Question

For a description in spherical polar coordinates with axial symmetry of the flow of a very viscous fluid, the components of the velocity field \(\mathbf{u}\) are given in terms of the stream function \(\psi\) by $$ u_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta}, \quad u_{\theta}=\frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r} $$ Find an explicit expression for the differential operator \(E\) defined by $$ E \psi=-(r \sin \theta)(\nabla \times \mathbf{u})_{\phi} $$ The stream function satisfies the equation of motion \(E^{2} \psi=0\) and, for the flow of a fluid past a sphere, takes the form \(\psi(r, \theta)=f(r) \sin ^{2} \theta\). Show that \(f(r)\) satisfies the (ordinary) differential equation $$ r^{4} f^{(4)}-4 r^{2} f^{\prime \prime}+8 r f^{\prime}-8 f=0 $$

Short answer

Find curl of \(\mathbf{u}\), substitute into \(E\psi\), and simplify with \(\psi(r, \theta) = f(r) \sin^2 \theta\) to find differential equation for \(f(r)\).

Step-by-step solution

- Understanding given velocity components

First, identify the expressions for the velocity components in terms of the stream function \(\psi\): \[ u_{r} = \frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta}, \quad u_{\theta} = \frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r} \]

- Curl of velocity field

Write the expression for the curl of the velocity field in spherical coordinates and focus on the \(\phi\)-component: \[ (abla \times \mathbf{u})_{\phi} = \frac{1}{r} \left[ \frac{\partial}{\partial r}(r u_{\theta}) - \frac{\partial u_{r}}{\partial \theta} \right] \]

- Substitute velocity components

Substitute the expressions for \(u_{r}\) and \(u_{\theta}\) into the equation for the curl: \[ (abla \times \mathbf{u})_{\phi} = \frac{1}{r} \left[ \frac{\partial}{\partial r} \left(-\frac{\partial \psi}{\partial r} \right) - \frac{\partial}{\partial \theta} \left( \frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta} \right) \right] \]

- Simplify the expression

Simplify the expression for \( (abla \times \mathbf{u})_{\phi} \):\end{} \[ (abla \times \mathbf{u})_{\phi} = -\frac{1}{r} \frac{\partial}{\partial r} \left( \frac{1}{r \sin \theta} \frac{\partial \psi}{\partial r} \right) - \frac{1}{r} \frac{1}{ \sin \theta} \left( \frac{1}{r^{2} \sin \theta} \frac{\partial^2 \psi}{\partial \theta^2} - \frac{\cos \theta}{r^{2} \sin^2 \theta} \frac{\partial \psi}{\partial \theta} \right) \]

- Define differential operator

The differential operator \(E\) is defined as \[-(r \sin \theta)(abla \times \mathbf{u})_{\phi}\]. Hence, \[-(r \sin \theta)(abla \times \mathbf{u})_{\phi}=E\psi\]

- Derive equations for the stream function

Substitute the known relationships into \(E \psi = 0\) using \(\psi(r, \theta) = f(r) \sin^2 \theta\). Simplify to find the differential equation satisfied by \(f(r)\).

Show differential equation for f(r)

Start from \(E \psi = 0\) written as \[-(r \sin \theta)(abla \times \mathbf{u})_{\phi}=E f(r) \sin^2 \theta\] and simplify the resulting expression, then isolate \(f(r) \) leading to: \[r^4 f^{(4)} - 4r^2 f'' + 8r f' - 8 f=0\]

Key concepts

Velocity Field

In fluid dynamics, the velocity field is a vector field that represents the velocity of fluid at different points in space. For spherical polar coordinates with axial symmetry, the components of the velocity field, denoted by \(u_r\) and \(u_{\theta}\), are expressed using the stream function \(\psi\).

The given velocity components are:

• \(u_r = \frac{1}{r^2 \sin \theta} \frac{\partial \psi}{\partial \theta} \)
• \(u_{\theta} = \frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r} \)

Here, \(u_r\) represents the radial component, while \(u_{\theta}\) represents the polar or angular component.

This representation simplifies the analysis of fluid flow, allowing us to solve for quantities like the curl of the velocity field more easily.

Stream Function

The stream function \(\psi\) is a scalar function used to describe the flow of an incompressible fluid, especially in cases with axial symmetry. For such systems, it simplifies the representation of the fluid's velocity components.

The given velocity components in terms of \(\psi\) are:

• Radial component: \(u_r = \frac{1}{r^2 \sin \theta} \frac{\partial \psi}{\partial \theta} \)
• Polar component: \(u_{\theta} = \frac{-1}{r \sin \theta} \frac{\partial \psi}{\partial r} \)

The stream function helps us convert partial differential equations for fluid flow into more manageable forms. Using \(\psi\), we can describe the patterns of flow and solve for other characteristics like the curl of the velocity field.

Differential Operator

In the given exercise, we need to find an explicit expression for the differential operator \(E\). This operator is used to describe the relationship between the stream function \(\psi\) and the fluid flow.

The operator \(E\) is defined by:
\[ E \psi = -(r \sin \theta)(abla \times \mathbf{u})_{\phi} \]
Here, \(abla \times \mathbf{u} (abla \times \mathbf{u})_{\phi}\) represents the \(\phi\)-component of the curl of the velocity field \(\mathbf{u}\). By substituting the velocity components \(u_r\) and \(u_{\theta}\), and simplifying, we derive the expression for \(E \psi\). This simplifies further analysis of the fluid flow and helps solve the equation of motion.

Equation of Motion

The equation of motion in fluid dynamics describes the behavior of fluid particles. For the flow of a fluid past a sphere with axial symmetry, the stream function must satisfy the equation \(E^2 \psi = 0\).

Given that the stream function is \(\psi(r, \theta) = f(r) \sin^2 \theta\), we can derive the equation for \(f(r)\). The differential operator \(E\) applied twice to \(\psi\) results in:

• \(r^4 f^{(4)} - 4 r^2 f'' + 8 r f' - 8 f = 0\)

This ordinary differential equation must be solved to find the function \(f(r)\) that describes the fluid behavior around a sphere. This form of the equation reflects the complexity of the flow while maintaining symmetry.