Solutions with Wright functions for time fractional convection flow near a heated vertical plate
- Abdul Shakeel^{1},
- Sohail Ahmad^{2}Email author,
- Hamid Khan^{1} and
- Dumitru Vieru^{3}
https://doi.org/10.1186/s13662-016-0775-9
© Shakeel et al. 2016
Received: 22 October 2015
Accepted: 28 January 2016
Published: 15 February 2016
Abstract
We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum and energy equations are considered as fractional differential equations with respect to the time t. Solutions of the initial-boundary values problem are determined by means of the Laplace transform technique and are represented by means of the Wright functions. The fundamental solution for the temperature field is obtained. This allows obtaining the temperature field for different conditions on the wall temperature. A numerical case is analyzed in order to obtain information regarding the influence of the fractional parameters on the temperature and velocity fields. Some physical aspects of the fluid behavior are presented by graphical illustrations.
Keywords
Unsteady free convection flow viscous fluid time-fractional derivativesMSC
80A20 76A101 Introduction
Fluid convection at vertical plates, resulting from buoyancy forces, has found applications in several industrial and technological fields such as heat exchangers, electronic cooling equipments, aeronautics, and nuclear reactors. Free convection flow occurs not only due to temperature difference, but also due to concentration difference or combination of these two. Many transport processes exist in nature and in industrial applications, in which the simultaneous heat and mass transfer occur as a result of combined buoyancy effects and diffusion of chemical species.
The flows of free convection are common in environmental heat transfer processes. As a result, many investigations have been made by considering a wide range of mechanical and thermal boundary conditions [1]. Using the Laplace transform technique, Soundalgekar [2] has obtained an exact solution to the flow of a viscous fluid past an impulsively started semiinfinite isothermal vertical plate. Effects of heating or cooling of the plate on the flow field were analyzed through Grashof number. Transient free convection flow past an infinite vertical plate has been studied by Ingham [3]. Merkin [4] gave the similarity solutions for the same problem. Seth and Ansari [5] analyzed the natural convection flows past an impulsively moving vertical plate with ramped wall temperature in the presence of thermal diffusion with heat absorption. Narahari and Nayan [6] studied the free convection flow past an impulsively started vertical plate with Newtonian heating in the presence of thermal radiation and mass diffusion. The problem of free convection under the influence of a magnetic field has attracted the interest of researchers in view of its applications in geophysics, astrophysics, petroleum industries, and cooling of nuclear reactors. Georgantopoulos et al. [7] studied the magnetohydrodynamic free convection flow past an impulsively started vertical plate with constant temperature. Raptis and Singh [8] have studied the effect of a uniform transverse magnetic field on the free convection flow of an electrically conducting fluid past an accelerated vertical plate. Tokis [9] have obtained a class of exact solutions of the unsteady free convection flow of an electrically conducting fluid near a moving infinite vertical plate in the presence of uniform transverse magnetic field fixed to the fluid or to the plate. Narahari and Debnath [10] studied the unsteady magnetohydrodynamic free convection flow past an accelerated vertical plate with constant heat flux and heat generation or absorption. Toki and Tokis [11] obtained an elegant exact solution for the unsteady free convection flows on a porous plate with time-dependent heating. In last time, the fractional calculus has become an interesting mathematical method for solution of diverse problems in mathematics, science, and engineering [12–14]. Fractional calculus involves the computation of integrals or derivatives of any real order. This calculus has applications in study of the heat flux, temperature and entropy generation, diffusion phenomena, modeling control systems, viscoelasticity, biology, etc. The advantage of the fractional derivatives in theory of viscoelasticity is that it affords possibilities for obtaining constitutive equations for elastic complex modulus of viscoelastic materials with only few experimentally determined parameters [15]. Debnath [16] obtained solutions for the Stokes and Rayleigh problems for a viscous fluid with time-fractional derivatives. For the same model of fluid, the unsteady Couette flow was analyzed. The well-known time-fractional diffusion equations have been treated in different contexts by many authors [17–20]. Other studies regarding the flow of complex fluids whose governing equations contain time-fractional derivatives can be found in [21–26].
In this paper we investigate the unsteady free convection flow of a Newtonian fluid near a vertical heated plate in the case of the time-fractional derivatives models. Solutions of the temperature and velocity fields are obtained, in the case of the oscillating motion of the vertical plate, by means of the Laplace transform technique. By using the Wright functions and the generalized G-Lorenzo-Hartley functions elegant closed forms for temperature and fluid velocity were obtained. The fundamental solution corresponding to the temperature is also obtained. This allows obtaining the temperature field for different conditions on the wall temperature. Some numerical examples were analyzed. To check the accuracy of the obtained results, we have used the Stehfest algorithm to the inverse Laplace transform [27]. The values found with the analytical solutions and with the Stehfest algorithm are in good agreement. Also, the influence of the fractional parameters on the temperature and velocity is studied.
2 Statement of the problem
For \(\alpha = \gamma = 1\), these coefficients are the Grashof and Prandtl numbers, respectively.
3 Solution of the problem
3.1 Temperature field
3.1.1 Fundamental solution
The expression of the temperature field was determined by the Laplace transform technique. It is useful, however, to determine the fundamental solution \(G_{T}(y,t)\) for the temperature field, namely, solution corresponding to the problem (10), the second condition (11), and the second condition (13), with the Dirichlet boundary condition \(T(0,t) = \delta (t)\), \(\delta (t)\) being the Dirac distribution.
3.1.2 The Nusselt number
3.1.3 Particular case \(\gamma = 1\)
3.2 Velocity field
3.2.1 Skin friction on the wall
4 Numerical results
Comparison of the temperature values at the time \(\pmb{t = 15}\) for \(\pmb{\alpha = 0.75}\)
y | γ = 0.628 | γ = 0.785 | ||
---|---|---|---|---|
T ( y , t ) Eq. ( 20 ) | T ( y , t ) Eq. ( 36 ) | T ( y , t ) Eq. ( 20 ) | T ( y , t ) Eq. ( 36 ) | |
0.0 | 15.000000 | 14.999974 | 15.000000 | 14.999974 |
0.1 | 12.389940 | 12.389940 | 13.161314 | 13.161330 |
0.2 | 10.205969 | 10.205947 | 11.523626 | 11.523640 |
0.3 | 8.384067 | 8.384039 | 10.068354 | 10.068355 |
0.4 | 6.868772 | 6.868765 | 8.778190 | 8.778163 |
0.5 | 5.612234 | 5.612234 | 7.637068 | 7.637067 |
0.6 | 4.573338 | 4.573334 | 6.630126 | 6.630121 |
0.7 | 3.716893 | 3.716888 | 5.743665 | 5.743671 |
0.8 | 2.435875 | 2.435674 | 4.965081 | 4.965072 |
0.9 | 1.964265 | 1.964266 | 4.282854 | 4.282853 |
1.0 | 1.579894 | 1.579892 | 3.686454 | 3.686456 |
Comparison of the velocity values at the time \(\pmb{t = 2}\) , for \(\pmb{\alpha = 0.873}\) and \(\pmb{\gamma = 0.628}\)
5 Concluding remarks
In this paper we studied the unsteady free convection flow of a viscous fluid near a heated vertical plate. Both governing equations, the momentum equation and energy equation, are considered as time-fractional differential equations of order \(\alpha \in (0,1]\) and \(\gamma \in (0,1]\), respectively. Based on the Laplace transform technique and using the Wright functions and G-Lorenzo-Hartley functions, the closed forms of the temperature and velocity fields were determined. The fundamental solution corresponding to the temperature, the Nusselt number, and the friction coefficient on the wall are obtained. Some physical aspects of the fluid behavior were studied by numerical simulation and graphical illustrations. As a general result, it should be noted that for decreasing values of the fractional coefficient α, the fluid velocity and boundary layer thickness are decreasing. The fluid temperature decreases if the values of the fractional coefficient γ decreases and increases for decreasing values of α.
Declarations
Acknowledgements
This work is supported by COMSATS Institute of Information Technology, Attock Campus.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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