Impact of Inclined Magnetic Field and Activation Energy on Stagnation Point Flow of Nano-fluid Over a Stretching Surface

Impact of Inclined Magnetic Field and Activation Energy (Part 3)

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Impact of Inclined Magnetic Field and Activation Energy (Part 1)

Impact of Inclined Magnetic Field and Activation Energy (Part 2)

Nanofluid flow is caused by shrinking and stretching surfaces

Introduction

Impact of Inclined: Nanofluid flow caused by shrinking and stretching surfaces will be the main focus of this post. Two-dimensional viscous and incompressible boundary layer equations which rule the physical phenomenon will be used. By examining mechanism of nanofluid, these two dimensional viscous and incompressible fluid will be placed in cartesian coordinate system. An appropriate similarity transformation transforms the complex coupled system of PDEs into a nondimensional form of ODEs. The bvp4c MATLAB function will be used to solve coupled ODEs numerically. The impact of physical factors such as the Reynold number will be investigated, and the findings will be presented.

Problem Development

An incompressible viscous nanofluid flows across a stretching/shrinking sheet in two dimensions (2D). The non-linear stretching sheet causes the flow. We use the Cartesian coordinate system with the x-axis following the stretching/shrinking sheet and the y-axis perpendicular to it, subject to a constant magnetic field. The sheet stretches along the x-axis with velocity uw(x) = axm, but the temperature Tw(x) varies.

The governing equations are as follows:

(1)   \begin{equation*}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\end{equation*}

The boundary conditions are as follow:

At y = 0

(2)   \begin{equation*}u = u_w(x) + \mu \frac{\partial u}{\partial y} M (x), \end{equation*}

(3)   \begin{equation*}T = T_w(x) + \mu \frac{\partial T}{\partial y} N(x),\end{equation*}

(4)   \begin{equation*}C = C_w(x) + \mu \frac{\partial C}{\partial y} N_1(x),\end{equation*}

    \[v = v_w(x)\]

At y → ∞
u → ue(x),                C → C∞              T → T∞

Where M(x) represents velocity, N(x) represents temperature, N1(x) indicates concentration slip parameters, µ represents dynamic viscosity, and C∞ represents ambient fluid concentration.

Impact of Inclined Magnetic Field and Activation Energy (Part 1)

Graphical Illustrations

λAM¯f”(0)
5.05.01.0-0.75797
 5.1 -0.74386
 5.2 -0.73028
 5.3 -0.71717
5.05.01.4-0.75884
  1.6-0.75925
  1.8-0.75965
Table 1: Analysis of reduced skin friction coefficient f”(0) for λ = 5.00.

The behaviour of the decreased skin friction coefficient f”(0) on different parameters is shown in Table 1.

Reduced skin friction coefficient f”(0) has a smaller effect on higher values of the magnetic parameter, but a larger effect on higher values of the velocity slip parameter A.

λAM¯f”(0)
-5.05.01.01.13498
 5.1 1.11393
 5.2 1.09365
 5.3 1.07409
-5.05.01.41.13648
  1.61.13718
  1.81.13786
Table 2: Analysis of reduced skin friction coefficient f”(0) for λ=-5.00

Table 2 shows the impact of a lower local Nusselt number θ‘(0) on a variety of factors.

NbNtLeM¯θ‘(0)
1.01.01.01.00.80029
1.2   0.76751
1.4   0.73474
1.6   0.70209
1.01.21.01.00.78574
 1.4  0.77112
 1.6  0.75645
1.01.01.41.00.78131
  1.6 0.77558
  1.8 0.77149
1.01.01.01.40.80011
   1.60.80003
   1.80.79995
Table 3: Analysis of reduced Nusselt number θ‘(0) for λ = 5.00

Table 3 depicts the behaviour of the reduced sherwood number φ‘(0) as a function of different factors.

NbNtLeM¯θ‘(0)
1.01.01.01.00.78596
1.2   0.75323
1.4   0.72055
1.6   0.68802
1.01.21.01.00.77135
 1.4  0.75669
 1.6  0.74198
1.01.01.41.00.76687
  1.6 0.76106
  1.8 0.75691
1.01.01.01.40.78629
   1.60.78644
   1.80.78658
Table 4: Analysis of reduced Nusselt number θ’(0) for λ = −5.00.
NbNtLeM¯φ‘(0)
1.01.01.01.01.19388
1.2   1.30876
1.4   1.39083
1.6   1.45230
1.01.21.01.01.10072
 1.4  1.01284
 1.6  0.93024
1.01.01.41.01.71923
  1.6 1.95452
  1.8 2.17482
1.01.01.01.41.19335
   1.61.19314
   1.81.19294
Table 5: Analysis of reduced Sherwood number φ‘(0) for λ = 5.0
NbNtLeM¯φ‘(0)
1.01.01.01.01.16142
1.2   1.27567
1.4   1.35716
1.6   1.41816
1.01.21.01.01.06985
 1.4  0.98345
 1.6  0.90221
1.01.01.41.01.67996
  1.6 1.91307
  1.8 2.13181
1.01.01.01.41.16220
   1.61.16266
   1.81.16292
Table 6: Analysis of reduced Sherwood number φ‘(0) for λ = −5.0

Graphical illustration of a velocity profile for stretching parameter λ = 5.00

Impact of Inclined Magnetic Field and Activation Energy (Part 3)

Figure 1

Graphical illustration of velocity profile for shrinking parameter λ = −5.0.

Impact of Inclined Magnetic Field and Activation Energy (Part 3)

Figure 2

Graphical illustration of temperature profile for stretching parameter λ = 5.00.

Graphical illustration of temperature profile for stretching parameter λ = 5.00.

Figure 3

Graphical illustration of temperature profile for stretching parameter λ = 5.00.

Figure 4

Conclusions

The effect of an inclined magnetic field and activation energy on nanofluid stagnation point flow across a stretched surface is investigated in this thesis. The bvp4c function in MATLAB is used to find solutions for the coupled nonlinear ODE’s. The following closing observations are based on the major effects of these factors on the current investigation.

  • Large values of magnetic parameter H show decreasing behaviour for velocity profile in case of stretching whereas it shows increasing behaviour in case of shrinking.
  • Higher values of γ imply a stronger velocity profile in the case of shrinking whereas it implies a weaker profile in the case of stretching.
  • In both stretching and shrinking cases, a larger Prandtl number Pr results in a weaker temperature and concentration profile.
  • Larger values of activation energy E results weaker concentration profile in both cases.
  • Increasing the values of Schmidt number Sc, σ, and Brownian motion Nb produces weaker concentration profile for both cases.
  • An increment in the Eckard number Ec improves the temperature profile for both cases.

Impact of Inclined Magnetic Field and Activation Energy (Part 1)

Impact of Inclined Magnetic Field and Activation Energy (Part 1)

Also Impact of Inclined Magnetic Field and Activation Energy (Part 1)

Impact of Inclined Magnetic Field and Activation Energy on Stagnation Point Flow of Nano-fluid Over a Stretching Surface

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