Investigation of variable thermo-physical properties of viscoelastic rheology: A Galerkin finite element approach

Investigation of variable thermo-physical properties of viscoelastic rheology: A Galerkin finite element approach Imran Haider Qureshi,1 M. Nawaz,1,a Shafia Rana,1 Umar Nazir,1 and Ali J. Chamkha2,3 1Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan 2Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia 3RAK Research and Innovation Center, American University of Ras Al Khaimah, P.O. Box 10021, Ras Al Khaimah, United Arab Emirate


I. INTRODUCTION
Transport of heat and mass in fluid flows plays a vital role in many processes which frequently occur at industry and engineering. There are many mechanisms for the transport of both heat and mass in fluids flows induced by moving surfaces (moving with constant or variable velocity). Flows induced by elastic sheets have been studied by the several researchers because such flows are encountered in many industrial and engineering applications. Stretching phenomenon in the extrusion of polymers, glass-fiber, paper production, drawing of plastic/elastic films is commonly cited in the literature. Stretching phenomenon also plays a significant role during the production of sheets. Therefore different stretching rates (linear, nonlinear, one dimensional and multidimensional wall stretching rates) are considered. [1][2][3][4][5] During manufacturing processes, fluids with different rheologies are used. Maxwell fluid is one of the viscoelastic fluids and possesses the dual property of both the viscosity and elasticity. Several studies on Maxwell fluids are available. Some recent and relevant investigations are described here. For instance, Nawaz et al. 6 theoretically discussed Hall and ion slip effects on the flow of Maxwellian of plasma over a vertical surface. The effect as discussed in Ref. 6 are studied by Hayat et al. 7 considering non-Newtonian rheology of Jaffrey fluid when Hall and ion slip effects are of considerable order of magnitude. Khan et al. 8 visualized the effects of microorganisms on the stratification in the Maxwellian fluid subjected to magnetic field. Awais et al. 9 examined three dimensional flow of Maxwell fluid over an exponentially stretching surface. The simultaneous effects including chemical reaction on mass transfer in the flow of Maxwell fluid are investigated by Awais et al. 10 The theoretical study on heat transfer in MHD Maxwellian fluid using Cattaneo-Christov heat flux model is carried out by Sleem et al. 11 and noted the significant effect of non-Fourier parameter (relaxation parameter associated with Cattaneo-Charistov model) on the temperature and heat flux.
Experimental and theoretical data shows that the viscosity, the thermal conductivity and the mass diffusion coefficient remain constant under some specific scenarios. Such scenarios are rare. However, there may be some real time physical situations where change in viscosity, thermal conductivity and mass diffusion coefficient are very small and negligible. In such cases assumption of constant physical properties (viscosity thermal conductivity, mass diffusion coefficient) may be considered but in a general sense, the assumption of constant properties is not valid and may lead to erroneous results. Due to this reason, researchers have given mathematical models for variable thermo-physical properties. However these studies are fewer and most of them are restricted to the Newtonian fluid. For example, Lai and kulacki 12 studied the effects of variable viscosity on the flow in the porous medium by considering all the physical properties to be constant except dynamic viscosity. Prasad et al. 13 analyzed heat characteristics in the flow of viscoelastic fluid in the presence of variable viscosity. The effects of variable Prandtl number on the flow of fluid of variable viscosity and variable thermal conductivity are discussed by Singh and Agarwal. 14 Mokhopadhyay and Layek 15 studied the effect of temperature on the viscosity of the fluid but they considered the thermal conductivity of the fluid to be constant. Pop et al. 16 analyzed the effects of variable viscosity on the flow of fluid over a moving of surface. Chaim. 17 analyzed the effects of temperature dependent thermal conductivity of the transfer of heat in the flow over stretching surface. Gary et al. 18 studied stagnation point flow of viscosity varying fluid. Mehta and Sood 19 examined the behavior of the variation of the dynamic viscosity of the fluid in the presence of porous medium. The effect of variable viscosity on the boundary layer flow over a stretching body is studied by Makhopadhyay et al. 20 Aziz 21 investigated the behavior of thermo-physical properties on the transport of heat and mass in three-dimensional flow subjected to magnetic fluid. Sandeep et al. 22 numerically investigated simultaneously the effects of Brownian motion and thermospheres on the transport of heat and mass in non-Newtonian Carreau liquid.
Literature review reveals that no study dealing with variable viscosity, variable thermal conductivity and variable diffusion coefficient in the flow of Maxwell fluid is available yet. This investigation fills this gap in the literature. The governing problems are solved by a powerful numerical technique called FEM and has been used by many researchers working in the field of engineering and science. Some most relevant studies can be mentioned through Refs. 5 and 23-25 and refs. therein. This investigation consists of five sections. Section I contains literature review. Section II is designated for discussion about rheological and thermo-physical properties of the Maxwell fluid. Problem statement and its mathematical formulation is also given in this section. Galerkin finite element derivation is given in Section III. The key observations are given in Section V.

A. Viscoelastic fluid model and its rheology
Serval models for non-Newtonian viscoelastic fluids have been proposed. Maxwell model is a viscoelastic model which exhibits dual property of viscosity and elasticity. Due to this nature, it behaves like solid as well as fluid. The constitutive equations for Maxwell (viscoelastic) fluid are given by 10,11,13,14 where p is the pressure, I is the identity tensor and S is the extra stress tensor defined by where A 1 is the first Rivilon-Ericksen tensor, D Dt is the convective derivative, λ 1 is the relaxation time and µ is the dynamic viscosity. For λ 1 = 0, the constitutive equation (2) reduces to Newtonian case.

B. Variable viscosity, variable thermal conductivity, variable mass diffusion coefficient
Most of the studies consider constant physical properties including viscosity, thermal conductivity and mass conductance. The assumption of constant viscosity, thermal conductivity and mass conductance is valid only in some rare cases. Especially when mass transport occurs in isothermal circumstances, when heat transfer takes place due to high temperature differences, the assumptions of constant viscosity, constant thermal conductivity and constant mass diffusion coefficient is not valid. Therefore, various models for variable viscosity, variable thermal conductivity and variable mass diffusion coefficient are in practice. The most common model exhibits the variation of viscosity, thermal conductivity and mass diffusion coefficient as a function of temperature. Experimental data shows that the viscosity varies as an inverse function of temperature difference T − T ∞ , where T is the temperature of the fluid and T ∞ is the temperature of the ambient fluid. However, thermal conductivity and mass diffusion are linear coefficient functions of temperature. Mathematical models 13 for viscosity and thermal conductivity are • where a = r/µ ∞ , T r = T ∞ − 1/r. Further a and T ∞ are constant and is very small constant. For liquids a > 0 whereas for gasses a < 0. K ∞ and µ ∞ , respectively, are the thermal conductivity and the viscosity of the fluid outside the boundary layer region. Diffusion of mass in liquids is an analogue of heat conduction in solid, liquids. Conduction of heat is governed by Fourier law of heat conduction whereas mass diffusion is governed by Fick's first law. Due to this analogy, mass diffusion coefficient may be consider as a linear function of temperature i.e.
where D ∞ is the mass diffusion coefficient for the solute in the fluid outside the boundary layer region and 1 is very small parameter. It is important to note that for 1 = = 0, D = D ∞ and K = K ∞ , the case of constant thermal conductivity and mass diffusion coefficient.

C. Physical model and coordinate system
Consider diffusion of solute with variable diffusion coefficient in the flow of Maxwell fluid of variable viscosity and variable thermal conductivity (see Eqs. (3) and (4)) over an elastic sheet. The sheet is stretchable and can be stretched with velocity U (x) = U 0 e +(x /L ) . Also consider that the sheet is permeable. Through pores of sheet, fluid can be sucked out or can be injected into the flow regime with velocity V (x) = V o e (x /2L ) As sheet is stretching exponentially, therefore, it is not reasonable to assume the temperature of the sheet to be constant and it seems to be logical to take the temperature of the sheet equal to T (x) = T ∞ + Be bx /L where U 0 and V 0 are the reference velocities L is the reference length and B is the constant which depend on the thermal property of the fluid. Further, V 0 < 0 is the case for which fluid is being sucked out from the flow regime. However, for V 0 > 0, Maxwell fluid can be injected into the flow regime. The concentration of solute at the surface of stretching surface is taken of the form C (x) = C ∞ + Be bx /L . Flow is induced by the stretching of the sheet and is fully developed. The solute of temperature dependent mass diffusion coefficient is being transported by both the convection and diffusion. Flow configuration is shown in Fig. 1.
Above stated scenario suggests the following steady two dimensional flow fields Moving through Eqs. (1)-(6) and using the boundary layer approximations, one gets the following boundary layer equations ∂u ∂x where ρ is the density and c p is the specific heat of the fluid.
No slip assumption suggests the following boundary conditions

D. Dimensional analysis
Since flow is two dimensional, therefore, single steam system exists and has direct relation with velocity components u and . Existence of stream function, reference velocities and reference length suggest the following new variables and reduce the boundary layer Eqs. (7)-(10) and the boundary conditions (11) into the couple non linear boundary value problems ( where k * is the Deborah number, θ r (= 1/r(T − T )) is the fluid viscosity parameter, Pr ∞ and Sc ∞ are Prandtl and Schamidt numbers for the ambient regime. s is suction/injection parameter. It is worth mentioning that = 0 is the case of constant diffusion coefficients (mass diffusion and thermal conductivity). Further θ r → ∞ is the case of constant viscosity. However, θ r = −1 is for liquids. This discussion concludes that 1 = = 0 and θ r → ∞, represents the case of constant thermophysical properties. It is also observed that k * = 0, = 1 = 0 and θ r → ∞, is the case of Newtonian fluid with constant properties.

E. Variable Prandtl and variable Schmidt numbers
According to the definitions of Prandtl and Schamidt numbers Pr = µc p /k and Sc = µ/ρD, here µ, k and D are variables and dependent upon temperature (see Eqs. (3)-(5)). Thus Eqs. (13)-(15) gives the following variable forms of Prandtl and Schamidt numbers. 13 using above expression in Eqs. (14) and (15), we have The Skin friction coefficient C f at the surface is defined by The dimensionless form of heat flux at the surface (Nusselt number) is defined by: The Sherwood number Sh (mass flux at the surface) is defined by where m = −D(1 ∂C/∂y y=0 and Re is the local Reynolds number.

B. Stiffness matrix elements computation
It is important to note that the elements of the stiffness matrix are functions of unknown nodal values. Thus assembly process gives birth to a nonlinear system of algebraic equations which are converted into the linear system of algebraic equation using Picard linearization scheme and the resulting linear system of algebraic equations is solved iteratively by Guass Seidal approach. The numerical experiments are performed to search η max with computational tolerance 10 −6 with 200 elements. Numerical computations showed that η max = 3 (see graphicall representations of velocity, temperature and concentration).

Grid independent study
The unknown nodal values also depend on the number elements (grid size) but this is not desirable. The computed values correspond to the realistic case only when these are grid independent. Therefore, grid independent analysis is performed and is presented in Table I. This table shows that only 200 elements of the computational domain [0, 3] are sufficient for the grid independent solutions.

IV. RESULTS AND DISCUSSION
Extensive simulations are carried out in order to examine the behavior of velocity, temperature and concentration when fluid parameters are varied. Fig. 2 shows the variation of velocity for different  The Fig. 3 also illustrates that the momentum boundary layer thickness for suction case is less than that in the case of injection. The effects of fluid viscosity parameter θ r on velocity is represented in Fig. 4. The effect of elastic nature of fluid on the temperature is given in Fig. 5 for both θ r = −1 and θ r → ∞. It is noted from this display of the temperature curves that the temperature of viscous fluid (k * = 0) is less than that of the visco elastic fluid (k * 0). From this Fig. it is also observed that temperature of the fluid with variable viscosity is less than that of the fluid with constant viscosity. Fig. 5 predicts that thermal boundary layer thickness is an increasing function of Deborah number. It is also clear from Fig. 5 that the thermal boundary layer thickness for viscous fluid is less than that of the Maxwell fluid. Fig. 6. demonstrates the behavior of the temperature of the fluid (for both the cases of the constant viscosity (θ r → ∞) and temperature dependent viscosity (θ r = −1)) by varying suction/injection parameter s. The behavior of temperature and thermal boundary layer thickness with respect to the variation of the power index parameter b for both constant and variable viscosity  is sketched in Fig. 7. As well as temperature and thermal boundary layer thickness decreases when power index parameter b is increased. The effect of Prandtl number Pr on the temperature of the fluid for both the cases θ r = −1 and θ r → ∞ is displayed in Fig. 8. As the Prandtl Pr is the ratio of the momentum diffusion to the thermal diffusion and an increase in Pr corresponds to the decrease in thermal diffusion coefficient. So there is a decrease in the temperature with respect to the Prandtl number and this for both the cases (θ r = −1 and θ r → ∞). The variation of temperature of the fluid for various values of for both the cases θ r = −1 and θ r → ∞ is given in Fig. 9. The = 0 is the case when thermal conductivity of the Maxwell fluid is constant i.e. does not depend upon the temperature. However for 0 is the case when thermal conductivity is variable and is a function of temperature. It is noted that Fig. 9 that the temperature increases when is increased. The effect of fluid viscosity parameter θ r on the dimensionless temperature is given in Fig. 10. This Fig. reflects that the temperature decreases as the fluid viscosity parameter θ r is increased. Eventually, this decrease in temperature causes a decrease in thermal boundary layer thickness. Thus, thermal boundary layer thickness can be controlled through the fluid viscosity parameter θ r . The effect of power index parameter b on the concentration of solute diffusing in the Maxwellian fluid is displayed in Fig. 11. It is clear from this Fig. that the effect of power index parameter b on the concentration for θ r = −1 is less than that of the concentration in the Maxwell fluid when viscosity is not varying with respect to temperature. There is a prominent effect of elasticity on the diffusion phenomenon of the solute in Maxwell fluid for both the cases θ r = −1 and θ r → ∞. This behavior of elasticity on the concentration is represented by the Fig. 12. The effect of suction parameter s on concentration profile is displayed in Fig. 13. It is found that concentration profile decreases when s is increased for both constant and variable viscosity. The variation of dimensionless concentration field under the influence of fluid viscosity parameter θ r is given in Fig. 14 Table II. From this it can be observed that shear stress at the surface increases when Prandtl number is increased whereas heat and mass fluxes are decreasing functions of Prandtl number. As an increase in Prandtl number is due to the decrease in the thermal diffusivity which causes less transfer of heat from the surface into the fluid regime. Therefore, a reduction in rate of heat transfer (when Prandtl number is increased) is observed (see Table II.) As Schmidt Sc number is an analogue of Prandtl number Pr, therefore, Schmidt number Sc has the effects on Sherwood as those of Pr on Nusselt number. The rate of heat and mass transfer when thermal conductivity and mass conductance increases due to the rise in temperature. In qualitative sense, those observations are same for the both cases of constant and temperature dependent viscosity.