This document discusses computational analyses of fluid flow with additional effects. Chapter 1 provides definitions of fluid mechanics concepts. Chapter 2 analyzes second grade fluid flow over Riga surfaces with chemical reaction and MHD effects. The mathematical models and assumptions are presented. Computational results are shown in tables and figures exploring the effects of varying parameters. Chapter 3 analyzes third grade fluid flow over a stretching surface with chemical reaction and MHD effects, following a similar structure.

2. Supervised by
Dr. Nadeem Abbas
Presented by
Anonymous
Computational Analysis of Second Grade Fluid Flow over
Riga Surfaces under Chemical Reaction and MHD Effects

3.  Chapter 1: Fundamental definitions and concepts
 Chapter 2: Computational Analysis of Second
Grade Fluid Flow Over Riga Surfaces under
Chemical Reaction and MHD Effects
 Chapter 3:Computational Analysis of Third Grade
Fluid Flow Over Stretching Surface Under
Chemical Reaction And MHD Effects

4. Chapter 1
Basic definitions

5. Fluid can be define as “Any gas or liquid or
material that cannot maintain a tangential or
searing force when at rest and that passes a
continuous change in shape when subject to such a
stress”. A material which cannot oppose the
applied shear force is called fluid.
 Gas or liquid is very simplest example of fluid.

6. It is a subdivision of physics that hand out with the
behavior of fluids at rest. Fluid mechanics has many
implementations in a vast range of chemical
engineering and biology. It has further three sub
classes.
 Fluid statistics
 Fluid dynamics
 Fluid kinematics

7.  Fluid statistics
A category of fluid that deals with fluids at rest are called fluid
statistics. It is also called hydro statistics. It also studies the
condition of submerged body and the condition of equilibrium.
 Fluid dynamics
In engineering and physics, it is a sub dichotomy of fluid
mechanics that discuss about the flow of fluid gases and liquids
 Fluid kinematics
It is a type of fluid which concerned with such types of flow of
fluid that cannot take into account the forces that cause the
motion. For example velocity, speeds acceleration and
displacement etc.

8. A fluid in which a linear relationship exists at every
point between deformation rate and viscous stress is
called Newtonian fluids.
Examples:
 Water is a Newtonian fluid because its viscosity is
not affected by shear rate.
 Air is also a Newtonian fluid.

9. Fluids which disobey the Newtonian law of viscosity
are called non Newtonian fluids i.e.
Constant viscosity is not depending on stress.
 Example
Ketchup, Yogurt, Slurries, Gels, Polymer solutions

10. “The mass of any system can neither be demolish
nor be generated it can just be interchanged by
one system to another"
Mathematically, it can be represented as:
𝜕ρ
𝜕t
+ 𝛻. ρV = 0.
Where,
ρ = density, V = velocity of fluid.
 If flow is incompressible then ρ = 0 .
So, it is,
𝛻. V = 0.

11. The body forces operating on a surface and fluid forces
is equal to the rate change in volume linear momentum.
The differential form is given as:
ρ
dV
dt
= ∆. τ + ρb.
Where,
t is time, V is velocity,
ρ is density and b is body force.
τ = Cauchy tensor = −pI + μA1.

12. The energy equation can be expressed as:
ρCp
dT
dt
= τ. L + k𝛻2T − divqr.
Where,
ρ = fluid density,
L = gradient of velocity,
k = thermal conductivity of fluid,
qr = Radiative heat transmissin.

13. The concentration equation for nanoparticles is mathematically
represented as
𝜕
𝜕t
+ v. 𝛻 c = 𝛻
DT
T∞
𝛻T + DB(𝛻v) .
where,
DT = coefficient of diffusivity,
C = nanopaticles concentration,
DB = Brownian motion,
T∞ = Ambient temprature,

14. Computational Analysis of Second Grade Fluid Flow Over
Riga Surfaces under Chemical Reaction and MHD Effects
 Assumptions:
 Second Grade Fluid over Riga Surface
 MHD Effect

15. Mathematical Formulation
∂u
∂x
+
∂v
∂y
= 0,
(3.1)
u
∂u
∂x
+ v
∂u
∂y
= U∞
du∞
dx
+ v
∂2u
∂y2
+
α1
ρ
u
∂3u
∂x ∂y2
+
∂u
` ∂x
∂2u
∂y2
+ 3
∂u
∂y
∂2u
∂x ∂y
+ ν
∂3u
∂y3
+
2α2
ρ
∂u
∂y
∂2u
∂x ∂y
+ 6
β3
ρ
∂u
∂y
2 ∂2u
∂y2 −
σB2
ρ
(U − u),
(3.2)
u
∂T
∂x
+ v
∂T
∂y
= k
∂2
T
∂y2
+
Dm
cs
KT
∂2
a
∂y2
,
(3.3)
u
∂a
∂x
+ ν
∂a
∂y
= D
∂2
a
∂y2
+
Dm
Tm
KT
∂2
T
∂y2
,
(3.4)

16. Where, Dm is mass diffusivity,Tm is fluid temperature across boundary layer, cs
is concentration succeptibility, KT is thermal diffusivity rate and Cp is specific
The modifications are explained as,
u → U∞ , T → T∞ → a∞ as y → ∞.
heat flux. The boundary conditions for equations (3.1) to (3.4) are given below
ν = 0, u = Uw , kc
∂T
∂y
= −Qk0ae
−E
RT , D
∂a
∂y
= k0ae−
E
RT , as y → 0,
(3.5)
ѱ =
𝑈∞ 𝜈
𝑙
1
2
𝑥𝑓(𝜂), 𝑇 − 𝑇∞ =
𝑅𝑇∞
2
𝐸
𝜃, 𝑎 − 𝑎∞ = 𝑎∞ 𝛷 ,
𝜂 = 𝑦
𝑈∞
𝜈𝑙
1
2
(3.6)

17. This gives:
𝑓′ 2
− 𝑓𝑓′′
= 1 + 𝑓′′′
+ 𝜆1 2𝑓′
𝑓′′′
+ 3𝑓′′ 2
+ 𝑓𝑓𝑖𝑣
+ 2(𝑓′′ )2
+ 𝜆2𝑓′′ 2
𝑓′′′
− 𝑀(1 − 𝑓′ )
(3.7)
1
𝑃𝑟
𝜃′′
+ 𝑓𝜃′
+ 𝛼1 𝛷′′
= 0
(3.8)
1
𝑆𝑐
𝛷′′
+ 𝑓𝛷′
+ 𝛼2𝜃′′
= 0
(3.9)
By putting values we get the following results
From Equ.(3.7) we get the value of highest derivative
By applying the numerical technique, nonlinear differentia equation may be converted
in the initial value problem as:
f (iv )
=
1
−λ1f
1 + f ′′′
− f ′ 2
+ ff ′′
+ λ1 2f ′
f ′′′
+ 5f ′′′ 2
+ λ2f ′′ 2
f ′′′
−
M(1 − f′)
(3.10)

18. From Equ. (3.9)
Φ′′
= −Sc(fΦ′
− α2θ′′)
Put it in (3.8) we get the final results for the value of θ′′
1
Pr
θ′′
+ fθ′
+ α1 −Sc(fΦ′
− α2θ′′ ) = 0
1
Pr
θ′′
+ fθ′
− α1Sc(fΦ′ ) + α1α2θ′′
= 0
fθ′
− α1Sc(fΦ′ ) = −
1
Pr
+ α1α2 θ′
This implies that
θ′′
=
1
(−
1
Pr
+α1α2)
(fθ′
− α1Sc(fΦ′) (3.12)
(3.11)
Φ′′
= −Sc(fΦ′
− α2θ′′ ) (3.13)

19. Table 3.1: Numerical values for Prantle number and Sherwood number
𝐏𝐫 𝐋 𝐍 𝐒𝐜 𝐌 𝛂𝟏 𝛂𝟐 𝐀 𝐟′′(𝟎) −𝛉′(𝟎) -𝛟′(𝟎)
1.0 0.3 1.0 0.3 0.3 0.6 0.2 2.0 0.0 0.33787 0.4410
1.5 0.3 1.0 0.3 0.3 0.6 0.2 2.0 0.0 0.35298 0.4493
2.0 - - - - - - - - 0.36262 0.4545
- 0.1 1.0 0.3 0.3 0.6 0.2 2.0 - 0.36025 0.4507
- 0.2 - - - - - - - 0.36158 0.4527
- 0.3 - - - - - - - 0.36262 0.4545
- - 0.6 0.3 0.3 0.6 0.2 2.0 - 0.36312 0.4555
- - 0.9 - - - - - - 0.36178 0.4530
- - 1.0 - - - - - - 0.36262 0.4545
- - - 0.3 0.3 0.6 0.2 2.0 - 0.36262 0.4545
-- -- -- 0.4 -- -- -- -- -- 0.36262 0.5149
-- -- -- 0.5 -- -- -- -- -- 0.36262 0.5708
-- -- -- -- 0.2 0.6 0.2 2.0 -- 0.3628 0.5713
-- -- -- -- 0.6 -- -- -- -- 0.36206 0.5693
-- -- -- -- 1.0 -- -- -- -- 0.36138 0.5674
-- -- -- -- -- 0.4 0.2 2.0 -- 0.45593 0.6451
-- -- -- -- -- 0.5 -- -- -- 0.40423 0.6029
-- -- -- -- -- 0.6 -- -- -- 0.36138 0.5674
-- -- -- -- -- -- 0.5 2.0 -- 0.47048 0.5170
-- -- -- -- -- -- 0.8 -- -- 0.43501 0.5335
-- -- -- -- -- -- 1.0 -- - - 0.41689 0.5419
-= -- -- -- -- -- -- 1.0 -
0.4932
0.4887 0.6706
-- -- -- -- -- -- -- 2.0 -- 0.58757 0.8424
-- -- -- -- -- -- -- 3.0 -- 0.69832 1.0251

20. Fig. (3.1): Effects of λ1
onf′(η).
Fig. (3.2): Effects of N
on f′(η).

21. Fig. (3.3): Effects of N
onf′(η).
Fig. (3.4): Effects of Pr on
θ η .

22. Fig .(3.5): Effects of α1 on
θ(η).
Fig.(3.6): Effects of α2 on
θ η .

23. Fig. (3.7): Effects of Sc on
Φ(η).
Fig. (3.8): Effects of α2on
Φ(η).

24. Computational Analysis of Third Grade Fluid Flow Over
Stretching Surface Under Chemical Reaction And MHD
Effects
 Assumption:
 Third Grade Fluid over Stretching Surface
 MHD Effect

25. Mathematical Formulation
𝜕u
𝜕x
+
𝜕v
𝜕y
=0,
(4.1)
u
𝜕u
𝜕x
+ v
𝜕u
𝜕y
= u∞ du∞
dx
+ v 𝜕
2u
𝜕
y2
+
α1
ρ
u 𝜕
3u
𝜕x𝜕
y2
+
𝜕u
`𝜕x
𝜕
2u
𝜕
y2
+
(4.2)

26. U
𝜕T
𝜕x
+ v
𝜕T
𝜕y
=
Kf
ρCp
𝜕2T
𝜕y2 +
DT
T∞
𝜕2C
𝜕y2 +
μ
ρCp
U2
+ R T − TW ,
(4.3)
u
𝜕C
𝜕x
+ v
𝜕C
𝜕y
= Dm
𝜕2
c
𝜕y2
+
DT
T∞
(
𝜕2
T
𝜕y2
)
(4.4)
Subject to the boundary conditions,
ν = 0, u = Uw, kc
𝜕T
𝜕y
= −Qk0ae
−E
RT ,
D
𝜕a
𝜕y
= k0ae−
E
RT on y = 0
u → U∞, T → T∞ a → a∞ as y → ∞
(4.5)
Where,
U∞ =
U0x
l
and Uw = λU∞
ѱ =
U∞ν
l
1
2
xf η , T − T∞ =
RT∞
2
E
θ, a − a∞ = a∞Φ ,
η = y
U∞
νl
1
2
(4.6)

27. This gives:
f′2
− ff′′
= 1 + f′′′
+ λ1 2f′
f′′′
+ 3f′′2
+ ffiv
+ 2 f′′ 2
+ λ2f′′2
f′′′ −
M 1 − f′
,
(4.7)
1
Pr
θ′′
+ fθ′
+ DuΦ′′
+ Ec f′′ 2
+ Qθ = 0
(4.8)
Φ′′
+ Sc′
fΦ′
+ Srθ′′
= 0 (4.9)
Where,
Sc = Schmidt number =
ν
Dm
, Ec =
μ2
CpUT
= Eckert number
From Equ.(4.9)
Φ′′ = −Sc(fΦ′
+ Srθ′′
),
(4.10)

28. Put Equ. (4.10) in Equ.(4.11)
1
Pr
θ′′
+ fθ′
+ Du(−Sc fΦ′
+ Srθ′′
) + Ec f′′ 2
+ Qθ = 0
θ′′
= −pr fθ′
− DuSc fΦ′
+ Srθ′′
+ Ec f′′ 2
+ Qθ ,
θ′′
− DuScPrθ′′
= −Pr fθ′
− DuScfΦ′
+ Ec f′′ 2
+ Qθ
θ′′
1 − DuScPr = − Pr fθ′
− DuScfΦ′
+ Ec f′′ 2
+ Qθ
θ′′
= −
Pr
1 − DuScPr
fθ′
− DuScfΦ′
+ Ec(f′′)2
+ Qθ
(4.11)
From Equ. (4.7)
f(iv)
=
1
−λ1f
1 + f′′′
− f′2
+ ff′′
+ λ1 2f′
f′′′
+ 5f′′′2
+ λ2f′′2
f′′′
− M 1 − f′
(4.12)

29. Table 3.1: Numerical values for Prantle number, Sherwood number, Eckert number,
Dufour number
𝐏𝐫 L N 𝐒𝐜 M Q 𝐒𝐫 A 𝐃𝐮 𝐄𝐜 𝐒𝐑 f’’(0
)
−𝛉′(𝟎) −𝛟′(𝟎)
1.0 0.
3
2.0 0.3 0.8 1.0 1.0 0.2 0.5 0.4 1.0 0.0 -1.1496 -
0.0083618
1.5 ---
-
---- ---- ----- ----
-
---- ---- ---- ---- ---- 0.0 -2.0476 -0.27583
2.0 ---
--
---- ---- ---- ----
-
---- ---- ---- ----- ----- 0.0 -3.5011 -0.70438
2.5 ---
--
----
-
----
-
---- ----
-
----
-
----- ----
-
----- ------ 0.0 -5.8301 -1.3912
---
-
0.
2
2.0 0.3 0.8 1.0 1.0 0.2 0.5 0.4 1.0 0.0 -1.1529 -
0.0099154
---
-
0.
4
---- ---- ---- ---- ---- ---- ----
-
---- ---- 0.0 -7.2835 -1.816
---
-
0.
6
---- ---- ---- ---- ---- ---- ---- ---- ---- 0.0 -1.15 -
0.0084594
---
-
0.
8
---- ---- ---- ---- ---- ---- ---- ---- ---- 0.0 -1.1496 -
0.0083618
---
-
---
-
0.5 0.3 0.8 1.0 1.0 0.2 0.5 0.4 1.0 0.0 -1.0895 0.013145
---
-
---
-
1.0 ---- ---- ---- ---- ---- ---- ---- ---- 0.0 -1.104 0.0078877
---
-
---
--
1.5 ---- ---- ---- ---- ---- ---- ---- ---- 0.0 -1.1232 0.0007286
5
---
-
---
-
2.0 ---- ---- ---- ---- ---- ---- ---- ---- 0.0 -1.1496 -
0.0083618
---
-
---
-
---- 0.3 0.8 1.0 1.0 0.2 0.5 0.4 1.0 0.0 -1.1496 -
0.0083618
--- --- ---- 0.5 ---- ---- ---- ---- ---- ---- ---- 1.322 -1.4568 -0.30914

30. Fig. (4.1): Effects of λ1
onf′(η).
Fig. (4.2): Effects of N
onf′
η

31. Fig. (4.3): Effects of M
on f′(η).
Fig. (4.4): Effects of Du on
θ η .

32. Fig (4.6): Effects of Pr on
θ(η).
Fig (4.5): Effects of Ec on
θ(η).

33.  The rate of heat transfer improves as the Dufour number
increase.
 When the Soret number rises, the rate of heat transfer
increases, but the concentration profile decreases.
 By expanding the value of dimensionless parameters
velocities shows reducing behavior while the temperature
profile and the concentration profile shows expanding
behavior.
 By expanding the value of Prantle number
Nusselt number θ′ 0 and the Sherwood number Φ′(0)
will also expand.