A scientific report on heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate

This scientific report investigates the heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate with constant wall temperature. The problem is modelled in terms of coupled partial differential equations with initial and boundary conditions. Some suitable non-dimensional variables are introduced in order to transform the governing problem into dimensionless form. The resulting problem is solved via Laplace transform method and exact solutions for velocity, shear stress and temperature are obtained. These solutions are greatly influenced with the variation of embedded parameters which include the Prandtl number and Grashof number for various times. In the absence of free convection, the corresponding solutions representing the mechanical part of velocity reduced to the well known solutions in the literature. The total velocity is presented as a sum of both cosine and sine velocities. The unsteady velocity in each case is arranged in the form of transient and post transient parts. It is found that the post transient parts are independent of time. The solutions corresponding to Newtonian fluids are recovered as a special case and comparison between Newtonian fluid and Maxwell fluid is shown graphically.

examined helices of fractionalized Maxwell fluid whereas Jamil 9 analyzed slip effects on oscillating fractionalized Maxwell fluid. Corina et al. 10 provided a short note on the second problem of Stokes for Maxwell fluids. Zheng et al. 11 , developed exact solutions for generalized Maxwell fluid for oscillatory and constantly accelerating plate motions, Zheng et al. 12 used the same fluid model for heat transfer study due to a hyperbolic sine accelerating plate. Qi and Liu 13 studied some duct flows of a fractional Maxwell fluid. Tripathi 14 applied fractional Maxwell model to study peristaltic transport in uniform tubes.
Fetecau and Fetecau 15 , established a new exact solution for the flow of a Maxwell fluid past an infinite plate. In an other investigation, Fetecau and Fetecau 16 19 , Vieru and Rauf 20 , Vieru and Zafar 21 and Khan et al. 22 . However, in all these investigations, heat transfer analysis was not considered. More exactly, phenomenon of heat transfer due to mixed convection was not incorporated in all the above studies. Therefore, the focal point of this work is to analyze Maxwell fluid over an oscillating vertical plate with constant wall temperature and to establish exact solutions using the Laplace transform method. The obtained results consideration of heat transfer analysis in Maxwell fluid has industrial importance since many problems of physical interest involve heat transfer such as automotive industry (radiator, cooling circuits, lamps), aerospace (de-icing system, cooling systems), in chemical process industry (heat recovery systems, heat exchangers), energy (kilns, boiler, cross flow heat exchangers, solar panels) and home appliance (ovens, household heaters) [23][24][25] .

Mathematical formulation of the problem
Let us consider unsteady mixed convection flow of an incompressible Maxwell fluid over an oscillating vertical flat plate moving with oscillating velocity in its own plane. Initially, at time t = 0, both the fluid and the plate are at rest with constant temperature T ∞ . At time t = 0 + the plate is subjected to sinusoidal oscillations so that the velocity on the wall is given by V = U 0 H(t)cos(ω t), resulting in the induced Maxwell fluid flow. More exactly, the plate begins to oscillate in its plane (y = 0) according to V = U 0 H(t)cos(ω t)i; where the constant U 0 is the amplitude of the motion, H(t) is the unit step function, i is the unit vector in the vertical flow direction and ω is the frequency of oscillation of the plate. At the same time t = 0 + , the temperature of the plate is raised or lowered to a constant value T w . The velocity decays to zero and temperature approaches to a constant value T ∞ , also known as free stream temperature. The equations governing the Maxwell fluid flow related with shear stress and heat transfer due to mixed convection are given by the following partial differential equations: The appropriate initial and boundary conditions are: w 0 Introducing the following non-dimensional quantities: Scientific RepoRts | 7:40147 | DOI: 10.1038/srep40147 with the corresponding initial and boundary conditions:

Solution of the problem
Temperature. Taking Laplace transform of Eqs (8), (10) 2 , (11) 2 and using initial condition (9) 2 , we obtain The solution of the partial differential equation (12) subject to conditions (13) is given as: Taking the inverse Laplace transform and using (A1), we obtain: Pr 2 (15) Velocity field. Taking the Laplace transform of Eqs (6), (10) 1 , (11) 1 and using initial conditions, we obtain Using Eq. (14) in Eq. (16), we have Solve the partial differential Eq. (18), we have: The last equality can be written in equivalent form as: Gr a a q aq a a q a y q q Gr a a q aq a a q a y q  Let    Taking the inverse Laplace of Eq. (25), we obtain Differentiate Eq. (19) with respect to spatial variable y, we obtain u y q y q q q q y q q G r G q q q y q q G r G q q y q 2 2 Put Eq. (30) into Eq. (29), we obtain Applying the inverse Laplace transform to Eqs (31), (32), (33) and (34), we obtain k t a k s ds a a k t a at s k s ds where * represents convolution product and ⋅ k ( ) is defined in Appendix (A3).

Solutions in the absence of Buoyancy force (limiting case)
In this case, when Gr = 0 the solution corresponding to oscillating boundary motion can easily be obtained from Eqs.

Numerical results and discussions
The geometry of the problem is given in Fig. 1. In order to get some physical insight of the results corresponding to oscillating velocity on the boundary, some numerical calculations have been carried out for different values of pertinent parameters that describe the flow characteristics. All physical quantities and profiles are dimensionless. Also all profiles are plotted versus y. Figure 2 presents the temperature profiles for different values of time t and Prandtl number Pr variation. The fluid temperature is a decreasing function with respect to Prandtl number Pr and tends to a steady state slowly as the time t increases. Figure 3 presents the velocity profiles for different values of time t and Grashof number Gr variation. For other constant we have λ = 0.7, ω = 2, Pr = 5. It is observed that the fluid velocity is increased by increasing the Grashof number Gr. By increasing the time t the difference between the velocities as well as the steady state increases. Figure 4 presents the velocity profiles for different values of time t and Prandtl number Pr variation. For other constants, we have λ = 0.7, ω = 2, Gr = 5. It is observed that the fluid velocity decreases by increasing the Prandtl number Pr. By increasing the time t, the difference between the velocities as well as the steady state increases. Figure 5 presents the shear stress profiles for different values of time t and Grashof number Gr variation. For other constants, we have λ ω = . = = . 0 5, 2, Pr 0 3. It is observed that near the boundary the shear stress increases by increasing the Grashof number Gr but after some critical value of y the shear stress is decreased by increasing Gr. By increasing the time t the critical value of y is increased it means that the critical point is far from the boundary. Figure 4 presents the shear stress profiles for different values of time and Prandtl number Pr variation. For other constants we have λ = 0.3, ω = 2, Gr = 10. It is observed that the region near the boundary, the shear stress is decreased by increasing the Prandtl number Pr. By increasing the time t, Fig. 6 has the same behavior like Fig. 4. A comparison between Maxwell fluid and Newtonian fluid is shown graphically in Fig. 7.

Conclusions
This study reports the first exact solution for unsteady mixed convection problem of Maxwell fluid via Laplace transform method. Expressions of velocity, shear stress and temperature are obtained and then plotted graphically for various embedded parameters. The solution corresponding to Newtonian fluid problem is recovered as a special case. Moreover, it is found that in the absence of free convection term, the already published results can be recovered as a special case. From the plotted results, it is found that temperature decreases with increasing Prandtl number; however, for large timethe temperature decays later. Velocity decreases with increasing Prandtl number whereas an oscillating behavior is observed for Grashof number.