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In this paper, we analyze unsteady two dimensional hydromagnetic forced convection boundary layer flow of a viscous incompressible fluid along flat plates with thermophoresis. The potential flow velocity has been taken as a function of the distance x and time t. The governing partial differential equations are transformed to ordinary differential equation by applying local similarity transformation. The resulting similarity equations are then solved numerically for unsteady case, applying Nachtsheim-Swigert shooting iteration technique with six order Runge-Kutta method. The variations in fluid velocity, fluid temperature and species concentration are displayed graphically and discussed for different material parameters entering into the analysis. The effects of the pertinent parameters on the skin-friction coefficient, wall heat transfer coefficient and wall deposition flux rate are also displayed in tabulated form and discussed them from the physical point of view. An analysis of the obtained results shows that the flow field is influenced appreciably by the magnetic field parameter and the thermophoresis particle deposition.

In recent years, the problems of forced convective boundary layer flow over flat plates with thermophoresis under the influence of a magnetic field have been attracted the attention of a number of researchers because of their possible applications in many branches of science and technology, such as its applications in transportation cooling of re-entry vehicles and rocket boosters, cross-hatching on ablative surfaces and film vaporization in combustion chambers. Forced convective boundary layer flows have a great interest from both theoretical and practical point of views because of its vast and significant applications in cosmic fluid dynamics, solar physics, geophysics, electronics, paper production, wire and fiver coating, composite processing and storage system of agricultural product etc.

A radiometric force by temperature gradient that enhances small micron sized particles moving toward a cold surface and away from the hot surface is termed as thermophoresis. The velocity caused by thermophoresis is called thermophoretic velocity. Thermophoresis plays a significant role of transporting particles from hot fluid region to the cold fluid region. This phenomenon has many engineering applications in removing small particles from gas streams, in determining exhaust gas particle trajectories from combustion devices, and in studying the particulate material deposition on turbine blades. Thermophoresis principle is utilized to manufacture graded index silicon dioxide and germanium dioxide in the fabrication of optical fiber used in the field of communications. Thermophoresis is also a key mechanism of study in semi-con- ductor technology especially in production of controlled high-quality wafer as well as in production of magnetohydrodynamic (MHD) energy. Thermophoretic deposition of radioactive particles is one of the major factors causing accidents in nuclear reactors. The magnitudes of the thermophoretic force and velocity are proportional to the temperature gradient and depend on many factors like thermal conductivity of aerosol particles and carrier gas. Duwairi and Damesh [^{th} order chemical reaction of a heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. Parvin et al. [

The objective of the present paper is to obtain a local similarity solution of an unsteady two dimensional hydromagnetic forced convective boundary layer flow of a viscous incompressible fluid over flat plates with thermophoresis and be attempted to investigate the effects of several involved parameters on the velocity, temperature, concentration and other flow parameters like the skin-friction coefficient, local Nusselt number (wall heat transfer coefficient) and local Stanton number (wall deposition flux) across the boundary layer.

In this work, we considered the unsteady, laminar, hydromagnetic combined heat and mass transfer by forced convection flow along a flat plate. With x-axis measured along the plate, a uniform magnetic field

where

thermophoretic velocity

where k is the thermophoretic coefficient and T_{ref} is the reference temperature.

The appropriate boundary conditions for the above model are as follows:

and

The continuity Equation (1) is satisfied by introducing the stream function

such that

The momentum, energy and diffusion Equations (2)-(4) can be transformed to the corresponding ordinary differential equations by introducing the following similarity transformations:

The momentum, energy and diffusion Equations (2)-(4) after some simplifications, reduce to the following forms:

where, Pr (Prandtl number) =

M (Modified Local Magnetic field Parameter) =

Q (Modified local heat generation Parameter) =

Sc (Schmidt number) =

Ta (Thermophoretic Parameter) =

The corresponding boundary conditions are:

and

where the prime (

The Prandtl number (Pr) or Prandtl group is a dimensionless number and is defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state. As such, the Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity. For most gases over a wide range of temperature and pressure, Pr is approximately constant. Therefore, it can be used to determine the thermal conductivity of gases at high temperatures, where it is difficult to measure experimentally due to the formation of convection currents. Small values of the Prandtl number,

Schmidt number (Sc) is a dimensionless number with important applications to transport phenomena and is defined as the ratio of momentum diffusivity (viscosity) and mass diffusivity and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. It physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer.

Thermophoretic parameter is a phenomenon observed in mixtures of mobile par- ticles where the different particle types exhibit different responses to the force of a temperature gradient. Thermophoretic force has been used in commercial precipitators for applications similar to electrostatic precipitators. It is exploited in the manu-fac- turing of optical fiber in vacuum deposition processes. It can be important as a transport mechanism in fouling.

Magnetic field Parameter is a dimensionless number used in magneto fluid dynamics, equal to the product of the square of the magnetic permeability, the square of the magnetic field strength, the electrical conductivity and a characteristic length, divided by the product of the mass density and the fluid velocity. Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end.

A heat generation site is a structure where heat is generated for distribution to other structures and its parameter is known as heat generation parameter. The power consumption and the heat generation in metal cutting processes are dependent on a combination of the physical and chemical properties of the work piece material and cutting tool material, cutting conditions and the cutting tool geometry.

Important Physical ParametersThe skin-friction coefficient, local Nusselt number (wall heat transfer coefficient), and local Stanton number (wall deposition flux):

The parameters of engineering interest for the present problem are the skin-friction coefficient, local Nusselt number and the local Stanton number which indicate physically wall shear stress, rate of heat transfer and wall deposition flux respectively. These can be obtained from the following expressions:

A dimensionless skin-friction coefficient expressing the proportionality between the frictional force per unit area, or the shearing stress exerted by the wind at the earth's surface, and the square of the surface wind speed. Skin friction coefficient is a component of parasitic drag that occurs differently depending on the type of flow over the lifting body (laminar or turbulent). Just like any other form of drag, the coefficient of skin friction drag is calculated with various equations and measurements depending on the flow and then added to coefficients of other forms of drag to calculate total drag.

The skin-friction coefficient is given by

In heat transfer at a boundary (surface) within a fluid, the local Nusselt number is the ratio of convective to conductive heat transfer across (normal division by its area then takes care of the local factor and the normalization). The local Nusselt number analog has on occasion been used to actually.

The local Nusselt number may be written as

The local Stanton number is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. This number arises in the consideration of the geometric similarity of the momentum boundary layer and the thermal boundary layer, where it can be used to express a relationship between the shear force at the wall (due to viscous drag) and the total heat transfer at the wall (due to thermal diffusivity).

The Stanton number can be written as

where,

The set of ordinary differential Equations (10) to (12) with boundary conditions (18) and (19) are nonlinear and coupled. A standard initial value solver i.e., the shooting method is used to solve these equations numerically. For this purpose we applied the Nacthsheim-Swigert iteration technique (Nachtsheim and Swigert, 1965) [

In order to investigate the physical representation of the problem, the numerical values of velocity field (

The dimensionless velocity profiles for the influence of various physical parameters are

presented in Figures 1-15.

resistive force tends to slow down the flow and hence the fluid velocity decreases with the increase of the local magnetic field parameter as observed in

force on the forced convection flow.

Due to forced convection flow, the Prandtl number Pr affects the velocity distribution.

Prandtl number Pr, the velocity along the plate decreases with increase in Pr. It is evident from the decrease in the velocity profiles along x-direction and y-direction respectively as Pr increases that, the fluid’s velocity components decrease at every point above

the surface.

0) causes the thermal boundary layer to become thicker and the fluid become warmer.

The influences of the Schmidt number, Sc on the velocity profiles are plotted in

The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity). As the Schmidt number

increases the velocity increases.

The dimensionless temperature profiles are presented in Figures 6-10.

For different values of the magnetic field parameter M on the temperature profiles are plotted in

Prandtl number defines the relative effectiveness of the momentum transport by diffusion in the hydrodynamic boundary layer to the energy transported by thermal diffusion in the thermal boundary layer. According to the definition of Prandtl number high Pr fluids possess lower thermal conductivities which reduce the conduction heat transfer and increases temperature variations at the wall. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. This figure reveals that an increase in Prandtl number Pr results in a decrease in the temperature distribution, because, thermal boundary layer thickness decreases with an increase in Prandtl number, Pr.

From

The influence of the Schmidt number on the temperature is presented in

The dimensionless temperature profiles along x-direction for different values of Ta are presented in the

In order to gain physical insight of the problem, the numerical results for the dimensionless concentration profiles have been presented graphically in Figures 11-15.

The effect of the local magnetic field parameter (M) on concentration fields have shown in

From

In

The effect of the thermophoretic parameter (Ta) on concentration field is presented in the

The skin-friction coefficient,

The numerical results are illustrated in Tables 1-5 for the effect of various parameters on the Skin-friction coefficient, local Nusselt number, and local Stanton number. From

M | |||
---|---|---|---|

0.25 | 1.0235046 | 0.84207159 | 0.9578218 |

1.50 | 1.4691122 | 1.51357670 | 0.9232827 |

3.50 | 2.0202887 | 3.22451048 | 0.8829878 |

5.50 | 2.4603553 | 7.07822666 | 0.8280733 |

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

0.71 | 1.4675186 | 1.51179868 | 0.9237007 |

1.10 | 1.4674663 | 1.02651915 | 0.9307509 |

4.00 | 1.4672230 | 0.18606617 | 0.9364918 |

7.00 | 1.4665497 | 0.13633085 | 0.9408721 |

Q | |||
---|---|---|---|

0.50 | 1.4685408 | 0.78715171 | 0.9274211 |

2.50 | 1.4615275 | 1.50462781 | 0.9253480 |

4.00 | 1.4590538 | 1.74655398 | 0.9126801 |

6.00 | 1.4498124 | 2.96491001 | 0.9077078 |

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

0.30 | 1.4287004 | 1.50871980 | 0.6104537 |

0.60 | 1.4460197 | 1.41001477 | 0.9241163 |

5.00 | 1.4610567 | 1.38990800 | 2.6736081 |

8.00 | 1.4890560 | 1.19356201 | 2.8391696 |

Ta | |||
---|---|---|---|

0.50 | 1.9630242 | 1.50642932 | 0.9249399 |

2.50 | 1.6714498 | 1.49926073 | 1.0377505 |

4.50 | 1.3148420 | 1.07243777 | 1.1858530 |

6.50 | 1.2630476 | 0.50280036 | 1.5711148 |

coefficient and the local Nusselt number decrease while the local Nusselt number increases with the increase of the Prandtl number (Pr) and the Thermophoretic parameter (Ta) respectively. For the the increases of local heat generation parameter (Q), the Skin-friction coefficient and the local Stanton number decrease while the local Nusselt number inccreases in

In the presence of a magnetic field, the fluid velocity is found to be decreased, associated with a reduction in the velocity gradient at the wall, and thus the local skin-fric- tion coefficient decreases. Also, the applied magnetic field tends to decrease the wall temperature gradient and concentration gradient, which yield a decrease the local Nusselt number and the local Stanton number. The local skin friction coefficient as well as rate of heat transfer in the micropolar fluid is lower compared to that of the Newtonian fluid. The above table is highly significant influences of on skin friction, the rate of heat transfer and wall deposition flux have been found which can be physically realizable.

In this paper, we have discussed the effects of thermophoresis on an unsteady two dimensional forced convective heat and mass transfer boundary layer flow over flat plates. The results are analyzed for various physical parameters such as local magnetic field parameter, Prandtl number, Schmidt number, local heat generation parameter, magnetic parameter, thermophoretic parameter, heat and mass transfer characteristics. The numerical results have been presented in the form of graphs and tables. The particular conclusions drawn from this study can be listed as follows:

・ Thermophoretic particle deposition velocity decreases with the increasing values of the thermophoretic coefficient and concentration ratio where as it increases with the increasing values of the thermophoresis parameter.

・ The plate couple stress increases with the increase values of the magnetic field parameter.

・ Magnetic field significantly controls the flow, rotation of micro-constituents and heat transfer characteristics of a micropolar fluid.

・ Magnetic field retards the motion of the fluid.

・ Thermophoresis is an important mechanism of micro-particles transport due to a temperature gradient in the surrounding medium and has found numerous applications, especially in the field of aerosol technology and industrial air pollution.

・ The Prandtl number as well as the Schmidt number varies significantly within the boundary layer when the thermal conductivity is temperature dependent.

・ In forced convection regime, the surface mass flux increases with the increase of the thermophoretic parameter.

・ The local skin friction coefficient, Nusselt number and Sherwood number are higher for the fluids of constant electric conductivity than those of the variable electric conductivity.

・ The velocity profiles increase whereas temperature profiles decrease with an increase of the forced convection current.

・ Concentration within the boundary-layer decreases with the increasing values of the thermophoretic parameter whereas it increases as increases the thermophoretic coefficient as well as the concentration ratio.

Uddin, M.J. and Ali, M.Y. (2016) Effects of Hydromagnetic and Thermophoresis of Unsteady Forced Convection Boundary Layer Flow over Flat Plates. Journal of Applied Mathematics and Physics, 4, 1756-1776. http://dx.doi.org/10.4236/jamp.2016.49182