Calculating the velocity field around a rigid sphere under creeping flow conditions

Question

A Rigid sphere is placed at the origin in the undisturbed linear flow u_i^∞=E_ijx_j +Ω_ijx_j where E_ij and Ω_ij are the constant rate of strain and vorticity tensors respectively. Assume the sphere is freely suspended under creeping flow conditions.

a) Find the total velocity, u_i and the dynamic pressure, p', everywhere.

b) Determine how the sphere moves, i.e. what is the center of mass velocity and rotation rate?

c) In the far field, what is the slowest decaying term in the disturbance velocity and what kind of flow is this?

Answer 1

The problem you've presented involves the flow of a viscous fluid around a rigid sphere placed at the origin, subject to external linear flow characterized by the tensors Eij and Ωij. We will address each part of the problem:

**a) Find the total velocity, ui, and the dynamic pressure, p', everywhere.**

The total velocity u at any point in the fluid can be obtained by adding the external linear flow to the flow induced by the sphere. Under creeping flow conditions, the governing equations are the Stokes equations. The total velocity field can be expressed as:

\[u_i = E_{ij}x_j + \Omega_{ij}x_j + u_i^s\]

Where:

- \(u_i\) is the total velocity at point (x,y,z).

- \(E_{ij}\) is the rate of strain tensor.

- \(Ω_{ij}\) is the vorticity tensor.

- \(u_i^s\) is the velocity field induced by the sphere.

To find \(u_i^s\), you can use the Stokes flow equations and the principle of superposition. It's essentially the velocity field generated by a single rigid sphere in a viscous fluid, which can be found using methods like the method of images.

The dynamic pressure p' is given by:

\[p' = -p_0 + 2\mu D_{ij}D_{ij}\]

Where:

- \(p_0\) is the reference pressure (usually zero for convenience).

- \(\mu\) is the dynamic viscosity of the fluid.

- \(D_{ij}\) is the rate of strain tensor (symmetric part of \(E_{ij}\)).

You can calculate \(D_{ij}\) from \(E_{ij}\) and then find p'.

**b) Determine how the sphere moves, i.e., what is the center of mass velocity and rotation rate?**

The center of mass velocity \(V_c\) of the sphere can be found by integrating the velocity field \(u_i\) over the surface of the sphere. It represents the translational motion of the sphere:

\[V_c = \frac{1}{V_s} \int_{S} u_i dS\]

Where:

- \(V_s\) is the volume of the sphere.

- \(S\) is the surface of the sphere.

The angular velocity \(\omega\) (rotation rate) can be found from the vorticity tensor:

\[\omega_i = \frac{1}{2} \epsilon_{ijk} \Omega_{jk}\]

Where:

- \(\epsilon_{ijk}\) is the Levi-Civita symbol.

**c) In the far field, what is the slowest decaying term in the disturbance velocity, and what kind of flow is this?**

In the far field, the disturbance velocity typically decays as \(1/r\), where r is the distance from the sphere. The slowest decaying term in the disturbance velocity is the far-field term, which has this \(1/r\) decay. This corresponds to a potential flow. The far-field behavior is often described using the concept of point sources and sinks to model the flow around the sphere.

Please note that solving the specific equations and tensors requires knowledge of the exact form of Eij and Ωij, and the sphere's size and boundary conditions, which may vary depending on the problem setup.