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Unsteady mixed convective boundary layer flow of viscous incompressible fluid along isothermal horizontal plate is analyzed through Similarity Solutions. The governing partial differential equations are transformed into ordinary differential equations using the similarity transformation and solved numerically along with shooting technique. The flow field for the fluid velocity, temperature and concentration at the plate surface are significantly influenced by the governing parameters such as unsteadiness parameter, permeability parameter, Prandtl number, Schmidt number and the other driving parameters. The results show that both fluid velocity and temperature decrease but no significant effect on concentration for the increasing values of Prandtl number. It is also exposed that velocity and concentration is higher at lower Schmidt number for low Prandtl fluid. Finally, the dependency of the Skin-friction co-efficient, Nusselt number and Sherwood number, which are of physical interest, are also illustrated in tabular form for the governing parameters.

The study of mixed convective boundary layer flows is generated much in many engineering processes and also in polymer industries for fiber-glass production and condensation process. This study is very practical and worthy enough to discuss perfectly. If the fluid flow is caused solely due to the density differences resulting from temperature gradients without assistance of external force like a pump or a fan is termed as natural or free convection flow. Such flow is generated due to the buoyancy effects which is observed in many heat transfer processes and is applied in many technological applications. A convection situation involving both free and forced convection is known as mixed convection. In mixed convection flows, the forced and free convection effects are of comparable magnitude. Many studies exist for the mixed convection boundary layer flow about vertical, inclined, horizontal and wedge surfaces immersed in a viscous fluid. Because of significant effects of buoyancy on the flow field pressure gradient changes through the depth of the layer. Former researchers verified the effect of buoyancy and they focused on flows over plates which are smoothly cooled or heated.

K. Stewartson [

Kumari et al. [

K. Vajravelu, K. V. Prasad and Chu [

Recently, Ali et al. [

So far as we know, no previous study has been made to analyze unsteady mixed convective boundary layer flow along isothermal horizontal plate. In the present study, an attempt is made to investigate unsteady mixed convective boundary layer flow of viscous incompressible fluid along isothermal horizontal plate.

The partial differential equations for governing the flow are transformed into ordinary differential equations using the similarity transformation and solved numerically. We have investigated the effect of several governing parameter such as Reynolds number, Prandtl number, unsteadiness parameter, permeability parameter and buoyancy parameter and other flow parameters like Skin-friction co-efficient, Nusselt number and Sherwood number on the flow field. Furthermore the effect of suction has been taken into consideration. The governing differential equations relevant to the problem have been solved by using the similarity technique. Secondly computed numerical results were exhibited and analyzed in detail for different values of the involving parameters in the similarity transformation.

We consider an unsteady two dimensional mixed convective boundary layer flow of a viscous incompressible fluid along horizontal isothermal plate. The plate surface generates inert specie which diffuses inside the boundary. The co-ordinate system is selected such that x-axis is in the horizontal direction. The unsteady fluid flows start at time t = 0. The potential flow is given by the velocity distribution

The boundary conditions are:

where x and y are the coordinates measured along the plate and normal to it. u, v are the velocity components along the x and y axes.

analysis of the problem we introduce the stream function _{ }

Introducing dimensionless similarity variables into the system of Equations (2)-(5) is reduced to system of ordinary differential equations;

and

It is observed that the Equation (1) is identically same. The Boundary conditions in Equation (6) are reduced to the corresponding boundary condition for velocity, temperature and concentration fields are as;

where,

Unsteadiness parameters

The above equations with boundary condition are solved numerically by using shooting method. The effect of various governing parameters on the fluid velocity, temperature, pressure, concentration is exhibited in Figures. Finally, the dependency of the Skin-friction co-efficient, Nusselt number and Sherwood number, which are of physical interest are also illustrated in tabular form and analyzed in details.

Skin-friction co-efficient at the plate is defined as

The rate of heat transfer in terms of Nusselt number at the plate is given by

The ratio of mass transfer in terms of the Sherwood number at the plate is given by

where, the wall shear stress

Thus the values of the local skin-friction co-efficient

The obtained systems of non-linear ordinary differential equations together with boundary conditions are transformed into simultaneous linear differential equations of first order and then solved numerically by applying the Shooting method namely Nachtsheim-Swigert (1965) iteration technique (guessing the missing value) along with Runge-Kutta integration scheme. Nachtsheim-Swigert iteration technique is used as the main tool for the numerical approach. The dimensionless similarity equations for momentum, temperature and concentration equations are solved numerically by this iteration technique. In shooting method, the missing (unspecified) initial condition at the initial point of the interval is assumed and the differential equation is also integrated numerically as an initial value problem to the terminal point. The accuracy of the assumed missing initial condition is then checked by comparing the calculated value of the dependent variable at the terminal point with its given value. If a difference exists, another value of the missing initial condition must be assumed thus the process is repeated. This process is continued until the agreement between the calculated and the given condition for the specified degree of accuracy. In the process of iteration, the Skin friction coefficient, the Nusselt number and the Sherwood number proportional to

Numerical computations are executed for several values of dimensionless parameters involved in the equations controlling the fluid dynamics in the flow regime. The values of the Prandtl number, Pr = 0.71, 01.00 and 07.00 correspond to air, electrolyte solution such as salt water and fresh water at 25˚C and 1atm pressure. The values of Schmidt number, Sc = 00.22, 00.62 and 02.62 represent diffusing chemical species of most common interest in air like hydrogen, water vapor, and Propel Benzene respectively at 25˚C and 1 atm pressure. The values of suction parameter, is considered to be 00.50. Here, local temperature Grashof number (Gr) corresponds to the cooling problem for the cooling of electronic components and nuclear reactors in engineering applications. Local concentration Grashof number (Gc) indicates the chemical species concentration in the free stream region. The velocity

The variations of dimensionless velocity profiles in the boundary layer are depicted in Figures 1-8. Generally it is observed in the fluid velocity is lowest at the plate surface then increases quickly to its free stream values far away from the plate surface and leads to 1 with the increases of _{1} and A_{2} on the dimensionless velocity. We see that the velocity profile decrease with the increase of A_{1} and A_{2}.

mixed convective flow, the effect of Schmidt number on the velocity profile is repre- sented in the

_{a}.

The behaviors of the dimensionless temperature profiles for several thermo physical parameters are illustrated in Figures 9-16. From these Figures it is observed that boundary conditions of the flow profile under consideration, the fluid temperature arrive at 1.0 then decreases exponentially to the leading edge and leads to zero far away from the plate with the with the increases of_{1} and A_{2} on the dimensionless temperature. It is observed that in both cases the temperature profile decreases with the increases of A_{1} and A_{2}.

values of K.

The effect of various values of the Schmidt number, Sc on the dimensionless temperature profile is displayed in

_{a}.

Figures 17-24 depict the variation on the chemical species concentration profiles for different governing parameters. From these Figures it is noted that the chemical species concentration is highest at the plate surface and decreases exponentially to zero far away from the plate satisfying the boundary conditions of the flow profile under consideration. _{1} and A_{2} on the dimensionless concentration. It is observed that in both cases the concentration profile decreases with the increases of A_{1} and A_{2}. The influence Gr and Gc on the concentration profile is illustrated in

that concentration increases monotonically for the increasing values of K.

The effects of the Schmidt number, Sc on the dimensionless concentration profile is represented in _{a}) is represented in _{a}.

Tables 1-8 illustrate numerical results to exhibit the effect of several parameters on the skin-friction co-efficient, Nusselt number and Sherwood number for the physical interest of the problem. _{1} & A_{2 }

A_{1} | _{ } | _{ } | _{ } |
---|---|---|---|

00.50 | 6.6684047 | 0.86623414 | 0.4638774 |

00.60 | 6.7096433 | 0.88887867 | 0.4703514 |

00.70 | 6.7503441 | 0.91178743 | 0.4768741 |

00.80 | 6.7835569 | 0.93478588 | 0.4843823 |

A_{2} | _{ } | _{ } | _{ } |
---|---|---|---|

00.50 | 6.6684047 | 0.86623414 | 0.4638774 |

00.60 | 6.5837669 | 0.88911971 | 0.4770399 |

00.70 | 6.5026486 | 0.91176817 | 0.4901558 |

00.80 | 6.4247905 | 0.93417005 | 0.5032540 |

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

00.71 | 6.6684047 | 0.86623414 | 0.4638774 |

01.00 | 6.4879364 | 1.03522239 | 0.4612288 |

05.00 | 5.7126763 | 2.56639249 | 0.4524284 |

07.00 | 5.5704654 | 3.18305329 | 0.4512952 |

Sc | |||
---|---|---|---|

00.22 | 6.6684047 | 0.86623414 | 0.4638774 |

00.62 | 5.9529296 | 0.82996263 | 0.7705062 |

02.62 | 5.0214731 | 0.79034800 | 1.6753632 |

05.00 | 4.6631048 | 0.77886475 | 2.4818305 |

Gr | |||
---|---|---|---|

00.50 | 4.6450273 | 0.81946908 | 0.4414440 |

02.00 | 5.3353043 | 0.83613835 | 0.4500845 |

05.00 | 6.6684047 | 0.86623414 | 0.4638774 |

07.00 | 7.5143810 | 0.88388685 | 0.4730272 |

Gc | |||
---|---|---|---|

00.50 | 3.8369068 | 0.77442428 | 0.4158638 |

02.00 | 4.8149205 | 0.80935822 | 0.4370507 |

05.00 | 6.6684047 | 0.86623414 | 0.4638774 |

07.00 | 7.8290644 | 0.89716045 | 0.4804288 |

K | |||
---|---|---|---|

00.60 | 6.6684047 | 0.86623414 | 0.4638774 |

01.00 | 6.4963956 | 0.85764149 | 0.4591429 |

03.00 | 6.0404735 | 0.82913084 | 0.4424230 |

07.00 | 5.9078417 | 0.80429643 | 0.4297748 |

while Skin-friction co-efficient increases for the increasing of A_{1} and decreases for the increasing of A_{2}. It is shown in

J_{a} | |||
---|---|---|---|

01.00 | 6.6684047 | 0.86623414 | 0.4638774 |

03.00 | 6.2020847 | 0.84179466 | 0.4489248 |

05.00 | 5.6632770 | 0.80856924 | 0.4321363 |

07.00 | 5.0722075 | 0.76434371 | 0.3974721 |

number and Sherwood number decrease with the increase of the Permeability parameter (K) and buoyancy parameter (J_{a}).

Mixed convective boundary layer flow of viscous incompressible fluid has been investigated for unsteady flow along horizontal isothermal plate through similarity solutions. It is concluded that the fluid velocity, temperature and concentration profiles decrease as the unsteady parameter increases. Both fluid velocity and temperature decrease as the Prandtl number (Pr) increases but no significant effect on concentration. It is noted that velocity and concentration profiles exhibit significant changes while temperature exhibits minor change for the variation of Schmidt number (Sc). It is also revealed that velocity and concentration are higher at lower Schmidt number for low Prandtl fluid. Both the skin-friction co-efficient and Nusselt number decrease whereas Sherwood number increases with the increases of the Prandtl number (Pr). Skin friction coefficient and Sherwood number decrease whereas Nusselt number increases with the increases of the Schmidt number (Sc). Moreover, Nusselt number and Sherwood number decrease but skin-friction co-efficient increases with the increase of buoyancy forces (Gr and Gc). Again, Nusselt number and Sherwood number increase moreover Skin friction coefficient decreases with the increase of the Permeability parameter (K) and buoyancy parameter (J_{a}). It is hoped that this study will serve as a complement to the previous studies in engineering and scientific research.

The authors are highly grateful to the authority of Chittagong University of Engineering and Technology (CUET) for providing technical supports during this research work at Simulation Lab, Department of Mathematics, CUET, Chittagong, Bangladesh.

Uddin, M.N., Ali, Md.Y., Zahed, N.M.R. and Uddin, Md.J. (2016) Similarity Solutions of Unsteady Mixed Convective Boundary Layer Flow of Viscous Incompressible Fluid along Isothermal Horizontal Plate. Open Journal of Fluid Dynamics, 6, 279-302. http://dx.doi.org/10.4236/ojfd.2016.64022