Insight into the Natural Convection Flow Through a Vertical Cylinder Using Caputo Time-Fractional Derivatives

Abstract

The variation in temperature distribution with time for the case of the fractional model which models the flow of fluid through a vertical cylinder is considered. This article provides an insight into the natural convective flow of a viscous fluid through a vertical heated cylinder using the fractional differential equation with Caputo derivatives. Analytical solutions for temperature and velocity functions were obtained using Laplace transform and finite Hankel integral transform methods. Stehfest’s algorithm was used to obtain the inverse Laplace transforms. Numerical simulations and graphical illustrations were carried out in order to analyze the influence of the time-fractional derivative on the transport phenomenon. The significant difference between the fractional fluid flow and ordinary fluid at various time ($t$) is unraveled. At the initial time, the flow of fractional fluid is faster than the ordinary fluid.

Introduction

It is worth noticing within the last few decades that a considerable attention has been devoted to exploring the heat and/or mass transfer in the flow of various fluids over cylindrical domains. Considering the relevance of cyclones in the industry, Zhao and Abrahamson [1] stated that the body of this kind of object may be either conical or cylindrical, or made of both geometries. Thereafter, heat transfer through cylindrical objects (i.e. domains) has been pointed out in building machines, materials processing, solar collectors, furnace designs, heat exchangers, storage tanks, and nuclear designs. Sharma et al. [2], Franke and Hutson [3], Roschina et al. [4] and Bairi [5] investigated the convection heat transfer in cylindrical domains by assuming that the density varies according to Boussinesq approximation and the other thermo-physical fluid properties are constant. In Ref. [2] it is revealed that the intensity of heat source decays exponentially with time and concluded in Ref. [3] that increase in heat transfer is based on the heat input required to maintain the inside surface of the cylinder at a constant temperature. According to the report of Morgan [6] in the year 1975 on the natural convection from smooth circular cylinders, it is also revealed that there is a wide dispersion in the results of experiments due to axial heat conduction losses to the supporting structures of the horizontal cylinders and velocity fields by convective fluid movements. Moreover, Lemembre and Petit [7] introduced a numerical investigation of heat transfer of the two-dimensional natural convection laminar flow in cylindrical enclosures of heated lateral walls at uniform heat flux and cooled at the same heat flux at the top surface by insulating the bottom surface. The steady natural convection laminar flow in rectangular domains has been presented by Chen and Humphrey [8]. Kim and Viskanta [9], and Vargas et al. [10] introduced both experimental and numerical investigations to discuss the natural convection steady flow in two-dimensional Newtonian and incompressible fluid-filled enclosures. Makinde and Animasaun [11, 12], Animasaun [13] and Koriko et al. [14] discussed three convection fluid flow modes within the boundary layer. In addition, experimental and numerical investigations have been introduced by Kee et al. [15] for steady flow natural convection of heat generating tritium gas in two-dimensional closed vertical cylinders and spheres with their bounding isothermal walls. For more reports on free convection flow of Cattaneo–Christov fluid, axisymmetric Powell–Eyring fluid, Jeffrey nanofluid, micropolar flow using a modified Boussinesq approximation, and unsteady mixed convection MHD; see Refs. [16,17,18,19,20,21,22,23].

An experiment has been performed by Bohn and Anderson [24] to study the model of heat transfer of natural convection flow between both perpendicular and parallel vertical walls. They considered three-dimensional cubic enclosure filled with water. Moreover, they assumed that the cubic enclosure has isothermal sides and adiabatic top and bottom. The two-dimensional conjugate natural convection laminar flow in an enclosure was introduced numerically by Liaqat and Baytas [25]. In addition, Kuznetsov and Sheremet [26] investigated numerically the two-dimensional conjugate convective-conductive heat transfer in a rectangular enclosure due to the local heat and contaminant sources. Fractional calculus can provide a concise model for the description of the dynamic events. Such a description is important for gaining an understanding of the underlying multiscale processes. The mathematics of fractional calculus has been applied successfully in physics, chemistry, and materials science to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency. In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between thermal or material gradients and the diffusion of heat or ions. Therefore fractional calculus is being applied to build new mathematical models. Suitable information on viscoelastic properties of polymer and elastomers is necessary for an accurate modeling and analysis of structures with dynamic and vibration problems. When experimental data is used to correlate with the fractional order model results, it shows good agreement and has the capability to reproduce the experimental data for given material; see Sasso et al. [27]. Viscoelastic models based on the noninteger order calculus was introduced by Sloninsky [28]. The mechanical responses predicted by such models were found to be consistent with the molecular theory of polymers by Bagley and Torvik [29]. Schiessel and Blumen [30] derived mechanical analogs to fractional derivative elements and models by assembling numerous springs and dashpots (elastic and viscous terms, respectively) in series and parallel. The fractional models of increasing complexity were proposed by Song and Jiang [31] to simulate the rheological behavior of synthetic polymers as well as biological cells and tissues see Djordjevic et al. [32], Heymans [33], and Liu et al. [34]. For more details about mathematical models for fractional derivatives applications; see Chatterjee [35], Pfitzenreiter [36], and Kawada [37]. Finally, the review of the literature reveals that natural convective flow with heat transfer of a viscous fluid through a vertical cylinder using fractional equations with Caputo derivatives is still an open question, hence this study.

Problem Formulation

In this report, unsteady, laminar and incompressible viscous flows through an infinite vertical cylinder of the radius are considered. Here, the $z$-axis is taken along the axis of the cylinder in the vertically upward direction, and the radial coordinate $r$ is taken normal to it. Initially, it is assumed that the cylinder and the fluid are at the same temperature $T_{\infty }$ and the concentration on cylinder surface is $C_{\infty }$.

At time $t > 0$, the temperature and concentration on the cylinder surface are raised to $T_{w}$ and $C_{w}$, respectively (see Fig. 1). Following Deka et al. [38] and using Boussinesq’s approximation, the governing equation is of the form

$$ \frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{{g\beta_{T} }}{\nu }(T - T_{\infty } ) + \frac{{g\beta_{C} }}{\nu }(C - C_{\infty } ) = \frac{1}{\nu }\frac{\partial u}{\partial t};\quad r \in (0,r_{0} ),\quad {\text{t}} > 0, $$

(1)

$$ \frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r} - \frac{1}{{\alpha_{0} }}\frac{\partial T}{\partial t} = 0;\quad r \in (0,r_{0} ),\quad {\text{t}} > 0, $$

(2)

$$ \frac{{\partial^{2} C}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial C}{\partial r} - \frac{1}{D}\frac{\partial C}{\partial t} = 0;\quad r \in (0,r_{0} ),\quad {\text{t}} > 0, $$

(3)

The appropriate initial and boundary conditions are:

$$ \begin{aligned} & u = 0,\quad T = T_{\infty } ,\quad C = C_{\infty } ;\quad r \in [0,r_{0} ],\quad t = 0, \\ & u = 0,\quad T = T_{w} ,\quad C = C_{w} ;\quad r = r_{0} ,\quad t > 0. \\ \end{aligned} $$

(4)

where $\alpha_{0}$ is the thermal diffusivity and $D$ is the mass diffusivity coefficient. Introducing the following dimensionless variables:

$$ \begin{aligned} & t^{*} = \frac{t\nu }{{r_{0}^{2} }},\quad r^{*} = \frac{r}{{r_{0} }},\quad u^{*} = \frac{{ur_{0} }}{\nu },\quad T^{ * } = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\quad C^{ * } = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }},\quad \Pr = \frac{\nu }{{\alpha_{0} }},\quad Sc = \frac{\nu }{D}, \\ & Gr = \frac{{g\beta_{T} (T_{w} - T_{\infty } )r_{0}^{3} }}{{\nu^{2} }},\quad Gm = \frac{{g\beta_{C} (C_{w} - C_{\infty } )r_{0}^{3} }}{{\nu^{2} }}, \\ \end{aligned} $$

(5)

where ${ \Pr }$ is the Prandtl number, $Sc$ is the Schmidt number, $Gr$ is the thermal Grashof number and $Gm$ is the solutal Grashof number. The governing Eqs. (1)–(4) reduce to

$$ \frac{{\partial u\left( {r,t} \right)}}{\partial t} = \frac{{\partial^{2} u\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial u\left( {r,t} \right)}}{\partial r} + GrT\left( {r,t} \right) + GmC\left( {r,t} \right);\quad r \in (0,1),\quad {\text{t}} > 0, $$

(6)

$$ \frac{{\partial T\left( {r,t} \right)}}{\partial t} = \frac{1}{\Pr }\left( {\frac{{\partial^{2} T\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial T\left( {r,t} \right)}}{\partial r}} \right);\quad r \in (0,1),{\text{t}} > 0, $$

(7)

$$ \frac{{\partial C\left( {r,t} \right)}}{\partial t} = \frac{1}{Sc}\left( {\frac{{\partial^{2} C\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial C\left( {r,t} \right)}}{\partial r}} \right);\quad r \in (0,1),\quad {\text{t}} > 0, $$

(8)

$$ u\left( {r,0} \right) = 0,\quad T(r,0) = 0\quad C(r,0) = 0;\quad r \in [0,1], $$

(9)

$$ u\left( {1,t} \right) = 0;\quad T(1,t) = 1,\quad C(1,t) = 1,\quad t > 0. $$

(10)

In order to develop a fractional model following Lorenzo and Hartley [39], the ordinary time-derivative is replaced with Caputo derivative of fractional order $\alpha$ for $0 < \alpha < 1$. This leads to

$$ D_{t}^{\alpha } u\left( {r,t} \right) = \frac{{\partial^{2} u\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial u\left( {r,t} \right)}}{\partial r} + GrT\left( {r,t} \right) + GmC\left( {r,t} \right);\quad r \in (0,1),\quad {\text{t}} > 0, $$

(11)

$$ D_{t}^{\alpha } T\left( {r,t} \right) = \frac{1}{\Pr }\left( {\frac{{\partial^{2} T\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial T\left( {r,t} \right)}}{\partial r}} \right);\quad r \in (0,1),\quad {\text{t}} > 0, $$

(12)

$$ D_{t}^{\alpha } C\left( {r,t} \right) = \frac{1}{Sc}\left( {\frac{{\partial^{2} C\left( {r,t} \right)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial C\left( {r,t} \right)}}{\partial r}} \right);\quad r \in (0,1),\quad {\text{t}} > 0, $$

(13)

where $D_{t}^{\alpha } u(y,t)$—the Caputo derivative of fractional order of the function $u(r,t)$ is given as:

$$ D_{t}^{\alpha } u(r,t) = \left\{ {\begin{array}{*{20}l} {\frac{1}{\varGamma (1 - \alpha )}\int\limits_{0}^{t} {\frac{{u^{\prime}(r,\tau )}}{{(t - \tau )^{\alpha } }}} d\tau } \hfill & {\quad 0 \le \alpha < 1} \hfill \\ {u^{\prime}(r,t),} \hfill & {\quad \alpha = 1} \hfill \\ \end{array} } \right., $$

(14)

where $\varGamma ( \cdot )$ is the Gamma function.

Fig. 1

The physical model

Method of Solution

Applying the Laplace transform to Eq. (12) and using the initial and boundary conditions Eqs. (9) and (10), leads to the following transformed problem:

$$ q^{\alpha } \bar{T}(r,q) = \frac{1}{\Pr }\left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r}} \right)\bar{T}(r,q), $$

(15)

$$ \bar{T}(1,q) = \frac{1}{q}, $$

(16)

where $\bar{T}(r,\,q)$ is the Laplace transform of the function $T(r,\,t)$ and $q$ is the transform variable. Applying the finite Hankel transform of order zero to Eq. (15) and using condition Eq. (16). This leads to

$$ \overline{T}_{H} (r_{n} ,q) = \frac{{r_{n} J_{1} (r_{n} )}}{\Pr }\frac{1}{{q\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{\Pr }} \right)}}, $$

(17)

where $\bar{T}_{H} (r_{n} ,q) = \int\nolimits_{0}^{1} {r\bar{T}(r,q)J_{0} (rr_{n} )dr}$ is the finite Hankel transform of the function $\bar{T}(r,\,q),$ $r_{n} ,n = 0,1, \ldots$ are the positive roots of the equation $J_{0} (x) = 0$, $J_{0}$ being the Bessel function of first kind and zero order. Equation (17) can be written in the equivalent form as

$$ \bar{T}(r_{n} ,q) = \frac{{J_{1} (r_{n} )}}{{r_{n} }}\frac{1}{q} - \frac{{J_{1} (r_{n} )}}{{r_{n} }}\frac{{q^{\alpha - 1} }}{{q^{\alpha } + \frac{{r_{n}^{2} }}{\Pr }}}. $$

(18)

Taking the inverse Laplace transform of Eq. (18), leads to

$$ T_{H} (r_{n} ,t) = \frac{{J_{1} (r_{n} )}}{{r_{n} }} - \frac{{J_{1} (r_{n} )}}{{r_{n} }}R_{\alpha ,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{\Pr }} \right), $$

(19)

where $R_{\sigma ,\nu } (t,a) = L^{ - 1} \left\{ {\frac{{q^{\nu } }}{{q^{\sigma } - a}}} \right\} = \sum\nolimits_{m = 0}^{\infty } {\frac{{a^{m} t^{(m + 1)\sigma - \nu - 1} }}{\varGamma [(m + 1)\sigma - \nu ]};\quad \text{Re} (\sigma - \nu ) > 0} ,$ is the Lorenzo–Hartley’s function [38]. Taking the inverse Hankel transform. Leads to

$$ T(r,\,t) = 1 - 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{r_{n} J_{1} (r_{n} )}}R_{\alpha ,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{\Pr }} \right)} ;\quad 0 < \alpha < 1. $$

(20)

For the case when $\alpha = 1,$ easy to deduce that $R_{1,0} \left( {t, - \frac{{r_{n}^{2} }}{\Pr }} \right) = \exp \left( { - \frac{{r_{n}^{2} }}{\Pr }t} \right),$ so the temperature profile become

$$ T(r,t) = 1 - 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{r_{n} J_{1} (r_{n} )}}\exp \left( { - \frac{{r_{n}^{2} }}{\Pr }t} \right)} ;\quad \alpha = 1. $$

(21)

In the same line as the temperature distribution, the species concentration is obtained as

$$ C(r,t) = 1 - 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{r_{n} J_{1} (r_{n} )}}R_{\alpha ,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{Sc}} \right)} ;\quad 0 < \alpha < 1, $$

(22)

$$ C(r,t) = 1 - 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{r_{n} J_{1} (r_{n} )}}\exp \left( { - \frac{{r_{n}^{2} }}{Sc}t} \right)} ;\quad \alpha = 1, $$

(23)

with Laplace and finite Hankel transform

$$ \bar{C}_{H} (r_{n} ,q) = \frac{{r_{n} J_{1} (r_{n} )}}{Sc}\frac{1}{{q\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{Sc}} \right)}}, $$

(24)

that will be used to obtaining the velocity field. Applying the Laplace transform to Eq. (11), using the initial and boundary conditions (9) and (10), leads to

$$ q^{\alpha } \bar{u}(r,q) = \frac{{\partial^{2} \bar{u}(r,q)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial \bar{u}(r,q)}}{\partial r} + Gr\bar{T}(r,q) + Gm\bar{C}(r,q), $$

(25)

$$ \bar{u}(1,q) = 0. $$

(26)

Applying finite Hankel transform to Eq. (25), using boundary condition (26) and Eqs. (17), (24), leads to

$$ \bar{u}_{H} (r_{n} ,q) = \frac{Gr}{\Pr }\frac{{r_{n} J_{1} (r_{n} )}}{{q\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{\Pr }} \right)\left( {q^{\alpha } + r_{n}^{2} } \right)}} + \frac{Gm}{Sc}\frac{{r_{n} J_{1} (r_{n} )}}{{q\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{Sc}} \right)\left( {q^{\alpha } + r_{n}^{2} } \right)}}, $$

(27)

where $\bar{u}_{H} (r_{n} ,\,q) = \int\nolimits_{0}^{1} {r\bar{u}_{H} (r,\,q)J_{0} (rr_{n} )dr}$ is the finite Hankel transform of the function $\bar{u}(r,\,q).$

Equation (27), can be written in the following equivalent form

$$ \bar{u}_{H} (r_{n} ,q) = \frac{{NJ_{1} (r_{n} )}}{{r_{n} }}\frac{1}{{q\left( {q^{\alpha } + r_{n}^{2} } \right)}} - \frac{{GrJ_{1} (r_{n} )}}{{r_{n} }}\frac{{q^{\alpha - 1} }}{{\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{\Pr }} \right)}}\frac{1}{{\left( {q^{\alpha } + r_{n}^{2} } \right)}} - \frac{{GmJ_{1} (r_{n} )}}{{r_{n} }}\frac{{q^{\alpha - 1} }}{{\left( {q^{\alpha } + \frac{{r_{n}^{2} }}{Sc}} \right)}}\frac{1}{{\left( {q^{\alpha } + r_{n}^{2} } \right)}}, $$

(28)

where $N = Gr + Gm$.

Taking inverse Laplace transform of the Eq. (28), leads to

$$ \begin{aligned} u_{H} (r_{n} ,q) & = \frac{{NJ_{1} (r_{n} )}}{{r_{n} }}H(t) * F_{\alpha } (t, - r_{n}^{2} ) - \frac{{GrJ_{1} (r_{n} )}}{{r_{n} }}R_{\alpha ,\,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{\Pr }} \right) * F_{\alpha } (t, - r_{n}^{2} ) \\ & \quad - \frac{{GmJ_{1} (r_{n} )}}{{r_{n} }}R_{\alpha ,\,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{Sc}} \right) * F_{\alpha } (t, - r_{n}^{2} ), \\ \end{aligned} $$

(29)

where $H(t)$ is Heaviside unit step function, $L^{ - 1} \left\{ {\frac{1}{{q^{a} + b}}} \right\} = F_{a} ( - b,t) = \sum\nolimits_{n = 0}^{\infty } {\frac{{( - b)^{n} t^{(n + 1)a - 1} }}{\varGamma [(n + 1)a]}}$ is the Robrtnov and Hartley’s function Ref. [38] and “*” represents the convolution product. Inverting the Hankel transform leads to

$$ \begin{aligned} u(r,t) & = 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{J_{1}^{2} (r_{n} )}}u_{H} (r_{n} ,t),} \\ & = 2\sum\limits_{n = 1}^{\infty } {\frac{{J_{0} (rr_{n} )}}{{r_{n} J_{1} (r_{n} )}}\left[ {\begin{array}{*{20}l} {N\;H(t) * F(t, - r_{n}^{2} ) - GrR_{\alpha ,\,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{\Pr }} \right) * F(t, - r_{n}^{2} )} \\ {\quad - Gm\,R_{\alpha ,\,\alpha - 1} \left( {t, - \frac{{r_{n}^{2} }}{Sc}} \right) * F(t, - r_{n}^{2} ),} \\ \end{array} } \right].} \\ \end{aligned} $$

(30)

In the case $\alpha = 1,F_{1} (t, - a_{0} ) = R_{1,\,0} (t, - a_{0} ) = e^{{ - a_{0} t}}$.

Semi-analytical Solution

Given that the velocity expression Eq. (30) is complicated, therefore, difficult to use in numerical calculations, in this section a semi-analytical solution for the velocity field is presented. It is observed that, the temperature and concentration are solutions of the partial differential equation

$$ k_{0} D_{t}^{\alpha } w(r,t) = \frac{{\partial^{2} w(r,t)}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial w(r,t)}{\partial r}, $$

(31)

along with the initial and boundary conditions

$$ w(r,0) = 0,\,\,\,w(1,t) = 1. $$

(32)

The temperature and concentration solutions are corresponding to $k_{0} = \Pr ,$ and $k_{0} = Sc$, respectively. By applying the Laplace transform with respect to $t$ to Eq. (31), the transformed problem can be written in the equivalent form:

$$ r^{2} \bar{w}^{\prime\prime}(r,q) + r\bar{w}^{\prime}(r,q) - \left( {r\sqrt {k_{0} q^{\alpha } } } \right)^{2} \bar{w}(r,q) = 0, $$

(33)

$$ \bar{w}(1,q) = \frac{1}{q}. $$

(34)

Equation (33) is a modified Bessel equation with the general solution (bounded for $r \in [0,1]$)

$$ \bar{w}(r,q) = C_{1} I_{0} \left( {r\sqrt {k_{0} q^{\alpha } } } \right), $$

where $I_{0} ( \cdot )$ is the modified Bessel function of first kind and order zero. Using the boundary condition (34), the solution of Eqs. (33)–(34) is of the form

$$ \bar{w}(r,q) = \frac{1}{q}\frac{{I_{0} \left( {r\sqrt {k_{0} q^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {k_{0} q^{\alpha } } } \right)}}. $$

(35)

Now, using Eq. (35) into Eq. (25) the transform problem of the velocity function is of the form

$$ q^{\alpha } \bar{u}(r,q) = \frac{{\partial^{2} \bar{u}(r,q)}}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial \bar{u}(r,q)}}{\partial r} + \frac{Gr}{q}\frac{{I_{0} \left( {r\sqrt {\Pr q^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {\Pr q^{\alpha } } } \right)}} + \frac{Gm}{q}\frac{{I_{0} \left( {r\sqrt {Scq^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {Scq^{\alpha } } } \right)}}, $$

(36)

$$ \bar{u}(1,q) = 0. $$

(37)

A particular solution of Eq. (36) is

$$ \bar{u}_{p} (r,q) = \frac{Gr}{1 - \Pr }\frac{{I_{0} \left( {r\sqrt {\Pr q^{\alpha } } } \right)}}{{q^{\alpha + 1} I_{0} \left( {\sqrt {\Pr q^{\alpha } } } \right)}} + \frac{Gm}{1 - Sc}\frac{{I_{0} \left( {r\sqrt {Scq^{\alpha } } } \right)}}{{q^{\alpha + 1} I_{0} \left( {\sqrt {Scq^{\alpha } } } \right)}},\quad \Pr \ne 1,\quad Sc \ne 1, $$

(38)

The general solution bounded for $r \in \left[ {0,\,1} \right]$ is

$$ \bar{u}(r,q) = C_{2} I_{0} \left( {r\sqrt {q^{\alpha } } } \right) + \bar{u}_{p} (r,q). $$

(39)

By using the boundary condition Eq. (37), the Laplace transform of the velocity field is of the form

$$ \bar{u}(r,q) = \frac{1}{{q^{\alpha + 1} }}\left[ {\frac{Gr}{1 - \Pr }\frac{{I_{0} \left( {r\sqrt {\Pr q^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {\Pr q^{\alpha } } } \right)}} + \frac{Gm}{1 - Sc}\frac{{I_{0} \left( {r\sqrt {Scq^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {Scq^{\alpha } } } \right)}} - \left( {\frac{Gr}{1 - \Pr } + \frac{Gm}{1 - Sc}} \right)\frac{{I_{0} \left( {r\sqrt {q^{\alpha } } } \right)}}{{I_{0} \left( {\sqrt {q^{\alpha } } } \right)}}} \right]. $$

(40)

Even if the analytical form of the inverse Laplace transform of function $\bar{u}(r,q)$ can be obtained (using the residue theorem, for example), in this study the Stehfest’s algorithm Refs. [40,41,42] is adopted for numerical inversion of the Laplace transform, namely, the velocity field $u(r,t)$ is approximated by

$$ u(r,t) \approx \frac{\ln (2)}{t}\sum\limits_{j = 1}^{2p} {k_{j} \bar{u}\left( {r,j\frac{\ln (2)}{t}} \right),} $$

(41)

where $k_{j} = \left( { - 1} \right)^{j + p} \sum\nolimits_{{i = \left[ {\frac{j + 1}{2}} \right]}}^{\text{min} (j,p)} {\frac{{i^{p} (2i)!}}{(p - i)!i!(i - 1)!(j - i)!(2i - j)!},}$ $p$ is a positive integer number, $\left[ {\frac{j + 1}{2}} \right]$ is the integer part of the real number $\frac{j + 1}{2}$ and $\hbox{min} (j,p) = \frac{1}{2}\left[ {i + j - \left| {i - j} \right|} \right]$. In order to verify the numerical results, the values obtaining with Eqs. (30) and (41) are presented in Table 1. It is observed that there is a good agreement between results obtained with both formulas. Also, the absolute errors for different numbers of terms in the series solution are given in Table 2.

Table 1 Numerical values of the Eqs. (30) and (41) at $\alpha = 0.5,\Pr = 2,Sc = 1.5,Gr = 1.4$ and $Gm = 2.5$ for different values of time $t$

Table 2 Absolute errors for values of velocity given by Eqs. (30) and (41)

Numerical Results and Discussion

In this paper we have studied convective flows of the unsteady, laminar and incompressible viscous flow through an infinite vertical cylinder with constant temperature and concentration on the surface. The fluid temperature, concentration and velocity fields are obtained as solutions of the fractional differential equations with time-fractional derivatives of Caputo type. Numerical results obtained with the software Mathcad are presented in graphical illustrations. Especially, we studied the influence of the fractional parameter $\alpha$ on the fluid temperature and velocity to compare it with the ordinary case $\alpha = 1$. In Figs. 2 and 3, the graphs of dimensionless temperature versus $r,$ for the variation of fractional parameter $\alpha ,$ and consider the different values of time $t$ are illustrated. Form Fig. 2, it is clear that the temperature of the fluid is decreases by increasing the value of $\alpha$. But the temperature of the fluid is increasing with increasing time $t$ and the temperature difference is decrease and we consider this is a small time. Figure 3 is sketch to study the effect of fractional parameter $\alpha ,$ for larger values of the time t. In this case, the fluid temperature decreases if the fractional parameter decreases. Note that, for small values of the time t the heat transfer decreases at small values of the fractional parameter, but, the heat transfer increases at larger values of the time t and decreasing values of the fractional parameter. These aspects are well highlighted by the curves plotted in Fig. 4. The effect of Prandtl number $\Pr ,$ on dimensionless temperature, for different values of time t is graphically studied in Fig. 5. As expected, for increasing values of the Prandtl number the temperature decreases, because, for large values of Prandtl number, the momentum diffusivity dominates so, the thermal diffusivity is small. The influence of the fractional parameter on the fluid velocity was analyzed in Figs. 6, 7 and 8. At small values of the time t, the velocity of fluids modeled by fractional derivatives is bigger than velocity of ordinary fluid. At larger values of the time t, the fractional fluids flow slower than the ordinary fluid. Comparing the numerical values of velocity, obtained with both solutions, it is found a good agreement (see Table 1). In order to analyze the influence of the time-fractional derivative on the fluid flow parameter, numerical results are extracted from the analytical expressions and are illustrated through graphs. It is found that fluids modeled with the fractional differential equations, behave differently from ordinary fluids. At large values of time, the fractional fluid model flows slower than the ordinary fluid model, while at small values of time, the fractional fluid model flows faster than the ordinary fluid model.

Fig. 2

Profiles of dimensionless temperature versus $r$, for $\alpha$ variation with $\Pr = 3$ and different values of time $t$

Fig. 3

Profiles of dimensionless temperature versus $r$, for $\alpha$ variation with $\Pr = 3$ and different values of time $t$

Fig. 4

Profiles of dimensionless temperature versus $t$, for $\alpha$ variation with $\Pr = 3$ and different values of time $r$

Fig. 5

Profiles of dimensionless temperature versus $r$, for $\Pr$ variation with $\alpha = 0.4$ and different values of time $t$

Fig. 6

Profiles of dimensionless velocity versus $r$, for $\alpha$ variation with $\Pr = 2,\,\,Sc = 1.5,$$Gr = 1.5,Gm = 2.5$, and different values of time $t$

Fig. 7

Profiles of dimensionless velocity versus $r$, for $\alpha$ variation with $\Pr = 2,Sc = 1.5,Gr = 1.5,Gm = 2.5$, and different values of time $t$

Fig. 8

Profiles of dimensionless velocity versus $t$, for $\alpha$ variation with $\Pr = 2,Sc = 1.5,Gr = 1.5,Gm = 2.5$, and different values of time $r$

Conclusion

In this paper we studied the convective flow with heat and mass transfer of a viscous fluid modeled by fractional differential equations with Caputo derivatives, through a vertical cylinder. Closed forms of the analytical solutions for temperature, concentration and velocity fields are obtained using the Laplace transform and finite Hankel transform. These solutions are expressed by the generalized functions R-Lorenzo–Hartley functions and T-Robotnov’s functions. Semi-analytical solutions were obtained by coupling the Laplace transform with Bessel equation. For semi-analytical solutions, the inverse Laplace transform was obtained by the Stehfest’s numerical algorithm. Numerical simulations and graphical illustrations are carried out in order to analyze the influence of the time-fractional derivative on the flow parameters. The significant difference between the fractional fluid flow and ordinary fluid is unraveled. At the initial time, the flow of fractional fluid is faster than the ordinary fluid. The temperature distribution in the flow of fractional fluid ($0.1 \le \alpha < 1$) at the initial time ($0.1 \le t \le 0.3$) is more substantial than that of the ordinary fluid ($\alpha = 1$). Reverse is the case at future values of time $0.7 \le t \le 0.9.$