Problem 3.1 The material description of a fluid motion is given by the pathline equations
$$
\begin{aligned}
& x_1=\xi_1, \
& x_2=k \xi_1^2 t^2+\xi_2, \
& x_3=\xi_3
\end{aligned}
$$
with $k$ as a constant having a dimension, such that the dimensional integrity of both sides of the above equation systems is preserved. Show that the Jacobian determinant $J=\operatorname{det}\left(\partial x_i / \partial \xi_j\right)$ does not vanish and obtain the transformation $\boldsymbol{\xi}=\boldsymbol{\xi}(x, t)$.

Problem 3.2 The fluid motion is described by:
$$
\begin{aligned}
x_1 & =\xi_1, \
x_2 & =\frac{1}{2}\left(\xi_2+\xi_3\right) e^{a t}+\frac{1}{2}\left(\xi_2-\xi_3\right) e^{-a t}, \
x_3 & =\frac{1}{2}\left(\xi_2+\xi_3\right) e^{a t}-\frac{1}{2}\left(\xi_2-\xi_3\right) e^{-a t}
\end{aligned}
$$
(a) Show that the Jacobian determinant does not vanish.
(b) Determine the velocity and acceleration components
(1) in material coordinates $V_i\left(\xi_j, t\right), A_i\left(\xi_j, t\right)$,
(2) in spatial coordinates $V_i\left(x_j, t\right), A_i\left(x_j, t\right)$.

Problem 3.4 The motion of a fluid is described in the material coordinate by:
$$
\begin{aligned}
& x_1=\xi_1 e^{a t}, \
& x_2=\xi_2 e^{a t}, \
& x_3=\xi_3 e^{-2 a t}
\end{aligned}
$$
with $a=$ const. and $\boldsymbol{\xi}=\boldsymbol{\xi}(x, t=0)$.
a) Calculate the velocity and acceleration components $V_i\left(\xi_j, t\right.$ and $A_i\left(\xi_j, t\right)$ in material coordinates.
b) Determine the spatial description of the velocity and acceleration components $V_i\left(x_k, t\right)$ and $A_i\left(x_k, t\right)$ by eliminating the material coordinate $\xi_j=\xi_j\left(x_k, t\right)$ in the results obtained in (a).
c) Find the acceleration components using the substantial derivatives of $V_i\left(x_k, t\right)$.
d) Is this a potential flow? If yes, find the potential function.

Problem 3.7 The components of a velocity flow field $V_i\left(\xi_j\right)$ are given by
$$
\begin{aligned}
& V_1=a\left(x_1+x_2\right) \
& V_2=a\left(x_1-x_2\right) \
& V_3=W
\end{aligned}
$$
with the constants $a$ and $W$. Determine
(a) the divergence $\nabla \cdot V$ of the flow field,
(b) the rotation $\nabla \times \mathbf{V}$,
(c) the parametric representation of the pathlines $x_i=x_i\left(\xi_j, t\right)$ with $\xi_j=x_j(t=0)$,
(d) nonparametric representation of the projection of the pathlines in $x_1, x_2$-plane by eliminating the curve parameter $t$,
(e) the projection of the streamlines in $x_1, x_2$-plane by integrating the differential equations for the streamlines.

Problem 4.1 Incompressible Newtonian fluid with constant density and viscosity flows between two parallel plates with infinite width. Body forces are neglected. Given are the plate height $h$, the components of the pressure gradient, $\frac{\partial p}{\partial x_1}=-K \frac{\partial p}{\partial x_2} \equiv 0, \frac{\partial p}{\partial x_3} \equiv 0$, the velocity field between the plates $u_1\left(x_2\right)=\frac{K}{2 \mu}\left(\frac{h^2}{4}-x_2^2\right), u_2 \equiv 0, u_3 \equiv 0$, the density $\varrho$ and the absolute viscosity $\mu$.
(a) Show that the given velocity field satisfies the continuity and the Navier-Stokes equation.
(b) Determine the components of the stress tensor.
(c) Calculate the dissipation function $\Phi$.
(d) Find the energy per unit depth, length, and time dissipated in heat within the gap.
(e) Calculate the principal stresses and their directions.

Problem 4.2 Newtonian fluid flows through the sketched channel, Fig. 4.4, with infinite extensions in $x_1$ – and $x_3$-direction and the height $h$. The plane flow is steady, the density $\varrho$ and the viscosity $\mu$ are assumed to be constant, and body forces are neglected. The top and bottom wall are porous such that a constant normal velocity component $V_W$ can be established at the walls. The pressure gradient in $x_1$-direction is constant $\left(\partial p / \partial x_1=-K\right)$. Because of the infinite extension of the channel, the velocity distribution does not depend upon $x_1$. The variables $\varrho, \mu, K, h, V_W$ are given.
(a) Using the continuity equation, calculate the distribution of the velocity component in $x_2$-direction $u_2\left(x_2\right)$.
(b) Simplify the $x_1$-component of the Navier-Stokes equation for this problem.
(c) Give the boundary condition for the velocity component $u_1$.
(d) Calculate the velocity distribution $u_1\left(x_2\right)$. (Hint: After solving the homogeneous differential equation, the particular solution of the inhomogeneous differential equation can be found setting $u_{1_p}=$ const. $x_2$ ).

Problem 4.3 A Newtonian fluid with constant density and viscosity flows steadily through a two dimensional vertically positioned channel with the width $h$ shown in Fig. 4.5. The motion of the fluid is described by the Navier Stokes equations. The flow is subjected to the gravitational acceleration $\mathbf{g}=e_1 g$ and a constant pressure gradient in flow direction $x_1$. Assume that $V_2=V_3=0$

(a) Determine the solution of the Navier-Stokes equations.
(b) Write a computer program; show the velocity distributions for the following cases: (a) For $K=0$, (b) $K>0$, and (c) $K<0$.
(c) For which $K$ there is no flow?

Problem 4.4 A Newtonian fluid with constant density and viscosity flows steadily through a two dimensional channel positioned at an angle $\alpha$ shown in Fig. 4.6 with the width $2 \mathrm{~h}$. The motion of the fluid is described by the Navier Stokes equations. The flow is subjected to the gravitational acceleration $\mathbf{g}=\mathbf{e}_{\mathbf{1}} \mathbf{g}_1+\mathbf{e}_2 \mathbf{g}_2$ and a constant pressure gradient in flow direction $x_1$. Assume that $V_2=V_3=0$.
(a) Determine the solution of the Navier-Stokes Equations. Write a computer program and plot the velocity distributions for: (a) For $K=0$, (b) $K>0$, and (c) $K<0$.
(b) For which $K$ there is no flow?