u = 2 + 6x - 6y\).
• The y-component of velocity, $v$, is \(v = 3x + cx - y\).

We need to find the value of the constant 'c' that makes this flow irrotational.

Calculating Partial Derivatives

To apply the irrotational condition, we need to calculate the partial derivatives of $u$ with respect to $y$ and $v$ with respect to $x$.

Let's calculate $\frac{\partial u}{\partial y}$:

\(\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(2 + 6x - 6y)\)

Treating $x$ as a constant during partial differentiation with respect to $y$:

\(\frac{\partial u}{\partial y} = 0 + 0 - 6 = -6\)

Now, let's calculate $\frac{\partial v}{\partial x}$:

\(\frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(3x + cx - y)\)

Treating $y$ and 'c' as constants during partial differentiation with respect to $x$:

\(\frac{\partial v}{\partial x} = 3 + c - 0 = 3 + c\)

Applying the Irrotational Condition

For the flow to be irrotational, the condition $\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}$ must be satisfied.

Substituting the calculated partial derivatives:

\(3 + c = -6\)

Solving for the Constant 'c'

Now, we solve this equation for 'c':

\(c = -6 - 3\)

\(c = -9\)

Thus, the value of the constant 'c' for which the given velocity field represents an irrotational flow is -9.

Summary of Steps

Here is a summary of the steps taken to find the value of 'c' for irrotational flow:

1. Identified the $u$ and $v$ components of the velocity field $\mathbf{v}$.
2. Recalled the condition for irrotational flow in 2D: \(\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}\).
3. Calculated the partial derivative \(\frac{\partial u}{\partial y}\).
4. Calculated the partial derivative \(\frac{\partial v}{\partial x}\).
5. Equated the two partial derivatives to set up an equation involving 'c'.
6. Solved the equation to find the value of 'c'.

The value of 'c' that satisfies the irrotational flow condition is -9.