MATH35001: EXAMPLE SHEET1 V
1.) The gure below shows a lm of Newtonian incompressible uid on an inclined belt which is moving
with constant velocity U . The no-slip condition applies on the surface of the belt, i.e. the uid
particles on the belt move wi
MATH35001: EXAMPLE SHEET1 IV
1.) The interface between two media is positioned along the plane x3 = 0. Medium (1) is a Newtonian
incompressible uid which occupies the region x3 0.
(i) Assuming that the velocity and pressure elds in the uid are given, de
MATH35001: EXAMPLE SHEET1 III
1.) A 2D ow eld is given by u1 = 1+x2 cos x1 and u2 = 3+Ax3 sin x1 . For which value of
the constant A is this ow eld consistent with the assumption of incompressibility?
2.) The observation of a 2D ow eld shows that th
MATH35001: EXAMPLE SHEET1 II
1.) A 2D ow eld is given by u1 = ax2 and u2 = ax1 , where a > 0 is a constant.
a) Sketch the ow eld (begin by considering the velocity vectors on the coordinate
axes and on the main diagonals).
b) Determine the trajectories
MATH35001: EXAMPLE SHEET1 I
1.) Which of these equations in index notation are valid? Remember the summation
a) c = ai bi
b) c = aij bi
c) ci = aij bi
d) ci = aij bj
e) ci = aji bj
f) ij = ij T + Eijkl ekl
g) ij = kl Ti + Eijkl eij
MT35001: SOLUTION FOR EXAMPLE SHEET1 I
1.) Which one of these equations in index notation are valid? Remember the summation
a) c = ai bi (OK, this is the dot product c = a b)
b) c = aij bi (Wrong, the free index j doesnt appear on LHS)