Homework 1
- Your homework solution has to be handed in as a group solution via Moodle.
- Clearly state name and matriculation number of each student
1 Einstein Summation, Vector Algebra
In a right-handed (dextral), orthonormal system of fixed basis vectors \(\mathbf{e}_1\), \(\mathbf{e}_2\) and \(\mathbf{e}_3\) of unit length, an arbitrary vector \(\mathbf{u}\) has the following co-ordinate (component) expression
\[ \mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 = \sum_{i=1}^{3} u_i \mathbf{e}_i \]
By dropping the summation symbol, we get the vector \(\mathbf{u}\) in Einstein summation convention as follows \[ \mathbf{u} = u_i \mathbf{e}_i \] where a summation is implied in an expression, whenever indices occur twice. The repeated indices are called free or dummy indices. In order to evaluate and simplify expressions in vector and tensor algebra we make use of
- The Kronecker delta
\[ \delta_{ij} = \begin{cases} 1,& \text{if } i=j\\ 0,& \text{if } i\not=j \end{cases} \]
- The Levi-Cevita symbol \[ \epsilon_{ijk} = \begin{cases} +1,& \text{if } (i, j, k) \text{ has even permutation of} (1, 2, 3)\\ -1,& \text{if } (i, j, k) \text{ has odd permutations of} (1, 2, 3)\\ 0 & \text{otherwise} \end{cases} \]
For the dot product of two orthonormal basis vectors we have \[ \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \]
Tasks
Task 1
Show that the projection of a vector \(\mathbf{u}\) on any of the basis vectors, \(\text{proj}_{\mathbf{e}_{i}}(\mathbf{u}) := \mathbf{u} \cdot \mathbf{e}_{i}\), is identical to the component of \(\mathbf{u}\) corresponding to that basis vector.
Task 2
Calculate the dot-product of two arbitrary vectors \(\mathbf{u}\) and \(\mathbf{v}\) in index notation
Task 3
Using index notation, calculate the cross-product of two arbitrary vectors \(\mathbf{u}\) and \(\mathbf{v}\)
Task 4
Show that, for three arbitrary vectors \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) the triple vector product \[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = \mathbf{v} (\mathbf{u} \cdot \mathbf{w}) - \mathbf{w} (\mathbf{u} \cdot \mathbf{v}) \]
Between the Kronecker delta and the Levi-Civita symbol, the following identity holds
\[ \epsilon_{ijk} \,\epsilon_{klm} = \delta_{il} \,\delta_{jm} - \delta_{im} \, \delta_{jl} \]
2 Second Order Tensors
Given two vectors \(\mathbf{a}\) and \(\mathbf{b}\), we can construct a second order tensor using a dyadic product
\[ \mathbf{T} = \mathbf{a} \otimes \mathbf{b} = a_i \, \mathbf{e}_i \otimes b_j \, \mathbf{e}_j = a_i b_j \mathbf{e}_i \otimes \mathbf{e}_j = T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \]
The tensor \(\mathbf{T}\) then describes a linear transformation of one vector \(\mathbf{u}\) to another \(\mathbf{v}\) $as follows
\[ \mathbf{v} = \mathbf{T} \mathbf{u} = (\mathbf{a} \otimes \mathbf{b}) \mathbf{u} = (\mathbf{b} \cdot \mathbf{u}) \mathbf{a} \]
The transpose of a second order tensor \(\mathbf{T}\) is given by transposing either the components or the tensor basis \[ \mathbf{T}^{T} = T_{ji} \, \mathbf{e}_i \otimes \mathbf{e}_j = T_{ij} \, \mathbf{e}_j \otimes \mathbf{e}_i \]
Tasks
Task 1
Show that the components of the tensor in the orthonormal basis {\(\mathbf{e}_1\), \(\mathbf{e}_3\), \(\mathbf{e}_2\)} can be calculated as follows \[ A_{ij} = \mathbf{e}_i \cdot \mathbf{A} \mathbf{e}_j \]
Task 2
Using index notation, show that \(\mathbf{v} \cdot \mathbf{A}^{T}\mathbf{u} = \mathbf{u} \cdot \mathbf{A} \mathbf{v}\).
3 Differential Operator
In tensor calculus, given an orthonormal basis {\(\mathbf{e}_1\), \(\mathbf{e}_2\), \(\mathbf{e}_3\)} using cartesian co-ordinates, the vector differential operator \(\nabla\) is denoted using index notation by
\[ \nabla (\bullet) = \mathbf{e}_i \dfrac{\partial(\bullet)}{\partial x_{i}} = \mathbf{e}_i \partial_i \]
However, for a cylindrical co-ordinate system with orthonormal basis {\(\mathbf{e}_{r}\), \(\mathbf{e}_{\theta}\), \(\mathbf{e}_{z}\)}, due to the change of co-ordinate systems, we have \[ \nabla (\bullet) = \mathbf{e}_{r} \dfrac{\partial(\bullet)}{\partial r} + \dfrac{1}{r}\mathbf{e}_{\theta} \dfrac{\partial(\bullet)}{\partial \theta} + \mathbf{e}_{z} \dfrac{\partial(\bullet)}{\partial z} \] and for a spherical co-ordinate system with orthonormal basis {\(\mathbf{e}_{r}\), \(\mathbf{e}_{\theta}\), \(\mathbf{e}_{\varphi}\)}, we have \[ \nabla (\bullet) = \mathbf{e}_{r} \dfrac{\partial(\bullet)}{\partial r} + \dfrac{1}{r}\mathbf{e}_{\theta} \dfrac{\partial(\bullet)}{\partial \theta} + \dfrac{1}{r \sin\theta} \mathbf{e}_{\varphi} \dfrac{\partial(\bullet)}{\partial \varphi} \]
Furthermore, in vector notation, the Laplacian vector operator \(\Delta\) is given by \[ \Delta(\bullet) = \nabla \cdot \nabla (\bullet) \]
Let \(u\) be a scalar field and \(\mathbf{v}\) be a vector field.
Tasks
Task 1
Using a cartesian co-ordinate system, show that \[ \nabla \times (\nabla u) = \mathbf{0} \]
Task 2
Using a cylindrical co-ordinate system, show that \[ \nabla \cdot (\nabla \times \mathbf{v}) = 0 \]
Task 3
Calculate in a spherical co-ordinate system the value of \(\Delta u\)
Task 4
Given another scalar field \(\alpha\), using cartesian co-ordinates, show that
\[ \nabla \cdot (\alpha \mathbf{v}) = \alpha \nabla \cdot \mathbf{v} + \mathbf{v} \cdot \nabla \alpha \]
4 Balance Laws in Component and Index Notation
In the context of continuum mechanics, the material time derivative is given by \[ \dfrac{D(\bullet)}{Dt} = \dfrac{\partial(\bullet)}{\partial t} + \mathbf{v} \cdot \nabla (\bullet) \] Furthermore, in vectorial notation using the Eulerian formulation the mass balance law is given by \[ \dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \] and the momentum balance law is given by \[ \dfrac{\partial\mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla (\mathbf{v}) = \dfrac{1}{\rho} \nabla \mathbf{\sigma} + b \]
Tasks
Task 1
Using the mass balance law, calculate the value of \[ \dfrac{1}{\rho} \dfrac{D(\rho)}{Dt} \]
Task 2
In the Eulerian formulation, write down the momentum balance law in component notation.
In the Lagrangian formulation, the component notation of the momentum balance law is given as \[ x: \dfrac{Dv_{x}}{Dt} = \dfrac{1}{\rho} \left(\dfrac{\partial {\sigma}_{xx}}{\partial x} + \dfrac{\partial {\sigma}_{xy}}{\partial y} + \dfrac{\partial {\sigma}_{xz}}{\partial z}\right) + b_x \] \[ y: \dfrac{Dv_{y}}{Dt} = \dfrac{1}{\rho} \left(\dfrac{\partial {\sigma}_{yx}}{\partial x} + \dfrac{\partial {\sigma}_{yy}}{\partial y} + \dfrac{\partial {\sigma}_{yz}}{\partial z}\right) + b_y \] \[ z: \dfrac{Dv_{x}}{Dt} = \dfrac{1}{\rho} \left(\dfrac{\partial {\sigma}_{zx}}{\partial x} + \dfrac{\partial {\sigma}_{zy}}{\partial y} + \dfrac{\partial {\sigma}_{zz}}{\partial z}\right) + b_z \]
Task 3
In the Eulerian formulation, write down the mass and momentum balance laws in index notation
5 Eulerian and Lagrangian Description
Given the trajectories \(\mathbf X(t, \mathbf x_0) = (X(t, \mathbf x_0), Y(t, \mathbf x_0), Z(t, \mathbf x_0))^T\)
\[\begin{align*} X(t, \mathbf x_0) &= x_0 \, cos(t) + y_0 \, sin(t) \\ Y(t, \mathbf x_0) &= -x_0 \, sin(t) + y_0 \, cos(t) \\ Z(t, \mathbf x_0) &= 0 \end{align*}\]
with \(\mathbf x_0 = (x_0, y_0, z_0)^T\) and the scalar field
\[ \phi(\mathbf X(t), t) = x_0^2 + y_0^2 \]
Tasks
Task 1
Sketch two trajectories for \(t \in (0, 2 \pi)\) as well as their respective field values \(\phi(\mathbf X(t), t)\) over time.
Task 2
Describe \(\phi(\mathbf X(t), t)\) in the Eulerian frame.
Task 3
Consider another field given by \(\psi(\mathbf X(t), t) = X^2 + Y^2\). Rewrite \(\psi\) in the Eulerian frame.
Task 4
Compute \(\frac{d\phi}{dt}\) and \(\frac{D\phi}{dt}\) using their definitions.
6 Streamlines, Pathlines and Streaklines
Consider the velocity field
\[\begin{align*} \mathbf{v}(t,x,y) = \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} - k x + \alpha t \\ k y \end{pmatrix} \end{align*}\]
with \(k, \alpha\) being positive constants.
Tasks
Task 1
Compute the pathlines.
Task 2
Compute the streamlines.
Task 3
Compute the streaklines.
Task 4
Sketch the velocity field for
- \(t_0 < \frac{k x}{\alpha}\)
- \(t_1 = \frac{k x}{\alpha}\)
- \(t_2 > \frac{k x}{\alpha}\)
and furthermore add
- the pathline for one particular \(\mathbf x_0\)
- one particular streamline in each figure
- the streakline starting at \(t_0\) for one particular \(\mathbf x_0\).