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The group-theorytic approach is applied for solving the problem of the unsteady MHD mixed convective flow past on a moving curved surface. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved numerically using the shooting method. The effects of varying parameters governing the problem are studied. A comparison with previous work is presented.

Applications of group-theory in fluid mechanics and boundary layer flow have received much attention by many researchers as the concepts of group theory are extensively used in similarity and non-similarity related problems. Group-theory method provides a powerful tool to nonlinear differential models. The transformation group theory approach is applied to present an analysis of the similarity problem of MHD mixed convective flow past on a moving curved surface with suction. The natural flow originates from body force variations in fluids, whereas the forced convection is generally introduced by moving a body through a quiescent fluid or by forcing a fluid past a stationary body. This flow regime is concerned with circumstances where in both the natural and forced mechanisms of the flow must be considered simultaneously. The laminar boundary layer flow due to such combined forced and natural convection i.e. mixed convection has received considerable attention for steady and unsteady situations in evaluating flow parameters for technical purposes. The problem of mixed convective boundary layer flow gained different dimensions in the manufacturing processes in industry. There has been great interest in the study of Magnetic Hydro-Dynamic (MHD) flow due to the effect of magnetic fields on the boundary layer flow control and on the performance of many systems using electrically conducting fields. This type of flow has attracted the interest of many researchers due to its applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extraction etc.

Sparrow, Eichorn and Gregg [

The mathematical technique used in the present analysis is two-parameter group transformation that leads to a similarity representation of the problem. Morgan [

In this work, the effect of MHD mixed convective flow past on a moving curved surface has been investigated. Problems are solved analytically using group methods and then numerically by Runge-Kutta shooting method. Under the application of two-parameter group, the governing partial differential equations are reduced to system of ordinary differential equations with the appropriate boundary conditions and then numerically using the sixth order Runge-Kutta shooting method known as Runge-Kutta-Butcher initial value solver of Butcher [

Attention has been taken on the evaluation of the velocity profiles as well as temperature profiles for selected values of parameters consisting ,magnetic parameter M, Prandtl number Pr, buoyancy parameter l_{1} and suction parameter E_{w}. The numerical results of the velocity profiles as well as temperature profiles are displayed graphically for different values of magnetic parameter M, Prandtl number Pr, buoyancy parameter l_{1} and suction parameter E_{w}. The post processing software TECPLOT has been used to display the numerical results. A comparison with previous work is presented.

We consider the flow direction along the x-axis and η-axis and be defined in the surface over which the boundary layer is flowing. For simplicity

The governing boundary layer equations of the flow field in general orthogonal curvilinear co-ordinates are:

Continuity equation

Momentum equations

Energy equation

With initial and boundary conditions

where

the thermal diffusivity.

From the continuity Equation (1), there exists two stream functions

Applying h_{1} = 1 and h_{2} = ξ in Equations (2)-(4) we have

With initial and boundary conditions

The problem is solved by applying a two parameter group transformation to the partial differential Equations (6)-(8). This transformation reduces the four independent variables

Define the procedure is initiated with the group G, a class of transformation of two parameters

S stands for t, x, h, z, Y, j, U_{e}, V_{e}, ∆T and q, C^{S} and K^{S} are real-valued and at least differentiable in their real arguments

The transformation of the dependent variables and their partial derivatives are obtained from G via chain-rule operations

where S stands for y, f, q. i.e.

Equation (6) is said to be invariantly transformed whenever

Substitution from Equations (10) & (11) into Equation (12) yields

where

In a similar manner, the invariant transform of (7) gives

where

Similarly equation (8) is invariantly transformed giving

where

The initial and boundary conditions being also invariant implies that k^{t} = 0, k^{z} = 0.

The invariant transformation of (6)-(8), the initial condition and the boundary conditions summarize in a group G of the form

Our aim is to make use of group methods to represent the problem in the form of an ordinary differential equation (Similarity representation) in a single independent variable (Similarity variable). Then we have to proceed in our analysis to obtain a complete set of absolute invariants.

The complete set of absolute invariants is:

a)

b) _{e}, V_{e},q, ∆T.

A function

Independent Variables as Absolute Invariants

The absolute invariant

A successive elimination of

where

Invoking the group given in Equation (13) and the definition of the α’s and β’s we get

From Equation (16b) we obtain

ζ; Solving Equations (16a) and (16b) implies

Dependent Variables as Absolute Invariants

Similarly the absolute invariants of the dependent variables; Y, j, U_{e}, V_{e}, q are obtained from the group trans- formation (13),

A function _{ }

The solution of Equation (17) gives

In similar manner, we get

Since U_{e}(γ) and V_{e}(γ) are independent of ζ, whereas γ depends on ζ, it follows that U_{e}(γ) and V_{e}(γ) must be equal to constant, say one. Without loss of generality, the χ’s in Equations (18)-(19) are selected to the identity functions. So we can write

Again ∆T is independent of ζ, whereas γ depends on ζ, it follows that G(γ) is equal to a constant, say G_{0}. Without loss of generality G_{0} is equated to one. So

The system of ordinary differential Equations (6)-(8) eventually reduces to

where

and c’s are constant

Let in (28) c_{9}/c_{10} = 1; then it follows that

By considering c_{5} may be taken to be unity, we get from (28) the following

Now if we consider c_{8} = 1, (28) implies

implies

Evaluation of c’s implies

Equations (25)-(27) gives u-momentum equation

v-momentum equation

Energy equation

with related boundary conditions:

Equations (29)-(31) together with the boundary condition (32) are solved numerically using the sixth order Runge-Kutta shooting method known as Runge-Kutta-Butcher initial value solver of Butcher (1974) together with the Nachtsheim-Swigert iteration scheme described by Nachtsheim and Swigert (1965).

The numerical results of the velocity profiles as well as temperature profiles for different values of magnetic parameter M, Prandtl number Pr, buoyancy parameter l_{1} and suction parameter E_{w} will be discussed and display graphically.

The main objective of the present study is to analyze the effect of MHD mixed-convection flow on a moving curved surface. _{1} = −13.29, λ_{2} = −0.76 and E_{w} = 1.53 with several values of M. Since magnetic parameter is inversely proportional with Reynold number, Re so increasing values of the magnetic parameter M decreases the flow rate in the velocity boundary layer thickness. It has been seen from

grate slowly for magnetic parameter M = 0.9, 0.09, 0.01. It is observed from

creases with the increasing values of suction parameter E_{w}. It has been seen from _{w} increases, the v-velocity profiles increases up to the position of g = 1.50, 1.50, 1.35, 1.30 and from that positions of g velocity profiles decreases with the increasing values of suction parameter E_{w}. From _{1} = −13.29, −12.80, −12, −11, u-velocity profiles decreases up to the position of g = 2.10, 1.85, 1.65, 1.45 and from those positions of g, u-velocities integrate rapidly and increases for increasing values of g. In _{1}. The maximum values of the v-velocity are found to be

1.9628, 1.9548, 1.9453 and 1.9334 for λ_{1} = −13.29, −12.80, −12, −11 respectively. It is noted that the v-velocity decreases by approximately1.5% as λ_{1} increases from −13.29 to −11. _{1} = −13.29, λ_{2} = −0.76 and E_{w} = 1.53 the temperature profiles decreases with the increasing values of magnetic parameter M. _{1}.

Pr | ||||||
---|---|---|---|---|---|---|

Maleque kh.A. [ | Present results | Maleque Kh.A. [ | Present results | Maleque Kh.A. [ | Present results | |

0.7 | 1.33198 | 1.35000 | 1.20973 | 1.21000 | −0.88811 | −0.90000 |

1.0 | 1.32807 | 1.31200 | 1.20539 | 1.20500 | −1.01137 | −1.08000 |

5.0 | 1.31467 | 1.30500 | 1.19094 | 1.19050 | −1.79200 | −1.78000 |

7.0 | 1.31229 | 1.30100 | 1.18842 | 1.19000 | −2.01368 | −2.00000 |

10.0 | 1.30993 | 1.29500 | 1.18594 | 1.17250 | −2.27706 | −2.10000 |

A comparison of the present numerical results of_{w} are igno red with different values of Prandtl number Pr. It is evidently seen from

Dipika Rani Dhar,Mohammad Abdul Alim,Laek Sazzad Andallah, (2016) Group Method Analysis of MHD Mixed Convective Flow Past on a Moving Curved Surface with Suction. American Journal of Computational Mathematics,06,74-87. doi: 10.4236/ajcm.2016.62009