FUTA Journal of Research in Sciences, Vol. 12 (2) 2016:287 - 298 BOUNDARY LAYER ANALYSIS OF UPPER CONVECTED MAXWELL FLUID FLOW WITH VARIABLE THERMO-PHYSICAL PROPERTIES OVER A MELTING THERMALLY STRATIFIED SURFACE

O. K. Koríko, K. S. Adegbie, A. J. Omowaye and † I.L. Animasaun Dept. of Mathematical Sciences, Federal University of Technology, Akure, Ondo State Nigeria [email protected], [email protected], [email protected], [email protected]

ABSTRACT In this article, the boundary layer analysis which can be used to educate engineers and scientists on the flow of some fluids (i.e. glossy paints) in the industry is presented. The influence of melting heat transfer at the wall and thermal stratification at the freestream on Upper Convected Maxwell (UCM) fluid flow with heat transfer is considered. In order to accurately achieve the objective of this study, classical boundary condition of temperature is investigated and modified. The corresponding influences of thermal radiation and internal heat source across the horizontal space on viscosity and thermal conductivity of UCM are 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 nonlinear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformations and solved numerically using the Runge-Kutta method along with shooting technique. The transverse velocity, longitudinal velocity and temperature of UCM are increasing functions of temperature dependent viscous and thermal conductivity parameters. This could increase the efficiency of some fluids flow in the industry. Effects of selected emerging parameters on the velocity and temperature fields are plotted and discussed.

Keywords: Upper convected Maxwell fluid, variable viscosity, Thermal stratification, thermal radiation, Melting surface

INTRODUCTION hotter than the back, and so on. Physically, The first statement of the second law heat flows constantly from our body to the of thermodynamics can be explained in air around us. All bodies constantly emit terms of the spontaneous heat flow from a energy by a process of electromagnetic hot to a cold body. This implies that an ice radiation. The intensity of such energy flux cube must melt on a hot surface rather than strongly depends on the temperature of the become colder. In the industry, radiative body and the nature of its surface. In order to heat transfer and thermal radiation are investigate this phenomenon, Lienhard-IV commonly used to describe the science of and Lienhard-V (2008) stated that objects heat transfer caused by electromagnetic that are cooler than the fire emit much less waves. Obvious everyday examples of energy because the energy emission varies as thermal radiation include the heating effect the fourth power of absolute temperature. of sunshine on a clear day. The fact that, Sandeep and Sugunamma (2014) considered when one is standing in front of a fire; the these facts and reported the side of the body facing the fire feels much

287 O.K. Koriko et al., FUTA J. of Res. in Sci., Vol. 12 (2) 2016:287 - 298

effect of thermal radiation on hydromagnetic Spectral Collocation Method to explain free convection flow of a viscous entropy generation on magneto incompressible electrically conducting fluid. hydrodynamics flow and heat transfer of a The analysis, description, theoretical Maxwell fluid over a stretching sheet. and experimental studies of boundary layer Internal energy generation can be flow together with heat transfer across a explained as a scientific method of special kind of fluid (i.e. having the generating heat energy within a body by properties of elasticity and viscosity when chemical, electrical or nuclear process. undergoing deformation) as it flows over a Natural convection induced by internal heat horizontal surface has gained the attention of generation is a common phenomenon in many researchers. The dynamics of such nature. In many situations, there may be material (Maxwell fluid) is a fundamental appreciable temperature difference between topic in fluid dynamics and it has attracted the surface and the ambient fluid. This the attention of many researchers due to its necessitates the consideration of temperature wide industrial and technical applications. dependent heat source(s) that may exert a Series of investigations have been carried out strong influence on the heat transfer towards the understanding of the dynamics characteristics Salem and El-Aziz (2007). of viscoelastic material since the contribution Salem and El-Aziz (2008) further stated that of Maxwell (1867) and Christopher (1993). exact modeling of internal heat generation or The Upper Convected Maxwell model can absorption is quite difficult and argued that be described as the generalization of the some simple mathematical models can Maxwell material for the case of large express its average behavior for most deformation using the upper-convected time physical situations. Recently, Animasaun et derivative (also known as Oldyrold al. (2015) reported that when the plastic derivative) which is the rate of change of dynamic viscosity and thermal conductivity some tensor properties of a small parcel of of non-Newtonian Casson fluid are fluid that is written in the coordinate system considered as temperature dependent, stretching with the fluid. It is worth pointing exponentially decaying internal heat out that this concept has not been extended generation parameter is an important to investigate the dynamics of UCM flow in dimensionless number that can be used to between melting surface and thermal increase velocity and temperature of the fluid stratification located at the freestream. as it flows. Motsa and Animasaun (2016) It is a common known fact in deliberated on the velocity of fluid flow in rheology that given enough time, even a the presence of space and temperature- solid-like material will flow; see Barnes et dependent internal heat source due to al. (1989). In view of this, it is required to impulsive. Koriko et al. (2016) adopted characterize the fluidity of materials under homotopy analysis method to investigate free specific flow conditions (i.e. a dimensionless Convective micropolar fluid flow in the number that incorporates both the elasticity presence of space dependent internal heat and viscosity of material is required). In the source. Effect of this internally generated same context, it was reported that at high heat energy on the surface may lead to Deborah number, UCM flow corresponds to melting of solid surface. solid-like behavior and low Deborah From the knowledge of kinetic theory numbers to fluid-like behavior (Steffe, 1996 of matter, some solids may melt if expose to and Sadeghy et al. 2005). Raju et al. (2016) a high temperature. In an earlier study, the explained the motion of viscoelastic fluid effect of melting on heat transfer was studied flow in the presence of induced magnetic by Yin-Chao and Tien (1963) for the field when homogeneous and heterogeneous Leveque problem. The corresponding effect chemical reaction is considered. Recently, of melting on forced convection heat transfer Shateyi et al. (2015) adopted Chebyshev was reported in Tien and Yen (1965). In

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addition, effect of melting on heat transfer particles past a vertical porous plate moving between melting body and surrounding fluid through a binary mixture in an optically thin qualitatively from the point of view of environment. Motivated by all the works boundary layer theory was investigated. This mentioned above, it is of interest to contribution to the existing body of contribute to the body of knowledge by knowledge attracted Epstein (1975) to studying the dynamics of UCM fluid flow present a note on a systematic method of considering a case in which the influence of calculating steady state melting rates in all temperature on viscosity and thermal circumstances involving the melting of solid conductivity are properly accounted for. In bodies immersed in streams of warmer fluid this study, we aim at investigating the of the same material. Many researchers have dynamics of UCM fluid flow in between investigated the effect of melting parameters; melting surface and thermal stratification and it was revealed that fluid viscosity and located at the freestream. This is achieved by thermal conductivity have been assumed to modifying and incorporating all the be constant functions of temperature within necessary term(s) into the boundary layer the boundary layer. However, it is known equation in-line with boundary layer theory that physical properties of the fluid may and heat transfer theory. In addition, the change significantly when expose to internal objective is to unravel effects of generated temperature. For lubricating fluids, corresponding parameters on the velocity, heat generated by the internal friction and the temperature, shear stress and temperature corresponding rise in temperature affect the gradient of UCM fluid flow with variable viscosity of the fluid and so the fluid thermo physical properties. The knowledge viscosity can no longer be assumed constant. will be useful to Engineers and scientist In a case of melting as reported by Hayat et when interested to increase the efficient of al. (2010a), Fakusako and Yamada (1999), UCM flow during industrial processes. Ishak et al. (2010) and Pop et al. (2010), it is worth mentioning that temperature of fluid MATHEMATICAL FORMULATION layers at free stream may also have We consider steady and incompressible significant effect on the intermolecular Upper Convected Maxwell (UCM) fluid forces of Upper Convected Maxwell fluid. flow with variable thermophysical properties According to Batchelor (1987), along a melting surface. The flow under Animasaun (2015a) and Meyers et al. consideration is assumed to occupy the (2006), it is a well-known fact that properties domain 0 y   as shown in Figure 1. which are most sensitive to temperature rise Following Dunn and Rajagopal (1995) and are viscosity and thermal conductivity. Sadeghy et al. (2005), an incompressible Recently, Mukhopadhyay (2013) considered fluid flow obeying Upper Convected this same fact in order to explain stagnation Maxwell model with temperature dependent point flow behavior on non-melting surface dynamics viscosity and thermal conductivity; while Animasaun (2015a) adopted the model the continuity equation, x  momentum and reported the dynamics of unsteady equation and the energy equation can be magnetohydrodynamic convective fluid flow simplified using the usual boundary layer with radiation and thermophoresis of theory approximations and written as uv 0, (1) xy

u  u 2 u  2 u  2 u1   u B 2 22 (2) u v  u22  v 2, uv   u x  y  x  y  x  y  y  y

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a ny TTT 11   QTToo()  qr u v   e   . (3) x  y cp  y  y  c p  c p  y

Figure 1: Physical Configuration The Rosseland approximation in the Energy Neglecting higher order Equation (3) is a simplification of the 4 3 4 radiative transport Equation (RTE) for the TTTT43mm (6) case of optically thick media (UCM) as stated in Brewster (1972). The Rosseland Also differentiating both sides of Equation approximation for radiation is defined as (6) partially with respect to T

4 T 4 4 T  4T 3 (7) q  . (4) m r 3ky*  T Hence Since radiation travels only a short distance before being scattered or absorbed; Equation 44 qr 44 TTT       (4) introduces a new diffusion term into the  **     . (8) energy transport equation with a strong y  y33 k  y   y  k  T  y  temperature dependent diffusion coefficient. Assuming that the temperature difference Substituting Equation (7) into Equation (8), within the flow is such that T 4 may be we obtain expanded in a Taylor Series and expanding q 16T 3 2T T 4 about T ; neglecting higher orders. r  m . (9) m y3 k*2 y dT4()TT 2 d 2 T 4 TTTT44 (  ) m  ... (5) Substituting Equation (9) into Energy mmdT2! dT 2 T m T m Equation (3)

a ny 3 2 TTTT 11  QTTTo(  o ) 16 m  u v   e  *2. (10) x  y cp  y  y  c p  c p 3 k  y

In this present study, it is important to state temperature in the energy equation. This that exponential heat source is adopted to concept can be traced to the idea of Salem account for internal distribution of and El-Aziz (2007, 2008), Abegunrin et al.

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(2015a) and Animasaun (2015b). Equation subject to the following boundary conditions (1), Equation (2) and Equation (10) are T  [ ** c ( T  T )] v ( x ,0), TT at (11a,b,c) u  0,  s m o m y  0 y

u  ue ( ax), TT  as y   (12a,b)

The symbol κ is the thermal conductivity, * properties of UCM fluid as it flows over a melting surface within the boundary layer. is the latent heat of the fluid and cs is the heat capacity of the solid surface. In order to However, it is known that physical properties solve the problem completely in unbounded of the fluid may change significantly in the domains, it is possible to augment the presence of space dependent internal boundary conditions by assuming certain generated temperature. For lubricating fluids, asymptotic structures for the solutions at heat generated by the internal friction and the infinity. The formulation of Equation (11b) corresponding rise in temperature affect the states that the heat conducted to the melting viscosity of the fluid and so the fluid surface is equal to the heat of melting plus viscosity can no longer be assumed constant. the sensible heat required raising solid In order to account for the variation in thermo-physical properties of the fluid as it temperature T * to its melting temperature o flows past a horizontal melting surface, it is T Epstein and Cho (1976). The increase of m valid to consider the mathematical model of temperature may also lead to a local increase temperature-dependent viscosity model used in the transport phenomena by reducing the in Koriko et al. (2015) and Mukhopadhyay viscosity across the momentum boundary (2013) which was developed using the layer and so the heat transfer rate at the wall experimental data of Batchelor (1987) may also be affected greatly. Due to this, it is together with the mathematical model of very important to account for the influence temperature-dependent thermal conductivity of temperature on the thermo-physical model used in Animasaun (2015a) as

(T )* [ a  b ( T  T )], (T )* [ a  b ( T  T )] valid accurately when TT (13a,b) 1 w 2  w 

Since TTm   , it is necessary to modify increase in the volume or quality of UCM Equation (13a,b) as suggested by Koriko et fluid as it flows is to assume parameter a 1. al. (2015). and Animasaun (2015c). In this These mathematical models together with theoretical study, the best way to avoid either classical similarity variables for temperature are modified to TT (T )* [1  b ( T  T )], (), m (T )* [1  b ( T  T )]. (14a,b,c) 1  TT 2 m  o Due to the relationship between dynamic The idea of Vimala and Loganthan (2015) viscosity, exponential internal heat source and Animasaun (2015b) was followed to within the fluid domain and free stream define thermal stratification (Tm ) at the temperature of UCM fluid as stated in melting wall ( y  0) and at the free stream Equation (14a). Using Equation (15b) and Equation (14b), we obtain ( y ) as

Tmo T m1 x, T  To m2 x. (15a,b) T T (1  ) T  To   m1 x From these models, it is valid to write the

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relation of the form Animasaun (2016), it is valid to define temperature dependent viscous parameter  b1 Tmo T b 1 m 1 x, b1T To   b1m2 x.(16a,b) as Equation (17a). This assumption is based on the fact that (i) the stress is a linear To is known as reference temperature. It is function of the strain rates as the fluid flows worth noticing from Equation (16a,b) that along a surface, and (ii) the UCM fluid is there exist two differences in temperature isotropic. The ratio of the two terms in due to stratification which occurs for all x at Equations (16a,b) can thus produce the fixed point y = 0 and the other that occurs for dimensionless thermal stratification all x as y . Following Omowaye and parameter ( St ).

m1   b1T To , b1  Tm T o  S t St  (17a,b,c) m2

Upon using Equation (14) - Equation (17), we obtain

u u  2u 2u 2u   u B2 u  v  u2  v2  2uv   *[a    S ] *  u, (18)  2 2  t x y  x y xy  y y 

2 T T  *[a  ](T  T )  2  *b    u  v   o  2 (T  T ) 2   2  o   x y c p y c p  y 

3 2 Qo (T To ) ny a / 1 16Tm  T  e  * 2 . (19) c p c p 3k y

The dimensionless governing equation is obtained by using the following non-dimensional quantities

1   a 2 v   , u  ,   y,   xf () a (20a,b) x y  It is important to note that Equations (20a,b) automatically satisfy continuity Equations (1). Then, Eqs. (18), (19), (11) and (12) becomes

32 2 df d dfdfdf    df [1    St   f ]32  f    2  f     M   0, (21) d d  d   d   d   d 

2 4 d d  d  df df d  n 1    Pr S t  P r   P r f  P r  e  0. (22) 3N d2 d  d  d  d  d 

The corresponding dimensionless boundary conditions take the form

df d  0, m P f 0,   0 at   0 (22) d d r df 1,  (1S ) as  . (23) d t

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Dimensionless viscoelastic parameter The calculated values for f () and () at (Deborah number)  a , temperature   4 are compared with the known dependent thermal conductivity boundary conditions in Equation (23) and the // / parameter b2 () T To , magnetic field estimated values f (0) and  (0) are adjusted parameter M B2 a , coefficient of to give a better approximation of the / thermal diffusivity  cp , Prandtl solution. Series of values for and  (0) are considered and applied with fourth-order number Pcrp , thermal radiation *3 classical Runge-Kutta method using step size parameter NT (4  m ), heat source  h 0.01. The above procedure is parameter  Q c a and melting op repeated until asymptotically converged **results are obtained within a tolerance level parameter m( T  To ) c p [  c s ( T m  T o )]. of 105 . METHOD OF SOLUTION Numerical solutions of the ordinary DISCUSSION differential Equation (21) and Equation (22) The numerical computations have been subject to Neumann boundary conditions carried out for various values of temperature (22) and (23) are obtained using classical dependent viscous parameter, stratification Runge-Kutta method with shooting parameter, Deborah number, magnetic field techniques. The BVP cannot be solved on an parameter, thermal radiation parameter, infinite interval and it would be impractical temperature dependent thermal conductivity to solve it for even a very large finite parameter, Prandtl number, space dependent interval. In this study, we imposed the heat source parameter, intensity of heat infinite boundary condition at a finite point distribution on space parameter and melting parameter using numerical scheme discussed   4. The set of coupled nonlinear  in the previous section. In order to illustrate ordinary differential equations along with the results graphically, the numerical values boundary conditions have been reduced to a are plotted in Figures 2 - 7. The variations of system of five simultaneous equations of f () and f / () along  with different values first order for five unknowns following the method of superposition Na (1979). In order of  are plotted in Figure 2 and 3 to integrate the corresponding IVP the values respectively. It is seen that an increase in the of f (0), f // (0) , and  / (0) are required, but magnitude of leads to the enhancement of no such values exist after the non- transverse velocity and velocity profiles. It is dimensionalization of the boundary further noticed that, increase in the conditions (Equations 11 and 12). The magnitude of has no significant effect on // velocity profiles near the melting wall. As suitable guess values for G1  f (0) , / shown in Figure 2, the parameter has an G  ()0 and G  f (0) are chosen and 2 3 evident effect on f (η) that the larger the then integration is carried out. In this method of solution, Newton-Raphson method is value of is, the greater the velocity. It is embedded as root finding of the non-linear also observed that temperature dependent equations of the corresponding systems of viscous parameter has profound effect on five first-order ordinary differential transverse velocity profiles within the fluid equations. The five initial conditions at domain compare to that of horizontal (  0 ) can now be defined as column velocity profiles. This can be traced to the high value assigned to melting parameter. vector with five elements as From Equation (22), it easy to deduce that melting parameter is a multiple of Guess(Pr G mG );0; G ;0; G .  3 2 1 2  temperature gradient term. It indicates that,

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as magnitude of  increases at a constant in temperature difference of Tm To .

value of b1 , there is a corresponding increase

0 1

0.9 -0.1  = 0  = 0 )  = 1

 0.8 -0.2  = 1

)  = 2

 = 2  0.7

-0.3  = 3 (

 = 3 / 0.6  = 4 -0.4  = 4 0.5  varies,  = 0.5,  = 0.5,  varies,  = 0.5,  = 0.5, -0.5 S = 0.2, N = 2, M = 0.5, P = 0.71, St = 0.2, N = 2, M = 0.5, Pr = 0.71, t r 0.4  = 0.25, n = 1, m = 4 -0.6  = 0.25, n = 1, m = 4 0.3 -0.7 Velocity profiles f 0.2

Hence, velocity Transverse profiles f ( the increase in temperature weakens illustrated in Figure 4. It is observed that the -0.8 the intermolecular forces that hold the shear 0.1stress profile decreases near freestream -0.9 0 molecules0 0.5 of UCM1 1.5 so tight.2 2.5 In view3 3.5 of this,4 and increases0 0.5 near1 1.5 the 2 melting2.5 3 surface.3.5 4 the dynamic viscosity  is gradually reduced Physically, this corresponding effect and corresponds to increase in velocity as (decrease in drag) as shown in Figure 4 can shownFigure in 2 : FiguresIncreasing 2 effect and 3.of temperature It is further be tracedFigure to 3 :the Increasing fact that effectthe stretching of temperature of the dependentobserved variable in Figure fluid 3 viscosity that increases parameter in over the freestreamdependent layer variable together fluid with viscosity the highparameter heat transversemagnitude velocity of temperature profiles of dependent Upper Convected viscous energyover which transverse exists velocitynear  profiles4. This of result Upper is parameter haveMaxwell negligible fluid effect on velocity realistic dueConvected to the factMaxwell that fluidS  0.2 .To profiles near the free stream. Physically, the t further unravel the dynamics of UCM fluid temperature of UCM fluid within few layers flow over melting surface, the effect of near freestream is almost the same; this account for the negligible increasing effect melting parameter (m) and temperature of over velocity profiles. In such a dependent variable fluid viscosity parameter () over local skin friction coefficient situation, the flow velocity approaches to the // possible maximum value. f (0) which is proportional to local heat In the presence of thermal radiation and high transfer rate is shown in Figure 5. magnitude of melting at the wall, the effect of temperature dependent variable fluid viscosity () over shear stress profiles is

2 0.07    1.8 varies, = 0.5, = 0.5, 0.06 S = 0.2, N = 2, M = 0.5, P = 0.71,  = 0 t r 1.6  = 0.25, n = 1, m varies  = 0  = 1

)  = 1  ( 0 ) 0.05

1.4  = 2 //  = 2 ( //  = 3  = 3 1.2  = 4  0.04 1  varies,  = 0.5,  = 0.5, 0.03 St = 0.2, N = 2, M = 0.5, Pr = 0.71, 0.8  = 0.25, n = 1, m = 4

0.6 0.02

Skin Friction Coefficients f Shear stressShear profiles f 0.4 0.01 0.2

 0 0 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Melting parameter (m)  Figure 5: Increasing effect of temperature

Figure 4: Increasing effect of temperature dependent dependent variable fluid viscosity parameter and

variable fluid viscosity parameter over Shear stress melting parameter over Shear stress profiles of

profiles of Upper Convected Maxwell fluid Upper Convected Maxwell fluid at the wall.

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It is observed that the rate of increase in parameters on shear stress profiles near f // (0) with () is significant at higher value freestream is presented in Table 1. of melting parameter. The effect of both Table 1: Increasing effect of temperature dependent variable fluid viscosity parameter and melting parameter (m) over Shear stress profiles f // (  4) of Upper Convected Maxwell fluid near the free stream. when when when when (  0) (  1) (  2) (  3) m  1 0.800497489277775 0.852705981783623 0.915866357976491 0.983437834478091 m 1.5 0.917299421257627 0.953179275701427 1.005858881864090 1.065568882695260 m  2 1.047440672263030 1.061787074416630 1.100999956353010 1.151835743690170 m  2.5 1.198445372440900 1.184445440836170 1.205548863444140 1.246373506726640 m  3 1.377604438847530 1.324973870536900 1.326217958134970 1.353667166929880 m  3.5 1.593361976072640 1.491424281205490 1.460604532294710 1.471820423608440 m  4 1.866834452426350 1.694291592072390 1.622345361664350 1.603310965833320

It is easy to deduce from Table 1 that at each temperature gradient also decreases value of , the coefficient f //(  4) is an significantly within the fluid domain with the increasing function of melting parameter increasing value of the radiation (m) . Physically, the significant increase in parameter N . The effect of radiation parameter ()N is to reduce the temperature the corresponding value of f //(  4) due to significantly in the flow region. Thermal increasing in the magnitude of melting radiation is found to be energy transfer by parameter at a constant value of can be the emission of electromagnetic waves which traced to the influence of substantial heat carry energy away as the fluid flows. Since energy which exists at the free stream. The the increase in radiation parameter means influence of thermal radiation is illustrated in increase in the rate at which heat energy is Figures 6 and 7. It is noticed that an increase released from the flow region; this account in the radiation parameter results in decrease for decrease in temperature distribution as of the temperature profiles of UCM within the magnitude of parameter N increases. the boundary layer. It is also noticed that the

0.8 0.45

0.7 N = 1 N = 1 0.4 N = 2 N = 2 ) 0.6  N = 3

) N = 3

 (

/ N = 4 

( N = 4 0.35  0.5 N = 5 N = 5 N 0.4 0.3  = 0.5,  = 0.5,  = 0.5, St = 0.2, N varies, M = 0.5, Pr = 0.71, 0.3 0.25  = 0.25, n = 1, m = 1 N

Temperature profiles Temperature 0.2  = 0.5,  = 0.5,  = 0.5, 0.2 St = 0.2, N varies, M = 0.5, Pr = 0.71, N 0.1  = 0.25, n = 1, m = 1 N Temperature Gradient profiles Gradient Temperature 0.15 0 0 0.5 1 1.5 2 2.5 3 3.5 4  0.1 Figure 6: Increasing effect of thermal radiation 0 0.5 1 1.5 2 2.5 3 3.5 4  parameter over temperature profile of Upper Convected Maxwell fluid Figure 7: Increasing effect of thermal radiation parameter over temperature gradient profiles

of Upper Convected Maxwell fluid

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