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The objective of this article is to present the dynamics of an Upper Convected Maxwell (UCM) fluid flow with heat and mass transfer over a melting surface. The influence of melting heat transfer, thermal and solutal stratification are properly accounted for by modifying the classical boundary conditions of temperature and concentration respectively. It is assumed that the ratio of inertia forces to viscous forces is high enough for boundary layer approximation to be valid. The corresponding influence of exponential space dependent internal heat source on viscosity and thermal conductivity of UCM is properly considered. The dynamic viscosity and thermal conductivity of UCM are temperature dependent. Classical temperature dependent viscosity and thermal conductivity models were modified to suit the case of both melting heat transfer and thermal stratification. The governing non-linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformations and completed the solution numerically using the Runge-Kutta method along with shooting technique. For accurate and correct analysis of the effect of variable viscosity on fluid flow in which (
*T*
* _{w}* or

*T*

_{m}) <

*T*

_{∞}, the mathematical models of variable viscosity and thermal conductivity must be modified.

Mass transfer can be described as the movement of mass (material) through a fluid-fluid interface or a fluid-solid interface. The term “mass transfer” is commonly used in engineering and in industry for physical processes that involve diffusive and convective transport of chemical species within physical systems. The three kinds of fluxes in relation to mass transfer have been explained in Asano [

It is a common known fact in rheology that given enough time, even a solid-like material will flow (see Barnes et al. [

Internal energy generation can be explained as a scientific method of generating heat energy within a body by chemical, electrical or nuclear process. Natural convection induced by internal heat generation is a common phenomenon in nature. Crepeau and Clarksean [

surface may lead to melting of solid surface. From the knowledge of kinetic theory of matter, some solids may melt if expose to a high temperature. In an earlier study, the effect of melting on heat transfer was studied by Yin-Chao and Tien [

In all of the above mentioned studies, fluid viscosity and thermal conductivity have been assumed to be constant function of temperature within the boundary layer. However, it is known that physical properties of the fluid may change significantly when expose to internal generated temperature. For lubricating fluids, heat generated by the internal friction and the corresponding rise in temperature affect the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. In a case of melting as reported by many researchers [

We consider steady and incompressible Upper Convected Maxwell (UCM) fluid flow with variable thermo- physical properties over a melting surface situated in hot environment. The flow under consideration is assumed to occupy the domain

where

In Equation (2) and Equation (3), elastic terms are

Using order of magnitude as introduced by Ludwig Prandtl and stated in Schichting [

and easy to show that in Equations (2) and (3), order of magnitude of the two elastic terms and order of magnitude of the two viscous terms are the same if

This condition can be explained following Sadeghy et al. [

In the presence of pressure gradient, the equations of motions together with continuity equation can be written as

In Equation (8) and Equation (9), there exist five unknowns (i.e. five dependent variables) which are u, v,

The time derivative

has been devised to satisfy the requirements of continuum mechanics (i.e., material objectivity and frame in difference; see Larson [

In Equation (11),

This can be differentiated and used to eliminate the pressure gradient Lienhard-IV and Lienhard-V [

Since the flow is along flat horizontal melting plate,

In this study on Maxwell fluid flow, it is assumed that the normal stress is of the same order of magnitude as that of the shear stress in addition to the usual boundary layer approximation for deriving the component of the

momentum boundary layer Equation (12). This is properly accounted for by introducing

momentum Equation (12); for details, see Motsa et al. [

Equations (8), (12), (13) and (14) are subject to the following boundary conditions

κ is the thermal conductivity,

The classical models in Equation (17) are valid when

It is worth mentioning that the first and fourth terms of Equation (18) are valid since

In this study, the idea of Vimala and Loganthan [

From these models, it is valid to write the relation of the form

Upon using Equation (18) - Equation (22), we obtain

In order to write the governing equations and the boundary conditions in dimensionless form, the following non-dimensional quantities are introduced,

It is important to note that the first two terms of Equation (25) automatically satisfy continuity Equation (8). Then, Equation (23) and Equation (24) becomes

The corresponding boundary conditions take the form

Here dimensionless viscoelastic parameter (Deborah number)

and melting parameter

tion coefficient

where the wall skin friction

Using Equation (25)

the local Reynolds number is defined as

Numerical solutions of the ordinary differential Equation (26) - Equation (28) with the Neumann boundary conditions Equation (29) and Equation (30) are obtained using classical Runge-Kutta method with shooting techniques. The BVP can not be solved on an infinite interval, and it would be impractical to solve it for even a very large finite interval. In this study, we impose the infinite boundary condition at a finite point

The calculated values for

In order to verify the accuracy of the present analysis, the results of Classical Runge-Kutta together with shooting (RK4SM) have been compared with that of bvp4c for the limiting cases when

The numerical computations have been carried out for various values of temperature dependent viscous parameter, thermal stratification parameter, solutal stratification parameter, Deborah number, magnetic field parameter, temperature dependent thermal conductivity parameter, Schmidt number, Prandtl number, space dependent heat source parameter, intensity of heat distribution on space parameter and melting parameter using numerical scheme discussed in the previous section. To avoid any corresponding effect(s) on the fluid flow (i.e. decrease in the volume and changing of state) of UCM due to high temperature when investigating the effect of dimensionless temperature dependent viscous and thermal conductivity parameters, variable “a_{1} = a_{2}” in Equations (26) and (27) have been considered as unity. In order to illustrate the results graphically, the numerical values are plotted in Figures 2-14.

0.084750549215246 | 0.084750612582614 | −0.216400317226149 | ||

0.085203684744618 | 0.085203475747002 | −0.220822799620452 | ||

0.085590730358658 | 0.085590497630018 | −0.227973593463212 | ||

0.086014042064422 | 0.086013790040591 | −0.240922640551570 | ||

0.039237792254884 | 0.039237496643905 | −0.192839256745501 | ||

0.054568561112953 | 0.054568458187466 | −0.198631705554188 | ||

0.062316129381638 | 0.062316082253395 | −0.206753991576532 | ||

0.067978247345109 | 0.067978247104503 | −0.220488109972874 | ||

−0.216400307739576 | −0.126425532660648 | −0.126425739785364 | ||

−0.220822718730358 | −0.126353607140818 | −0.126353292670234 | ||

−0.227973425611366 | −0.126290564565258 | −0.126290251750048 | ||

−0.240922368071198 | −0.126219831635247 | −0.126219525847626 | ||

−0.192839423367583 | −0.091023399837705 | −0.091023186146136 | ||

−0.198631742945421 | −0.103581405815614 | −0.103581365442893 | ||

−0.206753996042305 | −0.109309119916587 | −0.109309104668065 | ||

−0.220488109991622 | −0.113210624607363 | −0.113210624587697 | ||

concentration gradient profiles. The effect of

The variations of temperature profiles

In this study, setting m = 0 can seriously affect the melting processes at the wall. In addition to this fact, existence of melting at the wall together with an increase in thermal stratification parameter depicts anegligible increase in longitudinal velocity and significant increase in transverse velocity (see

We believe that this influence requires further investigation by replacing melting heat transfer model with suction model (i.e. to study the effect of suction on UCM fluid with variable thermo-physical properties subject to thermal and solutal stratification). It is also important to report that the influence of free stream temperature together with internal exponential heat source account for the increase in velocity and transverse velocity of UCM as it flows. In fact, these influences totally subdues the effect of increasing stratification which ought to decrease velocity profiles as reported in [

Hence, this increase in temperature weakens the intermolecular forces which hold the molecule of UCM so tight. In view of this, the dynamic viscosity is gradually reduced and corresponds to increase in velocity as shown in

Similarity solutions of steady UCM fluid flow over a melting surface; considering a case in which the flow is subjected to thermal and solutal stratification have been studied theoretically. The corresponding influence of thermal stratification, solutal stratification, variation in viscosity and thermal conductivity due to temperature is properly considered. The governing (dimensional) partial differential equations are converted into (dimensionless) nonlinear ordinary differential equations by using similarity transformation before being solved numerically using fourth order Runge-Kutta integration scheme along with shooting techniques. Results for the skin friction coefficient, local Nusselt number, local Sherwood number, transverse velocity profiles, velocity profiles, temperature profiles as well as concentration profiles are presented for different values of the pertinent parameters. Effects of Prandtl number, the melting parameter, temperature dependent viscous parameter, temperature dependent thermal conductivity parameter, solutal and thermal stratification on the flow and heat transfer characteristics are thoroughly examined. For accurate and correct analysis of fluid flow in which (

0 | −0.3548243390 | −0.4397880431 |

0.1 | −0.33413024290 | −0.4224901910 |

0.2 | −0.31343614671 | −0.4051923390 |

0.3 | −0.29274205052 | −0.3878944869 |

0.4 | −0.27204795433 | −0.3705966348 |

0.5 | −0.25135385814 | −0.3532987827 |

conductivity is to be investigated; the term

Authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to express their thanks to the anonymous Reviewer for his/her valuable and interesting comments.

Kolawole S.Adegbie,Adeola J.Omowaye,Akeem B.Disu,Isaac L.Animasaun, (2015) Heat and Mass Transfer of Upper Convected Maxwell Fluid Flow with Variable Thermo-Physical Properties over a Horizontal Melting Surface. Applied Mathematics,06,1362-1379. doi: 10.4236/am.2015.68129