The local skin-friction coefficient ?f00(0) and the rate of mass transfer ?0(0) increases while
the rate of heat transfer ?0(0) decreases with increasing the Schimdt number and Brownian
motion parameters have the same effect on the local skin-friction coefficient, the rate
of mass transfer and rate of heat transfer since increasing the Brownian motion parameter
increases ?f00(0), ?0(0) while ?0(0) decreases. The order of a chemical reaction, n and the
thermophoresis parameter, Nt have the same effect on the local skin coefficient, rate of heat
and rate of mass transfer since an increase in either two of the parameters leads to a decrease
in ?f00(0) and ?0(0) whereas an increase in ?(0). As the Prandtl number increase ?f00(0)
and ?0(0) also increase but with an encountered decrease in ?0(0).
Methodology 27
Parameter Symbol Value
Eckert number Ec (0:1; 3)
Prandtl Pr (4; 7:2)
Thermal buoyancy 1 (0; 1)
solutal buoyancy 2 (0; 1)
Magnetic field parameter M (0; 6)
Brownian motion parameter Nb (0:1; 5)
Thermophoresis parameter Nt (0:1; 0:5)
Schmidt number Sc (0:2; 2:62)
Order of chemical reaction n (1; 3)
Table 3.3: Parameters in the model and their values.
Brownian motion
The Brownian motion of nanoparticles is a potential factor in calculating the thermal conductivity
of nanofluids due to the fact of the size of nanaparticles, in the development of such
a theory for the thermal conductivity of nanofluids there is an assumption that the energy
transport in nanofluids involves four models which are the thermal diffusion in nanoparticles,
collision between base fluid molecules, thermal interaction of dynamic nanoparticles
with the base fluid molecules and also the collision between nanoparticles. Fig 3.2 shows
the impact of the Brownian motion on the velocity, temperature and concentration profiles.
The Brownian motion is the random movement of particles suspended in a fluid resulting
from the collision with the fast moving molecules of a fluid. An increase in the Brownian
motion parameter makes the momentum or velocity boundary layer to become thin as shown
in Fig3.2. The temperature increases with increasing the Brownian motion parameter and
as the Brownian motion increases we experience a decrease in the concentration of the fluid
and the velocity profile. The Brownian motion helps to heat the fluid in the boundary layer
and aggravates particle deposition away from the fluid on the surface thereby a reduction in
the concentration profile.
Methodology 28
Figure 3.2: The Brownian motion on the velocity, temparature and concentration profiles.
0 5 10
0
0.2
0.4
0.6
0.8
1
´
f?(´)
Nb = 0.1
Nb = 0.25
Nb = 0.5
(a) Velocity profile for Brownian motion
0 1 2 3
0
0.2
0.4
0.6
0.8
1
´
µ(´)
Nb = 0.1
Nb = 0.25
Nb = 0.5
(b) Temperature profile for Brownian motion
0 5 10 15
0
0.5
1
1.5
2
´
Á(´)
Nb = 0.1
Nb = 0.25
Nb = 0.5
(c) Concentration profile for Brownian motion
Thermophoresis
Thermophoresis is the movement of particles which arises due to some force resulting from
a temperature gradient that acts in the direction of high temperature to low temperature
regions. Thermophoresis is also known as thermo-migration or Ludwig-Soret effect or thermodiffusion,
thermophoresis is a phenomenon by which small sized particles suspended in a
non-isothermal gas acquire a velocity relative to the gas in the direction of decreasing temperature.
Increasing the thermophoresis parameter leads to an increase in the fluid concentration
and fluid velocity, Fig 3.3 depicts that the augmentation in the value of Nt improves the
concentration profile, however the temperature of the fluid lowers as the thermophoresis parameter
increases. The thermophoretic force for small particles is given as Ft = ?(pd2
prT)=T
where p is the gas pressure, is the gas mean free path, dp is the particle diameter, rT is
Methodology 29
the change in temperature and T is the absolute temperature of the particle. The impact of
thermophoresis parameter on velocity, concentration and temperature is shown in Fig 3.3.
Figure 3.3: The Brownian motion on the velocity, temparature and concentration profiles.
0 5 10
0
0.2
0.4
0.6
0.8
1
´
f?(´)
Nt = 0.1
Nt = 0.25
Nt = 0.4
(a) Velocity profile for Thermophoresis
0 1 2 3
0
0.2
0.4
0.6
0.8
1
´
µ(´)
Nt = 0.1
Nt = 0.25
Nt = 0.4
(b) Temperature profile for Thermophoresis
0 5 10 15
0
0.5
1
1.5
2
´
Á(´)
Nt = 0.1
Nt = 0.25
Nt = 0.4
(c) Concentration profile for Thermophoresis
Viscous Dissipation
Viscous dissipation is an important component in the flow of fluids having high velocities
and is of interest for many various industrial processes applications. Significant temperature
rises are observed in polymer processing flows such as injection molding or extrusion
of high rates. In aerodynamics, heating in a thin boundary layer around high speed aircraft
leads to the temperature rises in the skin. The Eckert number, Ec is associated with viscous
dissipation in our model. Viscous dissipation is important in fluid flows having high velocity
because this increases the temperature in the boundary layer flow, heat is generated as adjacent
fluid layers slide over each other in a process called viscous dissipation. This usually
Methodology 30
occur when a flowing viscous fluid converts some of its kinetic energy into thermal energy. In
Fig 3.4 it can be clearly shown that an increase in the Eckert number results in an increase
in the temperature. An increase in the fluid velocity can be attributed to a reduction in fluid
viscosity as a result of frictional heating, The increase in velocity is however insignificant as
the heat generation takes place in a smaller region compared to the rest of the fluid and this
explains the decrease in the temperature as we move further into the free stream away from
the surface.
Figure 3.4: The Viscous Dissipation on the velocity, temparature and concentration profiles.
0 5 10
2
0

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