UNSTEADY BOUNDARY LAYER FLOW AND HEAT TRANSFER OF MAXWELL VISCOELASTIC FLUID WITH TIME FRACTIONAL CATTANEO-CHRISTOV HEAT FLUX MODEL

The time fractional Cattaneo-Christov flux heat model is first introduced to investigate the flow and heat transfer of Maxwell viscoelastic fluid past a vertical flat plate. Fractional constitutive relation and Cattaneo-Christov heat flux model are applied to construct the governing boundary layer equations of momentum and energy, which are nondimensionalized by new dimensionless variables and solved numerically. The results indicate that there exist intersections on velocity and temperature profiles for different values of Prandtl number when the fractional Cattaneo-Christov flux heat model is considered.


INTRODUCTION
The study on heat transfer has attracted a considerable attention due to its widespread existence in many fields. The Fourier heat flux law (Grattan-Guinness, 2005) has been applied to investigate the features of heat transfer in the last two centuries. However, it leads to the paradox of infinite speed of propagation. In order to overcome this drawback, Cattaneo (2011) proposed a modified Fourier heat conduction law by adding a relaxation time term. So the diffusion equation is turned from a parabolic equation to a hyperbolic one, but the relation only involves partial time derivative. Recently Christov (2009) proposed an extension for the Cattaneo's law by using Oldroyd's upper-convected derivative, which successfully preserves the material-invariant formulation. This Cattaneo-Christov heat flux law is given by in which ξ, V, k and T represent the relaxation parameter, velocity vector, thermal conductivity and temperature respectively. The Cattaneo-Christov heat flux model has been employed to predict the heat transport behavior under different mechanical and thermal boundary conditions (Straughan, 2010;Waqas et al., 2016;Han et al., 2014;Sui et al., 2016). Hayat et al., (2017) have investigated the heat and mass transfer of the boundary-layer flow of Burgers nanofluid with Cattaneo-Christov double diffusion. Also, they have considered the 3D flow of Prandtl liquid by employing Cattaneo-Christov double diffusion models (Hayat et al., 2018). Viscoelastic fluids have gained tremendous attention of researchers Hayat et al., 2017) due to their wide application in different fields of engineering and industry, such as composite manufacturing process, polymer melts and solutions, tissue engineering and enhanced oil recovery. Constitutive equations with fractional derivatives have long played an important role in the description of complex dynamics in viscoelastic fluids (Song et al., 2000;Tan and Xu, 2002) as the fractional derivative is flexible. The Maxwell fluid is an important class of viscoelastic fluids, and the constitutive relation of Maxwell viscoelastic fluid written in terms of the fractional calculus has been shown to be consistent with thermodynamic principles (Friedrich, 1991), which is introduced as where λ is the relaxation time of heat conduction, α (0<α≤1) is the velocity fractional derivative parameter, σxy is the shear stress component, μ is the kinematic viscosity, ∂ α /∂t α is the Caputo fractional derivative operator and the fractional derivative of order α is defined as (Podlubny, 1999): where Γ(·) is the Gamma function. The fractional derivative is a global operator reflecting memory character (Du et al.,2012), which has been verified effective in different areas. The study for the application of fractional derivative operator has attracted much interest in recent years (Chen et al., 2013;Jiang and Qi, 2012;Yu et al., 2015). Ghazizadeh et al. (2010) studied the numerical solution of fractional order Cattaneo equation for describing anomalous diffusion. Fetecau et al. (2009) determined the velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Maxwell fluid by using Fourier sine and Laplace transforms. Tripathi et al. (2010) presents the transportation of viscoelastic fluid with fractional Maxwell model through a channel.
In the study of heat conduction processes, fractional calculus theory has also been applied to anomalous heat conduction owing to the

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org nonlocal nature of fractional operators. Povstenko (2009) formulated the theory of thermal stresses by the generalized Cattaneo-type equations with Caputo time fractional derivatives. Liu et al. (2016) proposed an improved constitutive model in which the space Riesz fractional Cattaneo-Christov model is used to characterize heat conduction phenomena, and then they (Liu et al., 2017a) presented a new time and space fractional Cattaneo-Christov upper-convective derivative flux heat conduction model where the space fractional derivative is characterized by the weight coefficient of forward versus backward transition probability. In the two papers，the velocity is considered as a constant for simplicity so they only dealt with the energy equation and presented the impacts of fractional parameters evolution on heat transfer characteristics.
Motivated by above discussions, we investigate the flow and heat transfer of Maxwell viscoelastic past a vertical flat plate with the time fractional Cattaneo-Christov heat flux, which can be rewritten as (Liu et al., 2017b) where τ is introduced to keep the dimension of constitutive equation balance and its dimension is "s", β (0<β≤1) is the velocity fractional derivative parameter, the symbol ∂ β /∂t β is the Caputo's time fractional derivative of order β. By the new dimensionless variables, the nonlinear governing equations with mixed time-space derivatives are nondimensionalized and solved numerically. The effects of embedded parameters on velocity and temperature profiles are presented graphically and analyzed in detail.

MATHEMATICAL FORMULATION
Consider two-dimensional unsteady boundary layer flow and heat transfer of Maxwell viscoelastic fluid past a vertical plate. The Cartesian coordinate system is considered in a way that the x-axis is along the plate and y-axis is perpendicular to the plate. It is also presumed that Tw is the temperature of the plate and the ambient temperature corresponds to T∞. The diagram of the physical model is described in Fig. 1.
where u, v and T are the velocity components and temperature respectively, ρ is the density of fluid, βT is the volumetric thermal expansion coefficient. It is worth noting that when α=0 the model is simplified as the classical Newtonian fluid, while α=1 is corresponding to the ordinary Maxwell model. Combining time fractional Cattaneo-Christov flux (4) with the following energy conservation equation (Povstenko, 2011) the fractional boundary layer energy equation can be obtained as where αf=k/cρ is the thermal diffusion coefficient, c is the specific heat capacity. By setting ξ=0, Eq. (8) reduces to the classical heat conduction model. The initial and boundary conditions are given as follows: In order to simplify the study, the dimensionless variables are introduced as follows: where L is the length of the vertical plate, Gr and Pr are the Grashof number and Prandtl number respectively. By omitting the dimensionless mart * for simplicity, the dimensionless governing equations can be obtained as below: The corresponding non-dimensional initial and boundary conditions become: , , 0, , , 0, as .

Iteration algorithm
According to the initial condition, we can acquire the values of u, v and θ in the specific domain at t=0. The variables of (k-1)-level are regarded as constants. The iteration equations can be written as tri-diagonal system of equations, then their solutions can be obtained by the Tomas algorithm (Carnahan et al., 1969). The values of (i-1)-level only influence the right side of the linear equations. When the absolute values of the difference between velocity u and temperature θ at all nodes within two consecutive time steps are less than 10 -   The results indicate that the numerical solutions in the paper are consistent with the previous published data.

RESULTS AND DISCUSSION
The present section aims to study the impacts of fractional derivative parameters α and β on the dimensionless velocity u and temperature θ. Moreover, we show the influence of Pr on velocity and temperature at both cases of classical heat flux ξ=0 and fractional Cattaneo-Christov heat flux model. Figures. 3 and 4 present the effects of fractional parameters α and β on the velocity distributions. It is shown from Fig. 3 that the maximum value of velocity profile reduces with α while the position of maximum value gets closer to the vertical plate. The thickness of the momentum boundary layer increases with α slightly, which indicates that the velocity fractional derivative parameter weakens the natural convection flow and boosts the elastic effect of Maxwell fluid. It is also worth noting that the velocity profiles intersect each other for different values of α, which implies that the fractional equation with relaxation times shows short-term memory for the previous moment and try to go back to the previous state. Unlike the influence of α on velocity, the maximum values of velocity remain mostly unchanged with different values of β as show in Fig. 4. The temperature fractional derivative parameter β has almost no influence on the velocity profiles near the vertical plate. However, the velocity increases remarkably with the increase of β .  The influences of α and β on the temperature distributions are illustrated in Figs. 5 and 6. It is shown in Fig. 5 that the temperature profiles rise with the increase of α. It is observed that the classical Newtonian fluid has the thinnest boundary layer, while the ordinary Maxwell fluid is corresponding to the thickest thermal boundary layer. The temperature distribution rises and the thermal boundary layer is thicker for larger values of α. On the contrary, the temperature profiles decline with the increase of β as depicted in Fig. 6, which demonstrate a loss of the thickness of thermal boundary layer. These results indicate that the temperature frictional parameter β enhances the efficiency of heat transfer.  The temperature declines and the thermal boundary layer becomes thinner as β increase, which implies that the temperature fractional parameter enhances the efficiency of heat transfer • The influence of fractional derivative parameter α on temperature distributions is opposite to β.
x y t x y t x y t x y t uv u x y t v x y t x y x y x y (13), the fractional derivatives become: where the truncation error is Ο(Δt 2-β +Δx) .
The iteration equations are achieved in the following forms: