Finite Difference method to solve Combined effects of viscous dissipation, radiation and heat generation on unsteady non-Newtonian fluid along a vertically stretched surface

1. Fluid Flow on Vertical Surface
By
Fahad Bin Mostafa, Md Abdus Samad
University of Dhaka @2016

2. Title of the dissertation
Combined effects of viscous dissipation,
radiation and heat generation on unsteady
non-Newtonian fluid along a vertically
stretched surface.

3. Abstract
In this thesis work, an analysis has been carried out to study the combined effect of unsteady free
convection boundary layer flow of non-Newtonian fluid along a vertically stretched surface with viscous
dissipation, thermal radiation and heat generation in the presence of magnetic field. The governing
nonlinear partial differential equations have been transformed to dimensionless equations with the help
of similarity consideration to determine similarity solutions. Transformed equations have been
discretized by implicit finite difference approximation to get solutions. The result of non-dimensional
velocity and temperature profiles are depicted and discussed for different parameter such as Prandtl
number 𝑃𝑟, Eckert number 𝐸𝑐, Grashof number 𝐺𝑟, magnetic parameter 𝑀, and radiation parameter 𝑁.
Moreover, for scientific interest the effects of skin friction coefficient 𝐶𝑓 and Nusselt number 𝑁 𝑢 are
presented in tables.

4. Introduction
Day by day research of non-Newtonian fluids is increasing because of its huge existence in the universe.
Applications of non-Newtonian fluids in many industrial implementation have been an interesting topic
to many researchers. Atmospheric elements, bio fluids such as blood, multiphase mixers, pharmaceutical
formulation, cosmetics and toiletries, paints, and beverage items are examples of non-Newtonian fluids.
Since the study of non-Newtonian fluid flow has become an important matter for the applicants. The
effort has arisen largely due to the need to optimize industrial system such as furnaces, ovens and
boilers and the interest in our environment and in non-conventional energy sources such as the use of
salt-gradient solar ponds for energy collection and storage. In particular, natural convection induced by
the simultaneous action of buoyancy forces resulting from thermal diffusion is of considerable interest in
nature and in many industrial applications such as geophysics, oceanography, drying processes and
solidification of binary alloy.

5. Mathematical Computation
Effect of viscous dissipation on unsteady non-Newtonian fluid along a
vertically stretched surface.
Consider, an unsteady free convection laminar boundary layer flow of non-Newtonian fluid
along a vertically stretched surface which is continuously moving. Here we will consider free
convection with viscous dissipation. Boundary layer has been generalized by the existence of
zero velocity at the surface of the plate. The frame of reference is chosen such that stretching
surface coincide with 𝑌 plane where flow is assumed only for 𝑌 > 0 .

6. For this particular research, the velocity component 𝑢 of the 𝑋 axis pertains along the
surface and the 𝑣 of the 𝑌 axis normal to the surface of the sheet. Equal and opposite forces
are introduced along 𝑋 − axis to keep the origin fixed for 𝑋 − axis in this case.

7. As we have taken non-Newtonian fluids, we can use the theory of power law fluid where 𝜏 is taken
as shear stress, 𝐾 is flow consistency index, stands for power law fluid Hence the relationship has
formulated by the equation 𝜏 = 𝐾
𝜕𝑢
𝜕𝑦
𝑛−1 𝜕𝑢
𝜕𝑦
Pseudo plastic fluid, Newtonian fluid, Dilatant Fluid are depended on changing value of 𝑛.
The continuity equation:
𝜕𝑣
𝜕𝑦
= 0
The momentum equation:
𝜕𝑢
𝜕𝑡
+ 𝑣
𝜕𝑢
𝜕𝑦
=
𝑘
𝜌
𝜕
𝜕𝑥
𝜕𝑢
𝜕𝑦
𝑛−1 𝜕𝑢
𝜕𝑦
+ 𝑔𝛽 𝑇 − 𝑇∞ −
𝜎𝐵2 𝑢
𝜌
The energy equation
𝜕𝑇
𝜕𝑡
+ 𝑣
𝜕𝑇
𝜕𝑦
= 𝛼
𝜕2 𝑇
𝜕𝑦2 +
𝑘
𝜌𝐶 𝑝
𝜕𝑢
𝜕𝑦
𝑛−1 𝜕𝑢
𝜕𝑦
2

8. Initial conditions: 𝑢 = 0, 𝑇 = 0, 𝑎𝑡 𝑡 = 0
Boundary conditions:
for 𝑡 > 0 𝑢 = 𝑈0 , 𝑇 = 𝑇 𝑤 , 𝑎𝑡 𝑦 = 0
𝑢 = 0 , 𝑇 = 𝑇∞ , 𝑎𝑡 𝑦 → ∞
For this study, 𝑢 and 𝑣 are the velocity components in the 𝑋 and 𝑌 direction respectively, 𝑇
stands for temperature profile. Here, 𝑈0 and 𝑇∞ are velocity and temperature profile of
free stream, 𝑇 𝑤 is the temperature at the surface. 𝑡 is for time and 𝜌 is density, 𝐵 is
magnetic field, n is power law index. 𝑘 is thermal conductivity of the fluid, 𝐶 𝑝 is the
specific heat at the constant pressure, 𝜎 is electric conductivity of the fluid.

9. Similarity Consideration
𝑡 =
𝑡𝑢0
𝐿
, 𝑋 =
𝑥
𝐿
𝑌 =
𝑦
𝐿
, 𝑈 =
𝑢
𝑈0
, 𝑣 =
𝑣
𝑈0
, 𝑇 =
𝑇 − 𝑇∞
𝑇 𝑤 − 𝑇∞
Where 𝐿 is the characteristic length and 𝑈0 is an arbitrary reference velocity
which is related to this problem. Since free stream conditions are quiescent in
free convection, there is no logical external reference velocity (𝑉𝑜𝑟 𝑈∞), as in
forced convection.

10. The dimensionless momentum equation is,
𝜕𝑈
𝜕 𝑡
+ 𝑉
𝜕𝑈
𝜕𝑌
=
1
𝑅𝑒
𝜕
𝜕𝑌
𝜕𝑈
𝜕𝑌
𝑛−1
𝜕𝑈
𝜕𝑌
+ 𝐺𝑟 𝑇 − 𝑀𝑈
The dimensionless energy equation is,
𝜕 𝑇
𝜕 𝑡
+ 𝑉
𝜕 𝑇
𝜕𝑌
=
1
𝑃𝑟𝑅𝑒
𝜕2
𝑇
𝜕𝑌2
+
𝐸𝑐
𝑅𝑒
𝜕𝑈
𝜕𝑌
𝑛−1
𝜕𝑈
𝜕𝑌
2
Here dimensionless numbers are
Reynold number 𝑅𝑒 =
𝑈2−𝑛 𝐿 𝑛
𝑘
𝜌
Eckert number 𝐸𝑐 =
𝑈 𝑛+1
𝐶 𝑝∆𝑇𝐿 𝑛
Prandlt Number Pr =
𝑘
𝜌
𝛼
𝑈0
𝐿
𝑛−1
Grashof number 𝐺𝑟 =
𝑔𝛽 𝑇 𝑤 − 𝑇∞ 𝐿
𝑈0
2
The magnetic field parameter 𝑀 =
𝜎𝐵2
𝐿
𝜌𝑈0

11. Initial condition
at 𝑡 = 0
𝑈 = 0, 𝑇 = 0
𝜕𝑈
𝜕𝑌
= 0,
𝜕 𝑇
𝜕𝑌
= 0
Boundary condition
at 𝑡 > 0
𝑈 = 1, 𝑇 = 1 𝑤ℎ𝑒𝑛 𝑌 = 0
𝑈 = 0, 𝑇 = 0 𝑤ℎ𝑒𝑛 𝑌 → ∞

12. Methodology of Numerical
computation
Because of its case of application, the finite-difference method is well suited. In contrast to an analytical
solution, which allows for temperature determination at any point of interest in a medium, a numerical
solution enables determination of velocity and temperature at only discrete points. The first step in any
numerical analysis must therefore be to select these points. The point frequently termed a nodal network,
grid, or mesh. The nodal points are designated by a numbering scheme that, for two dimensional system.
The x and y locations are designated by the 𝑖 and 𝑗 indices, respectively.
We will use implicit difference technique to approximate the solution of governing equations. Finally we will
discuss its convergences. Rectangular region of the flow field is chosen, the region is divided into a grid of
lines parallel to 𝑋 and 𝑌 axes, where 𝑋 is chosen along the stretching surface and 𝑌 is normal to the plate.

13. Let assume the length of the sheet is 𝑋 𝑚𝑎𝑥 = 50 which indicates that x varies from 0 to 50 where 𝑦 varies for
0 to 𝑌 𝑚𝑎𝑥 = 20. So the step size is calculated below.
ℎ =
𝑌 𝑚𝑎𝑥
𝑛 + 1
Here ∆𝑡 = 𝑘 = 0.000001 and for grid spacing 𝑝 𝑠𝑝𝑎𝑐𝑒 = 200 𝑎𝑛𝑑 𝑞(𝑡𝑖𝑚𝑒 = 1000000) in the 𝑋 and 𝑌 directions
respectively.

14. For numerical computation we have to use two point forward difference formula for determining time
derivatives such as
𝜕𝑈
𝜕 𝑡
𝑥𝑖, 𝑡𝑗 and
𝜕𝑈
𝜕 𝑡
(𝑥𝑖, 𝑡𝑗)
For space involving derivative, a two point central difference method has chosen to approximate the
derivatives
𝜕𝑈
𝜕𝑌
𝑥𝑖, 𝑡𝑗 and
𝜕 𝑇
𝜕 𝑡
𝑥𝑖, 𝑡𝑗
where double derivative is involved, there we have picked three point central difference approximation
to find
𝜕2 𝑈
𝜕𝑌2 𝑥𝑖, 𝑡𝑗 and
𝜕2 𝑇
𝜕𝑌2 𝑥𝑖, 𝑡𝑗
Momentum Equation
𝑈𝑖,𝑗+1 = 𝑈𝑖 + 𝑘 −𝑉
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
+
𝑛
𝑅𝑒
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
𝑛−1
𝑈𝑖−1,𝑗 − 2𝑈𝑖,𝑗 + 𝑈𝑖+1,𝑗
ℎ2
+ 𝐺𝑟 𝑇𝑖,𝑗 − 𝑀𝑈𝑖,𝑗

15. Energy Equation
𝑇𝑖,𝑗+1 = 𝑇𝑖,𝑗 + 𝑘 −𝑉
𝑇𝑖+1,𝑗 − 𝑇𝑖−1,𝑗
2ℎ
+
1
𝑃𝑟𝑅𝑒
𝑇𝑖−1,𝑗 − 2 𝑇𝑖,𝑗 + 𝑇𝑖+1,𝑗
ℎ2
+
𝐸𝑐
𝑅𝑒
∗ 𝑈𝐶
for 𝑖 = 1,2, … 𝑝 and 𝑗 = 1,2, … 𝑞
Initial condition
at 𝑡 = 0
𝑈𝑖,0= 0, 𝑇𝑖,0 = 0
𝜕𝑈
𝜕𝑌 𝑖,0
= 0 and
𝜕 𝑇
𝜕𝑌 𝑖,𝑗
= 0 for 𝑖 = 1,2, … 𝑝
Boundary condition
at 𝑡 > 0
𝑈0,𝑗= 1, 𝑇𝑖,0 = 1 where 𝑌 = 0
𝑈 𝑝+1,𝑗= 0, 𝑇𝑝+1,𝑗 = 0 where 𝑌 → ∞ 𝑗 = 1,2, … 𝑞

16. Result and its physical
explanation
In this section, numerical simulation of governing equations in case of various dimensionless parameters such as
𝐺𝑟, 𝑃𝑟, 𝑁, 𝑀 and 𝐸𝑐 have been discussed. Some standard values of parameters and dimensionless numbers are
chosen for its physical importance. Few parameters are chosen arbitrary. In this case graphs have been drawn
for time 𝑡 = 0. Skin friction coefficient 𝐶𝑓 and Nusselt number 𝑁 𝑢 are presented in tables (for each parameter
mentioned above) as well as shown graphically for some significant parameters.

24. Skin Friction Coefficient and Nusselt number for
Boundary Layers on a stretching surface
The skin friction coefficient (𝐶𝑓) is,
𝐶𝑓 =
𝜏 𝑤
1
2
𝜌𝑈0
2
Using forward difference approximation let us discretize the equation
𝐶𝑓=
2
𝑅𝑒
𝑈1,𝑞 − 𝑈0,𝑞
ℎ
𝑛
, if 𝑈1,𝑞 ≥ 𝑈0,𝑞
−
𝑈0,𝑞 − 𝑈1,𝑞
ℎ
𝑛
, if 𝑈1,𝑞 < 𝑈0,𝑞
Now the Nusselt number is given by,
𝑁 𝑢 =
ℎ𝐿
𝑘
Ultimately we will use it after transforming it into finite difference approximation.
𝑁 𝑢=
𝑇0,𝑞 − 𝑇1,𝑞
ℎ

25. Table 3.3: Variation of 𝐶𝑓 and 𝑁 𝑢 for different values of Eckert number 𝐸𝑐.
𝑛 𝐸𝑐 𝐶𝑓 𝑁 𝑢
0 2.997178797 0.3417135675
0.5
0.3 3.013198595 0.3370980749
0.4 3.029829601 0.3372002733
1.0 3.050152541 0.3269248016
0 3.242318136 0.3417453155
0.3 3.272981088 0.3371154413
1 0.4 3.304753560 0.3321830621
1.0 3.338210420 0.3269248016
0 3.449986021 0.3417121002
0.3 3.486650801 0.3370956729
1.5 0.4 3.534585038 0.3321728714
1.0 3.5824085101 0.3269248016

26. Our Main Focus
Combined effects of viscous dissipation,
radiation and heat generation on unsteady
non-Newtonian fluid along a vertically
stretched surface.

27. Mathematical Computation
The continuity equation:
𝜕𝑣
𝜕𝑦
= 0
The momentum equation:
𝜕𝑢
𝜕𝑡
+ 𝑣
𝜕𝑢
𝜕𝑦
=
𝑘
𝜌
𝜕
𝜕𝑥
𝜕𝑢
𝜕𝑦
𝑛−1 𝜕𝑢
𝜕𝑦
+ 𝑔𝛽 𝑇 − 𝑇∞ −
𝜎𝐵2 𝑢
𝜌
The energy equation:
𝜕𝑇
𝜕𝑡
+ 𝑣
𝜕𝑇
𝜕𝑦
= 𝛼
𝜕2 𝑇
𝜕𝑦2 +
1
𝜌𝐶 𝑝
4𝜎1
𝑘1
𝜕2 𝑇4
𝜕𝑦2 +
𝑘
𝜌𝐶 𝑝
𝜕𝑢
𝜕𝑦
𝑛−1 𝜕𝑢
𝜕𝑦
2
Initial conditions 𝑢 = 0, 𝑇 = 0 𝑎𝑡 𝑡 = 0
Boundary Conditions 𝑢 = 𝑈0 , 𝑇 = 𝑇 𝑤 , 𝑎𝑡 𝑦 = 0
𝑢 = 0 , 𝑇 = 𝑇∞ , 𝑎𝑡 𝑦 → ∞ for 𝑡 > 0

28. Value of 𝑛 Type of fluid
<1 Pseudo plastic fluid
1 Newtonian fluid
>1 Dilatant Fluid
If the surface and flow temperature differs, there will be a region where the fluid temperature varies from , 𝑇 =
𝑇 𝑤 , at 𝑦 = 0 𝑡𝑜 𝑇 = 𝑇∞ , 𝑎𝑡 𝑦 → ∞ , in the flow of outer region. This region is known as thermal boundary layer.
Suppose, no change of temperature profiles of the surface 𝑇 𝑊 as well as the free stream temperature profile 𝑇∞
are taken respectively.
For natural convection the flow is induced by a force known as buoyancy forces, which arise from density 𝜌
differences caused by the variation of temperature in the region. If we go through molecular level of the fluid, it
shows us the changes of temperature profiles such as 𝑇 − 𝑇∞ is arisen by the different values of temperature of the
fluid particle. The relative change is calculated as 𝛽 𝑇 − 𝑇∞ , where 𝛽 is designated as the volumetric coefficient
of thermal expansion.
Gravitational force is involved because of buoyancy force. In this case, 𝑔𝛽 𝑇 − 𝑇∞ is the lift force per unit volume
where gravitational acceleration 𝑔 is working through vertical axis. Hence, 𝑔𝛽 𝑇 − 𝑇∞ is active at 𝑋 direction. 𝐵 is
used as applied magnetic field which is depended on fluid’s characters. But it causes the growth of magnetic
force 𝑓. Where
𝑓 =
𝜎𝐵2 𝑢
𝜌

29. Similarity Analysis
The dimensionless momentum equation is,
𝜕𝑈
𝜕 𝑡
+ 𝑉
𝜕𝑈
𝜕𝑌
=
1
𝑅𝑒
𝜕
𝜕𝑌
𝜕𝑈
𝜕𝑌
𝑛−1
𝜕𝑈
𝜕𝑌
+ 𝐺𝑟 𝑇 − 𝑀𝑈
Dimensionless energy equation is
𝜕 𝑇
𝜕 𝑡
+ 𝑉
𝜕 𝑇
𝜕𝑌
=
1
𝑃𝑟𝑅𝑒
1 +
4
3𝑁
𝜕2
𝑇
𝜕𝑌2
+
𝐸𝑐
𝑅𝑒
𝜕𝑈
𝜕𝑌
𝑛−1
𝜕𝑈
𝜕𝑌
2
Initial condition at 𝑡 = 0
𝑈 = 0, 𝑇 = 0
𝜕𝑈
𝜕𝑌
= 0,
𝜕 𝑇
𝜕𝑌
= 0
Boundary condition at 𝑡 > 0 , 𝑈 = 1, 𝑇 = 1 when 𝑌 = 0
𝑈 = 0, 𝑇 = 0 when 𝑌 → ∞

30. Background calculations
𝜕 𝑇
𝜕 𝑡
+ 𝑉
𝜕 𝑇
𝜕𝑌
=
1
𝑈0
2−𝑛
𝐿 𝑛
𝑘
𝜌
×
1
𝑘
𝜌
𝛼
𝑈0
𝐿
𝑛−1
1 +
4 × 4𝜎1 𝑇∞
3
3𝑘1 𝑘
×
𝜕2
𝑇
𝜕𝑌2
+
𝐸𝑐
𝑅𝑒
𝜕𝑈
𝜕𝑌
𝑛−1
𝜕𝑈
𝜕𝑌
2
𝑈0 𝐿
𝛼
=
𝑈0
2−𝑛
𝐿 𝑛
𝑘
𝜌
×
𝑘
𝜌
𝛼
𝑈0
𝐿
𝑛−1
= 𝑃𝑟𝑅𝑒
𝑘
𝜌𝐶 𝑝
×
𝑈0
2𝑛
𝐿2𝑛 ×
𝐿
𝑈0 𝑇 𝑤 − 𝑇∞
=
𝑈 𝑛+1
𝐶 𝑝∆𝑇𝐿 𝑛 ×
𝑈0
𝑛−2 𝑘
𝜌
𝐿 𝑛 =
𝐸𝑐
𝑅𝑒
Radiation number,
𝑁 =
𝑘1 𝑘
4𝜎1 𝑇∞
3
Reynolds number 𝑅𝑒 =
𝑈2−𝑛 𝐿 𝑛
𝑘
𝜌
Eckert number 𝐸𝑐 =
𝑈 𝑛+1
𝐶 𝑝∆𝑇𝐿 𝑛
Prandlt Number Pr =
𝑘
𝜌
𝛼
𝑈0
𝐿
𝑛−1

31. The radiative heat flux 𝑞1 is described by the Rosseland approximation such that,
𝑞1= −
4𝜎1
3𝑘1
𝜕𝑇4
𝜕𝑦
Note that Rosseland approximation is valid for optically thick fluids.
Where, 𝜎1 is the Stefan-Boltzmann constant and 𝑘1 is the Rosseland mean absorption coefficient. It is
surmised that the temperature difference within the flow is sufficiently small such that that 𝑇4 can be can
be expressed in the Taylor series, where free stream temperature 𝑇∞ and neglecting the higher order terms
from Taylor series expansion.
𝑇4≈ 4𝑇∞
3 𝑇 − 3𝑇∞
4
And the thermal diffusivity
α =
𝑘
𝜌𝐶 𝑝

32. Methodology of Numerical
computation
Using FDM we obtain from Momentum Equation
𝑈𝑖,𝑗+1 = 𝑈𝑖 + 𝑘 −𝑉
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
+
𝑛
𝑅𝑒
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
𝑛−1
𝑈𝑖−1,𝑗 − 2𝑈𝑖,𝑗 + 𝑈𝑖+1,𝑗
ℎ2
+ 𝐺𝑟 𝑇𝑖,𝑗 − 𝑀𝑈𝑖,𝑗
Using FDM we obtain from Energy equation,
𝑇𝑖,𝑗+1 − 𝑇𝑖,𝑗
𝑘
+ 𝑉
𝑇𝑖+1,𝑗 − 𝑇𝑖−1,𝑗
2ℎ
=
1
𝑃𝑟𝑅𝑒
1 +
4
3𝑁
𝑇𝑖−1,𝑗 − 2 𝑇𝑖,𝑗 + 𝑇𝑖+1,𝑗
ℎ2
+
𝐸𝑐
𝑅𝑒
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
𝑛−1
𝑈𝑖+1,𝑗 − 𝑈𝑖−1,𝑗
2ℎ
2
for 𝑖 = 1,2, … 𝑝 and 𝑗 = 1,2, … 𝑞
The initial and boundary condition for this specific fluid flow is given below.
Initial condition at 𝑡 = 0.
𝑈𝑖,0 = 0, 𝑇𝑖,0 = 0
𝜕𝑈
𝜕𝑌 𝑖,0
= 0 and
𝜕 𝑇
𝜕𝑌 𝑖,𝑗
= 0 for 𝑖 = 1,2, … 𝑝

33. Boundary condition at 𝑡 > 0
𝑈0,𝑗= 1, 𝑇𝑖,0 = 1 where 𝑌 = 0
𝑈 𝑝+1,𝑗= 0, 𝑇𝑝+1,𝑗 = 0 where 𝑌 → ∞ 𝑗 = 1,2, … 𝑞
Here the subscripts 𝑖 designate the grid points with 𝑋 coordinate and the subscripts 𝑗 for time steps
at 𝑌 direction
we obtain the known value of 𝑈 at 𝑡 = 0. The updated velocity profile 𝑈 and temperature profile
𝑇 are obtained at all internal node points by successive application of the equations for any time step
𝑘.
The whole process will be done again and again and used time step will be sufficiently small. Finally
the desired value of 𝑈 and 𝑇 are obtained for the governing equations

34. Result and its physical explanation
The main aim of this study has been discussed in this part. After conducting numerical
computation we have reached at the situation where we describe the effect of dimensionless
numbers and parameters in the dimensionless equation.
Some standard values of parameters and dimensionless numbers are chosen for its physical
importance. Few parameters are chosen arbitrary. In this case graphs have been drawn for
time 𝑡 = 0. Skin friction coefficient 𝐶𝑓 and Nusselt number 𝑁 𝑢 are presented in tables (for
each parameter mentioned above) as well as shown graphically for some significant
parameters.

35. In case of free convection the
changes of velocity is different for
Newtonian and non-Newtonian
fluid. For 𝑛 = 1 the highest peak of
velocity profile is almost 3 but for
𝑛 = 0.5 the highest velocity profile
is significantly more than 3 in
vertical scale.
For dilatant fluid the velocity is not
increasing like pseudo plastic or
Newtonian fluid. The dimensionless
number 𝐺𝑟 exist in the momentum
equation, so it has great effect on
velocity profile 𝑈 . The increment
rate of velocity profile for pseudo
plastic fluid is maximum for the
increase of Grashof number
𝐺𝑟 between three types of fluid

36. From the Figure 4.4,
temperature profile 𝑇
doesn’t increase with the
increment of Grashof
number 𝐺𝑟. For each type of
fluid the temperature
profile is almost same but it
slowly reduce to zero at
different point of the
surface. The rate of
reduction is little different
for these three types of
fluid such as the difference
between pseudo plastic and
dilatant fluid is 0.10%.

37. Velocity profile decreases with the
increase of magnetic parameter 𝑀 in
each type of fluid. It has been
obtained that, introducing magnetic
field in the flow causes higher
restriction to the fluid. For
Newtonian and pseudo plastic fluid
the velocity profile reduces in same
rate but in case of dilatant fluid rate
of reduction is little bit slower.

38. But in case of temperature
profile there is no
significant effect found.
From the Figure 4.5, it is
obvious that temperature
profile 𝑇 slowly reduces to
zero along the sheet with
the increment of 𝑌 steps.

39. It is distinct from the above Figure
4.7 that velocity profile 𝑈
decreases with the increase of
radiation parameter 𝑁 . However
the rate of decreasing is almost
same at the beginning of the
surface. It is also clear that the
momentum boundary layer
thickness reduces rapidly for
decreasing 𝑁. For small changes of
𝑁 such as 0.5 𝑡𝑜 1 the rate of
decreasing velocity is about
0.94% for 𝑛 = 1, 0.77% for 𝑛 =
1.5 and 0.25% for 𝑛 = 0.5
respectively. But if the changes of
radiation parameter is from 1 𝑡𝑜 2,
then reducing rate decreases
because of free convection.
However in case of free
convection we cannot think huge
radiation from the fluid.

40. According to the Figure 4.8 the
temperature profile decreases with
in positive increment of the
magnitude of radiation parameter
𝑁. Form above result, we observe
that the small increment cause a lot
to the thermal boundary layer.
Temperature profile reduces to zero
slowly. So from the above illustration
it is evident that numerical outcomes
support the physical experiment
(Newton’s law of cooling). The rate
of heat transfer is thus increased. So
we can use the effect of radiation to
control the velocity and temperature
of boundary layer.

41. The effect of Prandtl number 𝑃𝑟 on
velocity profile shows significant
changes for the increase of its
magnitude. From the figure it can
be told the rate of reduction is
little for three types of fluid for
large Prandtl number but for small
Prandtl number such as 𝑃𝑟 =
0.71 𝑜𝑟 1 the reduction rate is
larger. However for 𝑃𝑟 = 0.71 there
is a sharp rise in the velocity
boundary layers near stretching
surface. Physically 𝑃𝑟 = 0.71
corresponds to air and 𝑃𝑟 = 1, 7,10
is for different values of water. For
pseudo plastic fluid the velocity
profile rise higher and slowly
decreases to zero at the wall.

42. It is clear from Figure 4.10 that
for small Prandtl number the
temperature profile reduces to
zero at the wall with the
increase of 𝑌 steps. But when
Pr = 7,10 𝑜𝑟 20 then profiles
raises a little at the beginning
and then slowly reduces to zero
at the wall. The rate of
reduction is almost same for
these three types of fluid.
Moreover, cross flows have been
found for each case.

43. According to Eckert number it implies the
ratio of advective transport and heat
dissipation potential. It provides the
connection between a flow's kinetic
energy and the boundary layer enthalpy
difference, and is used to characterize
heat dissipation. 𝐸𝑐 number shows small
effect on velocity and temperature profile
for its small changes. For Newtonian fluid
the Figure 4.11, peaks of the velocity
profile are just over 2.25 of velocity scale
where in case pseudo plastic fluid it is
almost 2.5 but for dilatant fluid it is just
under 2.25 of velocity scale. All these
velocity profile increases at the beginning
and slowly fall to zero with the increment
of 𝑌 steps. The rate of increment of
velocity and temperature profile with the
increase of Eckert number 𝐸𝑐 are almost
same. For each type of fluid momentum
boundary layer thickness and thermal
boundary layer thickness have changed over
the increment of 𝐸𝑐.

44. From Figure 4.12, there are not
any significant change of
temperature profiles 𝑇 at the
beginning and at the end for
different values of 𝐸𝑐. But at the
mid-point of each profile it shows
its variation. Although the rate of
change is very small and profile
slowly rises for the increment of
𝐸𝑐.

45. 𝑛 𝑁 𝐶𝑓 𝑁 𝑢
0.5 0.5572589081 0.4183440638
0.5
1 0.5556170156 0.4457585392
2 0.5542651235 0.4472651266
3 0.5536617753 0.4480264272
0.5 1.097097235 0.4183428075
1 1.068160893 0.4472638734
1 2 1.048716957 0.4472638734
3 1.041236803 0.4480258343
0.5 1.633654363 0.4163322648
1 1.56877755 0.43756029
1.5 2 1.525523567 0.4472626674
3 1.50737908 0.4480246383
The variation of skin friction
coefficient 𝐶𝑓 and local
Nusselt number 𝑁 𝑢 . If
radiation increases, the velocity
decreases, so for each type of
fluid 𝐶𝑓 decreases. Other
fixed parameters are 𝑟 =
5, 𝑃𝑟 = 0.71, 𝑅𝑒 = 10, 𝑀 =
0.5, 𝐸𝑐 = 0.3 . The rate of
reduction of 𝐶𝑓 is different
for 𝑁 = 0.5,1,2,3 . But in case
of Nusselt number 𝑁 𝑢 the
variation is little and increased
with increasing value of 𝑁. 𝐶𝑓
is highest for dilatant fluid and
minimum for pseudo plastic
fluid. However the behavior of
Nusselt number 𝑁 𝑢 is same for
each type of fluid.

46. 𝑛 𝐸𝑐 𝐶𝑓 𝑁 𝑢
0 0.5556174461 0.4375853918
0.5
0.3 0.5555449049 0.4376441039
0.6 0.5554660597 0.437488349
1.0 0.555343968 0.4385414382
0 1.068160893 0.4375853918
0.3 1.066456534 0.4376441039
1 0.6 1.064808736 0.4374877560
1.0 1.062432411 0.4385408453
0 1.568775453 0.4375853918
0.3 1.564971397 0.4376428979
1.5 0.6 1.561205349 0.4374843541
1.0 1.55617659 0.4385354484
If the Eckert number 𝐸𝑐
increases then velocity
profile increases a little bit.
Effect is smaller because of
having no concentration
equation. Other fixed
parameters are 𝐺𝑟 = 5, 𝑃𝑟 =
0.71, 𝑅𝑒 = 10, 𝑁 = 1, 𝑃𝑟 =
0.71 . Eckert number shows
significant variation for
particular concentration
[28]. According to Table 4.5 ,
Skin friction coefficient his
maximum for dilatant fluid
𝑛 = 1.5 and minimum for
pseudo plastic ( 𝑛 =

47. Conclusion
An analysis has been carried out to study the combined effect of viscous dissipation, radiation and heat
generation on unsteady non-Newtonian fluid flow along a vertically stretched surface particularly for free
convection. In this thesis work, governing equation has been transformed into dimensionless non-linear partial
differential equations. Then an implicit finite difference technique is used to get results. In chapter 3, effect of
viscous dissipation has been discussed for unsteady non-Newtonian fluid. Later, in chapter 4 combined effect
has been studied. Finally, the effects of various dimensionless numbers (𝑆𝑢𝑐ℎ 𝑎𝑠 𝐺𝑟, 𝑃𝑟, 𝐸𝑐) as well as
parameters 𝑠𝑢𝑐ℎ 𝑎𝑠 𝑁, 𝑀 has been depicted broadly for different power law index. Moreover skin friction 𝐶𝑓
and Nusselt number 𝑁 𝑢 have been computed for both cases and shown in table. From the present study the
following observation and conclusion can be drawn.
 Significant influence of Grashof number 𝐺𝑟 on velocity profile has been found. But for combined effect, the
increment rate of 𝑈 is highest for pseudo plastic fluid. Temperature profiles are almost same for each type of
fluid. Skin friction coefficient 𝐶𝑓 has increased but Nusselt number 𝑁 𝑢 remain unchanged for the growth
of 𝐺𝑟. Thus buoyancy force reduces the thickness of velocity boundary layer.

48.  Velocity profile decreases as 𝑀 increases in all types of fluids which reduces faster for Newtonian fluid
compared to non-Newtonian fluids. For combined effect velocity grows maximum for 𝑛 = 0.5 .Temperature
falling rate is similar to each type of fluid. 𝐶𝑓 reduces with the increment of 𝑀 but 𝑁 𝑢 remain unchanged.
 The velocity and temperature profile decreases with the positive increment of Prandtl number𝑃𝑟. Variation
of reduction is little different for dilatant and pseudo plastic fluid because of different choice of 𝑃𝑟. In case
of temperature profile the magnitude of temperature grows significantly small at the beginning then falls to
zero at wall.𝐶𝑓 and 𝑁 𝑢 decreased significantly for the increase of 𝑃𝑟. In the entire flow field Prandtl number
shows its effect significantly. In this case cross flow has been found.
 As 𝐸𝑐 comes from dissipation term, so that it is important for controlling boundary layer. Both velocity and
temperature profile increases with the positive increment of 𝐸𝑐 numbers. 𝐶𝑓 𝑎𝑛𝑑 𝑁 𝑢 shows opposite
scenario. Where skin friction increases slowly and Nusselt number decreases, although the effect is very
small.
 In case of radiation parameter 𝑁, velocity and temperature profile have decreased. So thermal radiation
can be used to control the boundary layer. Skin friction coefficient 𝐶𝑓 has decreased slowly but Nusselt
number 𝑁 𝑢 effect shows diversity for the increase of 𝑁 in case of different power law index 𝑛.

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52. Thanks