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Unformatted text preview: FLUID MECHANICS II
Assignment 2
Question 1
a) By following the motion of a ﬂuid parcel in a twodimensional, planar ﬂow (parcel moves from (x,y) to
(x+∆ x, y+∆ y) in time ∆ t), show that Lagrangian and Eulerian acceleration in 2 D (a = axˆ + ay ˆ) are
i
j
related by
Du(t)
∂u
∂u
∂u
=
+u
+v ,
Dt
∂t
∂x
∂y
Dv (t)
∂v
∂v
∂v
ay =
=
+u
+v ,
Dt
∂t
∂x
∂y ax = where u, v on RHS are the Eulerian velocities, i.e.
u(x, y, t) = u(x, y, t)ˆ + v (x, y, t)ˆ
i
j.
b) Work out the corresponding connection between the parcel rate of change of temperature
the Eulerian temperature ﬁeld T(x,y,t). DT (t)
Dt and Question 2
By expanding out the vector equality
a= Du(t)
∂u
=
+ (u. )u,
Dt
∂t where
= ∂ˆ
∂ˆ
∂ˆ
i+
j+
k
∂x
∂y
∂z work out the threedimensional equivalents of 1(a); i.e. ﬁnd ax , ay and az in terms of Eulerian quantities. Question 3: Problem 4.2
Question 4: Problem 4.3
a)b)d) Question 5
Suppose that the temperature ﬁeld T = 4 x2 − 3 y 3 , in arbitrary units, is associated with the velocity ﬁeld
of Prob 4.3. Compute the rate of change dT at (x, y ) = (2, 1).
dt Question 6: Problem 4.6
Question 7: Problem 4.8
Question 8: Problem 1.80
Consider only the lower portion of the positive quadrant. Plot up the following two streamlines:
i) the streamline that passes through (0,0).
ii) the streamline that passes through (2,1).
A small particle moves along the ﬂuid. Find its velocity and acceleration as it passes through (2,1). ...
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This note was uploaded on 09/14/2011 for the course ME 362 taught by Professor Ceciledevaud during the Winter '11 term at Waterloo.
 Winter '11
 CecileDevaud

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