ME108: Mechanical Behavior of Engineering Materials
Fall 2015 - Homework 4
Assigned: 09/25/2015, Due: 10/02/2015
Note: Homework may be submitted on plain, lined or graph paper (8.5 x 11 is preferred). Paper
must have clean edges (not be torn from spiral b

Top wall: U=U0 , V=0, =0
Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity
63
Bottom wall: U=V=0, =1
Right and Left wall: U=V=0,
X 0
Non-dimensional heat transfer parameter Nusselt number is stated as:
3.2 Numerical method
Firstly the proble

the fluid layer, one horizontal solid surface being at a temperature higher than the other. If
the upper plate is the hot surface, then the lower surface has heavier fluid and by virtue of
buoyancy the fluid would not come to the lower plate. Because in t

h1 a.V ; h2 b.V (12)
1231422314
1341223412
1;1;
22
1;1
22
axxxxayyyy
bxxxxbyyyy
(13)
Vectors a and b are the vectors that are limited by the middle points of opposite sides and h 1
and h2 are the projections of these vectors on the flow direction. The mos

Nusselt number curves. Here, one can note the time spent by the hot fluid coming from the
first source and traveling to the second one. For example, for Re = 100 and d = 1, 2, and 3,
the time shots are, respectively, around t = 1.4, 3.0, and 4.0. However,

transfer rate for a cycle on both surfaces has been calculated and it is found that both are the
same.
5.3 Effects of the aspect ratio on the flow response
The flow responses to the periodic thermal forcing for the other two aspect ratios are shown
in Fig

Re = 50
d=1
Re = 1
d=2
Re = 1000
d=1
Re = 100
d=1
Re = 100
d=2
Re = 50
d=2
Re = 10
d=2
Re = 10
d=3
Re = 1
d=3
Re = 1000
d=2
Re = 50
d=3
Re = 100
d=3
Re = 1000
d=3
Convection and Conduction Heat Transfer
22
Gr=103-Heater1
1 10 100 1000
0
2
4
6
8
10
12
14
d

0
0.5
1
Re = 1
= 00
Re = 10
= 00
Re = 50
= 00
Re = 100
= 00
Re = 1
= 450
Re = 10
= 450
Re = 50
= 450
Re = 100
= 450
Re = 1
= 900
Re = 10
= 900
Re = 50
= 900
Re = 100
= 900
Fig. 18. Isotherms for Gr = 105, Re = 1, 10, 50, 100 and = 0, 45, 90
Co

aspect ratios considered, above which the symmetric solutions are unstable to finite
perturbations and asymmetric solutions are instead obtained. Results are presented
detailing the occurance of the pitchfork bifurcation in each of the aspect ratios consi

recirculation gets further from the module allowing a stronger contact of the cold fluid with
the heat module.
A Mixed Convection Study in Inclined Channels with Discrete Heat Sources
13
(a)
(b)
(c)
Fig. 5. Velocity distributions for Gr = 105, = 0, and Re

, (7)
and the heating-up or cooling-down time scale of the enclosure under the same boundary
conditions from Saha et al. (2010a) is
tfr=
_
h(1 + A2)1/2 Ax1
_2
A1/2Ra1/4(1 + A2)5/4, (8)
where x1 is given by
x1 L
1
_
1 A1/2Ra1/4(1 + A2)1/4
h2 t p
1/2
, (9)

Salmun H. (1995b). The stability of a single-cell steady-state solution in a triangular enclosure.
Intl. J. Heat Mass Transfer., Vol. 18, 363369.
54ConvectionandConductionHeatTransfer
3
Analysis of Mixed Convection in a
Lid Driven Trapezoidal Cavity
M. A.

this barrier to communication.
2. Stereotyping - Stereotyping causes us to typify a person, a group, an event or a thing on
oversimplified conceptions, beliefs, or opinions. Thus, basketball players can be
stereotyped as tall, green equipment as better th

VI Contents
Chapter 8 Non-Linear Radiative-Conductive
Heat Transfer in a Heterogeneous
Gray Plane-Parallel Participating Medium 177
Marco T.M.B. de Vilhena,
Bardo E.J. Bodmann and Cynthia F. Segatto
Chapter 9 Optimization of the Effective
Thermal Conducti

T temperature of the fluid
T0 temperature of the ambient fluid
TA the amplitude
u, v velocity components along the x- and y- axes respectively
x, y cartesian coordinates
Greek letters
Volumetric coefficient of thermal expansion
Kinematic viscosity
Dens

temperature profiles evaluated along the line DE shown in Fig. 2 at different time instances of
the third thermal forcing cycle are depicted in Fig. 4. At the beginning of the cycle (t = 2.00P)
the velocity is the highest near the roof of the attic (see F

difference in temperature on the pair of heat sources. In a general way, as Re is raised, the
recirculations tend to cease, making it possible to the cold fluid be more in contact with the
module surfaces, hence, invigorating the heat transfer. For high v

Problem 4, part a
Problem 4, part b
(i)
Strain hardening
Cold working, including rolling, extruding, forging or drawing, will increase the
dislocation density both through creation of new dislocations during deformation or
from dislocation pileup at a gra

1 f I
130ij
Boblml _ ._
E
AT (REM? 80066 ,m Hg E;
I Pth-H l
I (mgt Mg? m 7wc o
we 45%" We Wm/C)
I 1w -
(249w -5 pYDHW 6
I tmicttog'fvndl.-1re, ' Mac/MW Pro, . ~04
I PMrIIM CMVSC. 1+.th lamallae Lmtg, 11W WWW?! Mammy
i PQW ML he) Mn! HUM cmhewidrh,

ME 108 Homework 4 Solutions
Problem 1
(a) 50% fine pearlite, 50% martensite.
!
(b)
!
!
!
!
1
Problem 2
(a) + Fe3 C (Bainite for this specific heat treatment process)
(b)
(c) Greater than 90% martensite
(d)
2
Problem 3
The microstructure of tempered marten

ME 108 Homework 6 Solutions
Problem 1
Ve =
4r3
D3
=
3
6
dVe
D2
dD
=
dVe =
2
Ve
We know
D2
dD
2
D3
6
=3
dD
D
pr
pD
=
, z = 0
2t
4t
dD
d(D)
=
x = y =
D
D
x = y =
From Hookes law
x =
Hence
x (1 )
1
pD(1 )
x (y + z ) =
=
E
E
4tE
dD
3pD(1 )
dVe
=3
= 3x =
Ve

homogeneous boundary conditions. We introduce these conditions directly in the matrix of
the system and/or in the basis trial functions. For this reason, it is easy to compute
collocation procedure. Let us explain this method for the steady dynamic proble

6. Environmental forces can inhibit growth and trust. Likewise, a persons environment
can nurture and sustain growth-producing behaviors such as creativity, high
learning, group productivity, personal growth, and group vitality.
Growth occurs when a perso

_
(2)
v
t
+ u v
x
+ v v
y
= 1
p
y
+
_
2v
x2 +
2v
y2
_
+ g (T T0) (3)
T
t
+ u T
x
+ v T
y
=
_
2T
x2 +
2T
y2
_
(4)
where u and v are the velocity components along x and ydirections, t is the time, p is
the pressure, , , and are kinematic viscosity, density

obtained in experimental and numerical investigations. The first comparison is accomplished
not only by using experimental results (Lee & Mateescu, 1998; Armaly et al., 1983), but also
by numerical ones (Lee & Mateescu, 1998; Gartling, 1990; Kim & Moin, 1

convective terms in all throughout the chapter. 5980 four-noded elements were used to
discretize the spatial domain. Comparisons were performed to validate the
computational code. It was observed from the results of the present problem that the
effect of

conditions in the transverse direction was considered for analysis where identical
disposition and heat generation of the ribs on each board were assumed. The governing
equations were discretised using a control volume approach on a staggered mesh and a
p

subcritical pitchfork bifurcation is created giving rise to an asymmetric plume occurring at
a critical Rayleigh number, Ra = 1.42 105. The steady state laminar natural convection in
right triangular and quarter circular enclosures is investigated by Kent

Int. J. Num. Meth. in Fluids, Vol. 8, pp. 1469-1490.
Comini, G.; Manzam, M. & Cortella, G. Open Boundary Conditions for the Streamfunction
Vorticity Formulation of Unsteady Laminar Convection, Num. Heat Transfer Part B,
Vol. 31, pp. 217-234.
Guimares, P.M

10
12
14
16
NUH1
NUH2
NUH3
Gr=105,
Fig. 20. Average Nusselt number vs Reynolds number for Gr = 10 3, 104, 105, = 0, 45, 90
Convection and Conduction Heat Transfer
28
much difference between them. An exception would be the case where Gr = 10 5, Re = 1000,

A supervisor may give instructions from the driver's seat of a pick-up truck. Talking
through an open window and down to an employee makes the truck door a barrier. A
person sitting behind a desk, especially if sitting in a large chair, talking across the

[10] E. Papanicolaou, Y. Jaluria, Mixed convection from and isolated heat source in a
rectangular enclosure, Numer. Heat Transfer, Part A 18 (1990) 427-461
[11] E. Papanicolaou, Y. Jaluria, Transition to a periodic regime in mixed convection in a
square c

flow and in that case the top moving lid is moving in the positive direction at a specified
rotational angle [figure 1]. The second condition is the aiding flow condition where the
shear driven flow aids the natural convective flow and the moving top lid