Understanding Velocity Potential for a Line Source

In fluid mechanics, a line source is a theoretical concept representing fluid flowing outwards uniformly in all radial directions from a line, typically considered infinite in length, in a two-dimensional flow field. This type of flow is often analyzed using potential flow theory, which assumes the fluid is incompressible and inviscid, and the flow is irrotational.

For irrotational flow, a velocity potential function, denoted by \(\phi\), exists such that the velocity vector \(\vec{V}\) is the gradient of the velocity potential function:

\(\vec{V} = \nabla \phi\)

In a two-dimensional polar coordinate system \((r, \theta)\), the velocity vector \(\vec{V}\) can be expressed in terms of radial velocity \(v_r\) and tangential velocity \(v_{\theta}\):

\(\vec{V} = v_r \hat{r} + v_{\theta} \hat{\theta}\)

The gradient of the velocity potential \(\phi\) in polar coordinates is given by:

\(\nabla \phi = \frac{\partial \phi}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial \phi}{\partial \theta} \hat{\theta}\)

Comparing the components of the velocity vector and the gradient of the potential function, we get:

• Radial velocity: \(v_r = \frac{\partial \phi}{\partial r}\)
• Tangential velocity: \(v_{\theta} = \frac{1}{r} \frac{\partial \phi}{\partial \theta}\)

Analyzing Flow from a Line Source

For a simple line source of strength \(Q\) (volume flow rate per unit length), the flow is purely radial. This means there is no tangential velocity component:

\(v_{\theta} = 0\)

The radial velocity \(v_r\) at a radial distance \(r\) from the source is determined by considering the flow through a cylindrical surface of radius \(r\) and unit length. The total flow rate through this surface must equal the source strength \(Q\). The area of this surface (per unit length) is \(2\pi r\). Therefore, the radial velocity is:

\(v_r = \frac{\text{Source Strength}}{\text{Area}} = \frac{Q}{2\pi r}\)

Deriving the Velocity Potential Function

Now we use the relationship between velocity components and the partial derivatives of the velocity potential \(\phi\).

Since \(v_{\theta} = 0\), we have:

\(\frac{1}{r} \frac{\partial \phi}{\partial \theta} = 0\)

This implies that \(\frac{\partial \phi}{\partial \theta} = 0\), which means the velocity potential \(\phi\) is independent of the angular coordinate \(\theta\). It is only a function of the radial distance \(r\), i.e., \(\phi = \phi(r)\).

Using the radial velocity component:

\(v_r = \frac{\partial \phi}{\partial r} = \frac{Q}{2\pi r}\)

To find \(\phi(r)\), we need to integrate the expression for \(\frac{\partial \phi}{\partial r}\) with respect to \(r\):

\(\phi(r) = \int \frac{Q}{2\pi r} dr\)

We can take the constant \(\frac{Q}{2\pi}\) out of the integral:

\(\phi(r) = \frac{Q}{2\pi} \int \frac{1}{r} dr\)

The integral of \(\frac{1}{r}\) with respect to \(r\) is \(\ln|r|\) (natural logarithm of the absolute value of r). Since radial distance \(r\) is always positive, we can write \(\ln(r)\).

\(\phi(r) = \frac{Q}{2\pi} \ln(r) + C\)

where \(C\) is the integration constant. In potential flow problems, the absolute value of the potential is not as important as its derivatives (which give velocity), so the constant \(C\) is typically ignored or set to zero.

Thus, the velocity potential function for a line source varies with the natural logarithm of the radial distance \(r\).

Based on the derivation, the velocity potential function \(\phi\) is proportional to \(\ln r\).

Comparison with Options

Let's compare our finding with the given options:

The derived velocity potential function \(\phi\) varies with \(\ln r\).

Conclusion

The velocity potential function for a line source varies with the natural logarithm of the radial distance, \(r\), specifically as \(\ln r\).

Revision Table: Velocity Potential and Line Source

Additional Information on Potential Flow

Potential flow theory simplifies the analysis of fluid motion under certain assumptions (inviscid, incompressible, irrotational). Besides line sources, other elementary flow patterns include:

• Uniform Flow: Flow with constant velocity in a single direction. Velocity potential \(\phi = Ux + Vy\) in Cartesian coordinates.
• Line Sink: Opposite of a line source, fluid flows radially inwards. Velocity potential \(\phi = -\frac{Q}{2\pi} \ln r\).
• Doublet: Formed by placing a source and a sink of equal strength very close together. Velocity potential \(\phi = -\frac{K \cos \theta}{r}\).
• Vortex: Flow where streamlines are concentric circles. This flow is irrotational everywhere except at the origin. Velocity potential \(\phi = -\frac{\Gamma}{2\pi} \theta\), where \(\Gamma\) is circulation. Note that \(\phi\) is multi-valued.

Complex potential \(W(z) = \phi + i\psi\) is often used in 2D potential flow, where \(\psi\) is the stream function. For a line source, the complex potential is \(W(z) = \frac{Q}{2\pi} \ln z\), where \(z = x + iy = re^{i\theta}\). Then \(W(z) = \frac{Q}{2\pi} (\ln r + i\theta)\), so \(\phi = \frac{Q}{2\pi} \ln r\) and \(\psi = \frac{Q}{2\pi} \theta\).

The stream function \(\psi\) represents streamlines, curves along which \(\psi\) is constant. For a line source, \(\psi = \text{constant}\) implies \(\theta = \text{constant}\), indicating that streamlines are straight lines radiating from the source, which is consistent with the radial flow.