Tutorial Two
[BTME2213 FLUID MECHANICS]
State all necessary assumptions, show all necessary steps and analyse the result. Give your
comment/write justification.
Q1.
⃗ = (4 + 𝑥𝑦 + 2𝑡)𝑖 + 6𝑥 3 𝑗 + (3𝑥𝑡 2 + 𝑧)𝑘. Find the acceleration of a fluid
Given a velocity field 𝑉
particle at (2 m, 4 m, −4 m) and time 𝑡 = 3 s.
[ANS: 1426.78 m/s2]
Q2.
A steady, incompressible, two-dimensional velocity field is given by the following components in the
𝑥-𝑦 plane:
𝑢 = 0.205 + 0.97𝑥 + 0.851𝑦
𝑣 = −0.509 + 0.953𝑥 − 0.97𝑦
a) Calculate the acceleration field.
b) Calculate the acceleration at the point (2 m, 1.5m).
Q3.
⃗ = 3𝑥𝑦 2 𝑖 + 2𝑥𝑦𝑗 + (2𝑦𝑧 + 3𝑡)𝑘 . At
The velocity field in a fluid medium is given by 𝑉
(1 m, 2 m, 1 m) and at time 𝑡 = 3 s, find
a) translational velocity,
b) rotational velocity, and
c) the vorticity of a fluid element.
[ANS: 18.14 m/s; 4.12 rad/s; 8.25 rad/s]
Q4.
The velocity of a viscous fluid flowing between the parallel surfaces as shown in Figure Q4 is defined
by 𝑢 = 0.002[1 − 10 × 103 𝑦 2 ] m/s, where 𝑦 is in metres. Determine the Vorticity and shear-strain
rate of a fluid element located at 𝑦 = 5 mm within the flow.
[0.2 rad/s; -0.2 rad/s]
Figure Q4
Q5.
Figure Q5
Flow through the converging nozzle in Figure Q5 can be approximated by the one-dimensional
velocity distribution
2𝑥
𝑢 ≈ 𝑉𝑜 (1 + )
𝑣=𝑤=0
𝐿
a) Find a general expression for the fluid acceleration in the nozzle.
b) For the specific case 𝑉𝑜 = 10 m/s and 𝐿 = 6 cm, compute the acceleration at the entrance and at
the exit.
[ANS:
Q6.
2𝑉𝑜2
𝐿
(1 +
2𝑥
𝐿
); 3333 m/s2; 10000 m/s2]
According to potential theory for the flow approaching a rounded
two-dimensional body, as in Figure Q6, the velocity approaching the
stagnation point is given by
𝑎2
𝑢 = 𝑈∞ (1 − 2 )
𝑥
where 𝑎 is the nose radius and 𝑈∞ is the velocity far upstream.
a) Compute the value and position of the maximum deceleration
along this streamline.
𝜕𝑢
Figure Q6
b) The viscous normal stress is given by 𝜏𝑥𝑥 = 2𝜇 𝜕𝑥 . Evaluate the
-3
maximum viscous normal stress if the density of flow medium is 917 kg.m , dynamic viscosity is
2
𝑈∞
5
0.97 kg.m-1.s-1 with 𝑈∞ = 2 m/s and 𝑎 = 6 cm.
[ANS: -0.372
; −√3 𝑎; 39 N/m2]
a
Department of Mechanical Engineering[Rev1:Oct2016]
pkq\dmee\fm\t\Kinematics of Fluid
T2-1