Planetesimal-Erosion

Flow around a planetesimal

This project aims to investigate the flow around and erosion of a planetisimal by a viscous fluid under physical conditions that typically occur in protoplanetary discs. As a starting point, the erosion model presented in Cedenblad et al. (2021) will be numerically reproduced and systematically extended in perspective.

A first step is to investigate the stresses acting on the surface of a stationary sphere flown around by a gas. In particular, the influence of the Mach number in the range \(0<\mathcal{M}<2\) on the shear and normal stress distribution is to be investigated in order to gain a basic understanding of the flow conditions relevant for erosion.

Theoretical Framework

Required is the distribution of normal and shear stresses acting on the surface of a sphere at rest with radius \(R\) as a function of the Mach number \(\mathcal{M}\). The Mach number is defined as the ratio between the velocity \(\boldsymbol{u}\) of the flowing fluid (or an object moving through the fluid, depending on the chosen reference system) and the speed of sound \(c_\mathrm{S}\) of the considered medium:

\begin{align} \mathcal{M} = \frac{|\boldsymbol{u}|}{c_\mathrm{S}} \end{align}

The normal stress describes the component of the mechanical stress acting perpendicular to the surface and contains in particular the contribution of the static pressure. Shear stress, on the other hand, acts tangentially to the surface and is closely linked to velocity gradients in the fluid. Both quantities are components of the stress tensor \(\boldsymbol{\sigma}\) and play a central role in the description of force transmissions between fluid and body, especially with regard to possible erosion processes on the object surface.

Normal and shear stress

In fluid mechanics, the mechanical stress acting on a surface element is described by the stress tensor \(\boldsymbol{\sigma}\). It specifies the internal stresses acting at each point of a continuum due to external forces. The stress tensor is a second-rank tensor whose components \(\sigma_{ij}\) represent the stresses acting on a surface with a normal in the direction of the \(j\)-th coordinate axis due to forces in the direction of the \(i\)-th coordinate axis. The stress tensor describes both the normal stress \(\sigma_{\mathrm{n}}\) and the shear stress \(\tau_{\mathrm{n}}\) acting on a surface element with unit normal vector \(\boldsymbol{n}\).

The mechanical stress that a fluid exerts on a certain surface within the continuum is described by the so-called traction vector \(\boldsymbol{T}^{(\boldsymbol{n})}\). This specifies the force per unit area that acts on a surface with a unit normal vector \(\boldsymbol{n}\). Mathematically, the traction vector results from the transpose of the application of the transposed normal vector to the stress tensor:

\begin{align} \boldsymbol{T}^{(\boldsymbol{n})} = (\boldsymbol{n}^\mathsf{T} \cdot \boldsymbol{\sigma})^\mathsf{T} \end{align}

The normal stress is the projection of this traction vector onto the normal coordinate axis:

\begin{align} \sigma_{\mathrm{n}} = \boldsymbol{T}^{(\boldsymbol{n})} \cdot \boldsymbol{n}= (\boldsymbol{n}^\mathsf{T} \cdot \boldsymbol{\sigma})^\mathsf{T} \cdot \boldsymbol{n} \end{align}

In component-wise notation this results in:

\begin{align} \sigma_{\mathrm{n}} = \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)_j \, n_j = \sigma_{ij} \, n_i \, n_j \end{align}

The shear stress corresponds to the magnitude of the shear stress vector \(\boldsymbol{\tau}_{\mathrm{n}}\), which results from the subtraction of the normal stress vector \(\boldsymbol{\sigma}_{\mathrm{n}}=\sigma_{\mathrm{n}}\boldsymbol{n}\) from the traction vector: \begin{align} \boldsymbol{\tau}_{\mathrm{n}} = \boldsymbol{T}^{(\boldsymbol{n})} - \sigma_{\mathrm{n}}\boldsymbol{n} \end{align}

In component-wise notation this results in:

\begin{align} \tau_{\mathrm{n},j} = \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)_j - \sigma_{\mathrm{n}} \, n_j = \sigma_{ij} \, n_i - (\sigma_{kl} \, n_k \, n_l) \, n_j \end{align}

The shear stress is then given by :

\begin{align} \phantom{\Rightarrow}\;& \tau_{\mathrm{n}}^2 = |\boldsymbol{\tau}_{\mathrm{n}}|^2 = \tau_{\mathrm{n},j} \, \tau_{\mathrm{n},j} \\[6pt] \Leftrightarrow\;& \tau_{\mathrm{n}}^2 = (\sigma_{ij} \, n_i - \sigma_{\mathrm{n}} \, n_j) (\sigma_{mj} \, n_m - \sigma_{\mathrm{n}} \, n_j) \\[6pt] \Leftrightarrow\;& \tau_{\mathrm{n}}^2 = \sigma_{ij} \, \sigma_{mj} \, n_i \, n_m - \sigma_{ij} \, \sigma_{\mathrm{n}} \, n_i \, n_j - \sigma_{mj} \, \sigma_{\mathrm{n}} \, n_m \, n_j + \sigma_{\mathrm{n}}^2 \\[6pt] \Leftrightarrow\;& \tau_{\mathrm{n}}^2 = (\sigma_{ij} \, n_i) (\sigma_{mj} \, n_m) - (\sigma_{ij} \, n_i \, n_j) \, \sigma_{\mathrm{n}} - (\sigma_{mj} \, n_m \, n_j) \, \sigma_{\mathrm{n}} + \sigma_{\mathrm{n}}^2 \\[6pt] \Leftrightarrow\;& \tau_{\mathrm{n}}^2 = \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)_j \, \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)_j - \sigma_{\mathrm{n}}^2 \\[6pt] \Leftrightarrow\;& \tau_{\mathrm{n}}^2 = \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)^2 - \boldsymbol{\sigma}_{\mathrm{n}}^2 \\[6pt] \Rightarrow\;& \tau_{\mathrm{n}} = \sqrt{ \big(\boldsymbol{T}^{(\boldsymbol{n})}\big)^2 - \boldsymbol{\sigma}_{\mathrm{n}}^2 } \end{align}