1 Introduction

Over the years, the study of transient free convective flow of viscous incompressible fluid in a vertical infinite concentric annulus has been investigated substantially by many researchers due to its industrial and technological applications such as in nuclear reactor cooling system, rapid cooling process, motion of fluid in computer equipment, radiators, storage devices, cooling of electronic equipment inside computers, furnaces, the growth of crystals for semiconductor industry most importantly when the convection current is induced by internal heat generation/absorption in addition to the heating at the outer surface of the inner cylinder. Many research work has been committed to study the effect of heat generation/absorption on convective flow in a channel, cylinder as well as annulus. In an attempt to generally examine the natural convective flow of a heat generating/absorbing fluid in an annulus (Jha and Yusuf 2017a) presented semi-analytical solution of transient natural convective flow of heat generation/absorption fluid in an annular porous medium. Later on, Yusuf and Jha (2018) investigated fully developed transient laminar natural convective flow of viscous, incompressible and heat generating/absorbing fluid in vertical concentric cylinders filled with porous material where the outer surface of the inner cylinder is heated constantly while the inner surface of the outer cylinder is maintained at constant temperature. In their work, it was observed that the effect of heat generation/absorption can be controlled by increasing/decreasing the viscosity ratio and there is an indication that heat generating fluid is desirable for optimum mass flux in the annular gap most importantly when the convection current is enhanced by constant heat flux. In another article, Yusuf (2017) analytically examined the laminar fully developed natural convective flow of an electrically conducting fluid in the vertical concentric annulus when the inner cylinder is heated or cooled either isothermally or at a constant heat flux while the inner surface of the outer cylinder is maintained at ambient temperature, (Al-Nimr and Darabseh 1995) presented the closed forms on transient fully developed free convection solutions, corresponding to four fundamental thermal boundary conditions in vertical concentric annulus. Recently, (Jha and Babatunde 2018) studied the natural convective flow with internal heat generation/absorption and found interesting results that skin friction and rate of heat transfer at the outer surface of the inner cylinder and inner surface of outer cylinder increases with increase in heat generation parameter while the reverse trend is found in the case of heat absorption parameter. In another related work, (Jha and Yusuf 2016) semi-analytically investigated the transient fully developed natural convection in an annular gap when the fluid is uniformly saturated with porous material. Similarly, (Jha and Odengle 2015) examined unsteady Couette flow in a composite channel partially filled with porous material. In order to establish the accuracy of the Riemann-sum approach used in their study, a numerical comparison with the implicit finite difference approach was presented with the aid of a table. In recent past, (Jha and Ajibade 2009) examined the influence of periodic heat input in fluid flowing in a vertical porous plate and later on considered the unsteady natural convective Couette flow of heat generating/absorbing fluid, (Jha and Ajibade 2010).

(Rachedi and Korti 2017) presented a numerical simulation of melting and solidification of different kinds of phase change material (PCM) encapsulated in spherical nodules in a water flow. The results indicate that natural convection dominates heat transfer during the charging mode while the solidification phase is conduction dominated in the discharging mode. Using numerical shooting technique with a fourth–fifth order Runge–Kutta scheme, (Eid 2016) carried out an investigation to study a problem of the chemical reaction and heat generation or absorption effects on MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet. It was found that Nusselt number is a decreasing function of the heat generation/absorption parameter and the chemical reaction parameter. Later, (Eid and Mahny 2017) examined the combined effects of magnetic field and heat generation/absorption on unsteady boundary-layer convective heat and mass transfer of a non-Newtonian nanofluid over a permeable stretching wall and concluded that the thermal and concentration boundary-layer thickness has higher values with the increasing of magnetic field and heat generation in the case of a pseudoplastic nanofluid than others. (Eid et al. 2017) studied the characteristics of heat transfer of gold nanoparticles (Au-NPs) in flow past a power-law stretching surface. Sisko bio-nanofluid flow (with blood as a base fluid) in existence of non-linear thermal radiation. The study revealed that a better heat transfer is achieved as a result of the thermal radiation accretion which leads to deficiency in the thermal boundary-layer thickness. Finally, some remarkable problems studied by different researchers in this field can be seen in Eid and Mahny (2017), Inman (1962), Mahmoudi (2015), Jha and Yusuf (2017b), Jha et al. (2016), Moalem (1976) and Miyatake and Fujii (1972).

Hence, the purpose of this paper is to study the case when the fluid considered in the work of Jha and Yusuf (2017b) is filled with porous material. Retardation due to Lorentz force opposing fluid flow is negligible which is therefore neglected. In addition, for effective utilization of the available heat supplied to the system, the inner surface of the outer cylinder is insulated against heat this is to retain heat within the system.

2 Mathematical analysis

Consider a fully developed transient natural convective flow of an incompressible, viscous and heat generating/absorbing fluid within a vertical concentric annulus of infinite length. The convective current is as a result of the constant heating at the outer surface of the inner cylinder as well as the heat generating/absorbing nature of the fluid. The z′-axis is taken along the axis of the cylinder in the vertically upward direction opposite the gravitational force. The radius of inner and outer cylinders are denoted by \( r_{1} \) and \( r_{2} \) respectively as depicted in Fig. 1. Initially, at time \( t^{{\prime }} \le 0 \) the fluid and the two cylinders were assumed be at ambient temperature \( T_{0} \). At \( t^{{\prime }} > 0 \), the temperature at the outer surface of the inner cylinder changes to \( T_{w} \) while the outer surface of the outer cylinder is thermally insulated. The flow is thereby prompted by the buoyancy force arising from the temperature difference of the fluid in the annulus. Thus, under the usual Boussinesq approximation, the dimensionless form of the governing equations for the model under consideration can be written as

$$ \frac{\partial U}{\partial t} = \gamma \left[ {\frac{{\partial^{2} U}}{{\partial R^{2} }} + \frac{1}{R}\frac{\partial U}{\partial R}} \right] - \frac{U}{Da} + \theta $$
(1)
$$ \frac{\partial \theta }{\partial t} = \frac{1}{Pr}\left[ {\frac{{\partial^{2} \theta }}{{\partial R^{2} }} + \frac{1}{R}\frac{\partial \theta }{\partial R}} \right] + \frac{H\theta }{Pr} $$
(2)

The above equations have been rendered dimensionless using the following non-dimensional quantities:

$$ \begin{aligned} &t = \frac{{t^{{\prime }} \nu }}{{ r_{1}^{2} }},\quad R = \frac{{r^{{\prime }} }}{{r_{1} }} \lambda = \frac{{r_{2} }}{{r_{1} }}\quad \theta = \frac{{\left( {T^{{\prime }} - T_{0} } \right)}}{{T_{w} - T_{0} }},\quad Pr = \frac{{\mu c_{p} }}{ k} \hfill \\ &\gamma = \frac{{V_{eff} }}{v}, \quad Da = \frac{K}{{r_{1}^{2} }},\quad U = \frac{{U^{{\prime }} }}{{U_{0} }},\quad U_{0} = \frac{{g\beta \left( {Tw - T_{0} } \right)r_{1}^{2} }}{\nu },\quad H = \frac{{Q_{0} r_{1}^{2} }}{k} \hfill \\ \end{aligned} $$
$$ \gamma = \frac{{V_{eff} }}{v}, Da = \frac{K}{{r_{1}^{2} }} , U = \frac{U'}{{U_{0} }} , U_{0} = \frac{{g\beta \left( {Tw - T_{0} } \right)r_{1}^{2} }}{\nu },\quad H = \frac{{Q_{0} r_{1}^{2} }}{k} $$
(3)

While the initial and boundary conditions in dimensionless form are given as:

$$ t \le 0 , U = \theta = 0 , 1 \le R \le \lambda $$
(4)
$$ t > 0:\left\{ {\begin{array}{*{20}l} {U = 0,\;\theta = 1\quad at\;R = 1} \hfill \\ {U = 0,\;\frac{\partial \theta }{\partial R} = 0\quad at\;R = \lambda } \hfill \\ \end{array} } \right. $$
(5)

Using \( \bar{U}(R,\eta ) = \int_{0}^{\infty } {U(R,t)e^{ - \eta t} dt} \) and \( \bar{\theta }(R,\eta ) = \int_{0}^{\infty } {\theta (R,t)e^{ - \eta t} dt} \) where \( \eta \) is the Laplace parameter and \( \eta > 0 \).

Fig. 1
figure 1

Schematic diagram of the problem

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The Laplace transforms of Eqs. (1) to (5) are given as

$$ \frac{{d^{2} \bar{U}}}{{dR^{2} }} + \frac{1}{R}\frac{{d\bar{U}}}{dR} - \frac{1}{\gamma }\left[ {\frac{1}{Da} + \eta } \right]\bar{U} + \frac{{\bar{\theta }}}{\gamma } = 0 $$
(6)
$$ \frac{{d^{2} \bar{\theta }}}{{dR^{2} }} + \frac{1}{R}\frac{{d\bar{\theta }}}{dR} - \left[ {\eta \Pr - H} \right]\bar{\theta } = 0 $$
(7)

Subject to

$$ \bar{U} = 0 ;\; \bar{\theta } = \frac{1}{\eta }\quad {\text{at}}\;t = 1 $$
(8)
$$ \bar{U} = 0; \;\frac{{d\bar{\theta }}}{dR} = 0\quad {\text{at}}\;R = \lambda $$
(9)

The solutions of the Bessel ordinary differential equations in Eq. (6) and (7) in the Laplace domain subject to the boundary conditions (8) and (9) are given below

$$ \bar{\theta }(R,\eta ) = A_{1} I_{0} (R\delta ) + A_{2} K_{0} (R\delta ) $$
(10)
$$ \bar{U}(R,\eta ) = B_{1} I_{0} (R\alpha ) + B_{2} K_{0} (R\alpha ) - \frac{1}{\gamma }\left[ {\frac{{A_{1} I_{0} (R\delta ) + A_{2} K_{0} (R\delta )}}{{(\delta^{2} - \alpha^{2} )}}} \right] $$
(11)

where

$$ \begin{aligned}& \delta = \sqrt {\eta Pr - H} ,\quad \alpha = \left[ {\frac{1}{\gamma }\left( {\frac{1}{Da} + \eta } \right)} \right]^{1/2} ,\quad A_{1} = \frac{{K_{1} \left( {\lambda \delta } \right)}}{{\eta \left[ {I_{1} \left( {\lambda \delta } \right)K_{0} \left( \delta \right) + I_{0} \left( \delta \right)K_{1} \left( {\lambda \delta } \right)} \right]}}, \hfill \\& A_{2} = \frac{{I_{1} \left( {\lambda \delta } \right)}}{{\eta \left[ {I_{1} \left( {\lambda \delta } \right)K_{0} \left( \delta \right) + I_{0} \left( \delta \right)K_{1} \left( {\lambda \delta } \right)} \right]}},\quad B_{1} = \frac{{M_{2} K_{0} \left( \alpha \right) - M_{1} K_{0} \left( {\lambda \alpha } \right)}}{{I_{0} \left( {\lambda \alpha } \right)K_{0} \left( \alpha \right) - I_{0} \left( \alpha \right)K_{0} \left( {\lambda \alpha } \right)}}, \hfill \\& B_{2} = \frac{{M_{1} I_{0} \left( {\lambda \alpha } \right) - M_{2} I_{0} \left( \alpha \right)}}{{I_{0} \left( {\lambda \alpha } \right)K_{0} \left( \alpha \right) - I_{0} \left( \alpha \right)K_{0} \left( {\lambda \alpha } \right)}}. \hfill \\ \end{aligned} $$

2.1 Skin frictions, Nusselt number and mass flux

Using Eq. (11), the skin friction at the surface of the cylinders in the Laplace domain at \( R = 1 \) and \( R = \lambda \) are given respectively as follows:

$$ \tau_{1} (1,\eta ) = \left. {\frac{{d\bar{U}}}{dR}} \right|_{R = 1} = \alpha \left\{ {B_{1} I_{1} (\alpha ) - B_{2} K_{1} (\alpha )} \right\} - \frac{\delta }{\varUpsilon }\frac{{\left[ {A_{1} I_{1} (\delta ) - A_{2} K_{1} (\delta )} \right]}}{{(\delta^{2} - \alpha^{2} )}} $$
(12)
$$ \tau_{\lambda } (\lambda ,\eta ) = \left. {\frac{{d\bar{U}}}{dR}} \right|_{R = \lambda } = \alpha \left\{ {B_{1} I_{1} (\lambda \alpha ) - B_{2} K_{1} (\lambda \alpha )} \right\} - \frac{\delta }{\varUpsilon }\frac{{\left[ {A_{1} I_{1} (\lambda \delta ) - A_{2} K_{1} (\lambda \delta )} \right]}}{{(\delta^{2} - \alpha^{2} )}} $$
(13)

Using Eq. (10) the rate of heat transfer at the surface of the inner cylinder in the Laplace domain is given by:

$$ \bar{N}u_{1} = \left. {\frac{{d\bar{\theta }}}{dR}} \right|_{R = 1} = \delta [A_{1} I_{1} (\delta ) - A_{2} K_{1} (\delta )] $$
(14)

Thus, the solution of the mass flow rate in the Laplace domain through the annular region is given by:

$$\begin{aligned} \bar{V} =\,& 2\pi \mathop \int \limits_{1}^{\lambda } R\bar{U}dR = 2\pi \Bigg\{ \left[ {\frac{{B_{1} }}{\alpha }\left( {\lambda I_{1} (\lambda \alpha ) - I_{1} (\alpha )} \right) - \frac{{B_{2} }}{\alpha }\left( {\lambda K_{1} (\lambda \alpha ) - K_{1} (\alpha )} \right)} \right]\\ & - \frac{1}{{\delta \varUpsilon (\delta^{2} - \alpha^{2} )}}\left[ {A_{1} \left( {\lambda I_{1} (\lambda \delta ) - I_{1} (\delta )} \right) - A_{2} \left( {\lambda K_{1} (\lambda \delta ) - K_{1} (\delta )} \right)} \right] \Bigg\} \end{aligned} $$
(15)

It is pertinent to note that the given solutions above are in Laplace domain. Equations (10)–(15) is to be transformed to the time domain. Due to the complex nature of these solutions, the Riemann-sum approximation approach was adopted. Following the work of Jha and Yusuf (2016), any function in the Laplace domain can be inverted to the time domain as follows:

$$ Z(R,t) = \frac{{e^{\eta t} }}{t}\left[ {\frac{1}{2}\bar{Z}(R,\eta ) + \text{Re} \left( {\mathop \sum \limits_{n = 1}^{M} \bar{Z}\left( {R,\varepsilon + \frac{in\pi }{t}} \right)} \right)( - 1)^{n} } \right] $$
(16)

where Z represents \( \theta , U, Q \) or \( Nu \) as the case may be, Re refers to the real part of the summation, \( i = \sqrt { - 1} \), M is the number of terms used in the Riemann-sum approximation and \( \varepsilon \) is the real part of the Bromwich contour that is used in inverting Laplace transforms. The Riemann-sum approximation for the Laplace inversion involves a single summation for the numerical process its accuracy depends on the value of \( \varepsilon \) and the truncation error dictated by M. According to (Tzou 1997), the value of \( \varepsilon t \) that best satisfied the result is 4.7.

2.2 Validation of the method

To validate the numerical technique employed in inverting Eqs. (10)–(15) to the time domain, the steady state solution of Eqs. (1) and (2) is computed, this is achieved by computing the steady state velocity field analytically which is expected to be in sync with transient state solution at large time. This is achieved by taking \( \frac{\partial (\,)}{\partial t} = 0 \) in the non-dimensional Eqs. (1) and (2) which then reduces to the ordinary differential equations given by:

$$ \gamma \left[ {\frac{{d^{2} U}}{{dR^{2} }} + \frac{1}{R}\frac{dU}{dR}} \right] - \frac{U}{Da} + \theta = 0 $$
(17)
$$ \frac{{d^{2} \theta }}{{dR^{2} }} + \frac{1}{R}\frac{d\theta }{dR} + H\theta = 0 $$
(18)

with the given boundary conditions

$$ U = 0;\; \theta = 1\quad {\text{at}}\;R = 1 $$
(19)
$$ U = 0;\; \frac{d\theta }{dR} = 0\quad {\text{at}}\;R = \lambda $$
(20)

Equations (17) and (18) are solved under the boundary conditions (19) and (20) to obtain the solution of the steady state temperature field, velocity field, skin friction on the surface of the cylinders, Nusselt number and mass flow rate. The solutions are given as;

$$ \theta = C_{1} J_{0} \left( {R\sqrt H } \right) + C_{2} Y_{0} \left( {R\sqrt H } \right) $$
(21)
$$ U = D_{1} I_{0} \left( {R\sqrt {\frac{1}{\gamma Da}} } \right) + D_{2} K_{0} \left( {R\sqrt {\frac{1}{\gamma Da}} } \right) + \frac{{\left[ {C_{1} J_{0} \left( {R\sqrt H } \right) + C_{2} Y_{0} \left( {R\sqrt H } \right)} \right]Da}}{{\left( {H\gamma Da + 1} \right)}} $$
(22)
$$ \tau_{1} = \left. {\frac{dU}{dR}} \right|_{R = 1} = \sqrt {\frac{1}{\gamma Da}} \left\{ {D_{1} I_{1} \left( {\sqrt {\frac{1}{\gamma Da}} } \right) - D_{2} K_{1} \left( {\sqrt {\frac{1}{\gamma Da}} } \right)} \right\} + \sqrt { - H} \frac{{\left[ {C_{1} J_{1} \left( {\sqrt H } \right) - C_{2} Y_{1} \left( {\sqrt H } \right)} \right]Da}}{{\left( {H\gamma Da + 1} \right)}} $$
(23)
$$\begin{aligned} \tau_{\lambda } = - \left. {\frac{dU}{dR}} \right|_{R = \lambda } =\,& - \Bigg[ \sqrt {\frac{1}{\gamma Da}} \left\{ {D_{1} I_{1} \left( {\lambda \sqrt {\frac{1}{\gamma Da}} } \right) - D_{2} K_{1} \left( {\lambda \sqrt {\frac{1}{\gamma Da}} } \right)} \right\}\\ &+ \sqrt { - H} \frac{{\left[ {C_{1} J\left( {\lambda \sqrt H } \right) - C_{2} Y_{1} \left( {\lambda \sqrt H } \right)} \right]Da}}{{\left( {H\gamma Da + 1} \right)}} \Bigg] \end{aligned} $$
(24)
$$ Nu_{1} = \left. {\frac{d\theta }{dR}} \right|_{R = 1} = \sqrt { - H} \left\{ {C_{1} I_{1} \left( { \sqrt { - H} } \right) - C_{2} K_{1} \left( {\sqrt { - H} } \right)} \right\} $$
(25)
$$\begin{aligned} V = 2\pi \mathop \int \limits_{1}^{\lambda } RUdR =\,& 2\pi \Bigg\{ D_{1} \sqrt {\gamma Da} \left[ {\lambda I_{1} \left( {\lambda \sqrt {\frac{1}{\gamma Da}} } \right) - I_{1} \left( {\sqrt {\frac{1}{\gamma Da}} } \right)} \right]\\ &- D_{2} \sqrt {\gamma Da} \left[ {\lambda K_{1} \left( {\lambda \sqrt {\frac{1}{\gamma Da}} } \right) - K_{1} \left( {\sqrt {\frac{1}{\gamma Da}} } \right)} \right]\\ & + \frac{Da}{{\sqrt { - H} \left( {H\gamma Da + 1} \right)}}C_{1} \left[ {\lambda I_{1} \left( {\lambda \sqrt { - H} } \right) - I_{1} \left( {\sqrt { - H} } \right)} \right]\\ & - C_{2} \left[ {\lambda K_{1} \left( {\lambda \sqrt { - H} } \right) - K_{1} \left( {\sqrt { - H} } \right)} \right] \Bigg\} \end{aligned}$$
(26)

where

$$ \begin{aligned} &C_{1} = \frac{{K_{1} \left( {\lambda \sqrt { - H} } \right)}}{{I_{0} \left( {\sqrt { - H} } \right)K_{1} \left( {\lambda \sqrt { - H} } \right) + I_{1} \left( {\lambda \sqrt { - H} } \right)K_{0} \left( {\sqrt { - H} } \right)}} \hfill \\& C_{2} = \frac{{I_{1} \left( {\lambda \sqrt { - H} } \right)}}{{I_{0} \left( {\sqrt { - H} } \right)K_{1} \left( {\lambda \sqrt { - H} } \right) + I_{1} \left( {\lambda \sqrt { - H} } \right)K_{0} \left( {\sqrt { - H} } \right)}} \hfill \\ &D_{1} = \frac{{C_{1} Da}}{H\gamma Da + 1},\quad D_{2} = \frac{{C_{2} Da}}{H\gamma Da + 1} \hfill \\ \end{aligned} $$

3 Results and discussion

The solution of the present study obtained in the previous section are seen to be governed by heat generating/absorbing parameter (H), Darcy number (Da), Prandtl number (Pr), radius ratio \( (\lambda ) \) and viscosity ratio \( (\gamma ) \). The effect of these parameters on the fluid temperature, velocity, skin-friction are shown using the line graph while the numerical values for rate of heat transfer at the surface of the inner cylinder and mass flow rate were generated using the MATLAB program. The present parametric study has been carried out over a reasonable range \( 0.01 \le Da \le 1.0 \), \( 0.5 \le \gamma \le 1.5 \), \( - 4 \le H \le 4 \) where \( \lambda = 2 \) is also taken as reference point. All through this work, two values of Prandtl number \( (Pr = 0.71) \) and \( (Pr = 7.0) \) were considered which corresponds to air and water respectively. In the course of this research work, positive values of H shows that the fluid is generating heat and is specified as (a) in Figs. 2, 3, 4, 5, 6, 7, 8 and 9, while negatives values of H shows that the fluid is absorbing heat and is specified as (b) in Figs. 2, 3, 4, 5, 6, 7, 8 and 9.

Fig. 2
figure 2

Temperature variation for different values of \( t(H = 2.0) \)

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Fig. 3
figure 3

Velocity variation for different values of \( t(H = 2.0, Da = 0.1, \gamma = 0.5) \)

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Fig. 4
figure 4

Velocity variation for different values of \( Da \left( {H = 2.0, t = 0.4, \gamma = 0.5} \right) \)

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Fig. 5
figure 5

Velocity variation for different values of \( \gamma \left( {H = 2.0, t = 0.4, Da = 0.1} \right) \)

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Fig. 6
figure 6

Variation of skin friction \( \left( {\tau_{1} } \right) \) for different values of \( Da \left( {H = 2.0, \gamma = 0.5} \right) \)

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Fig. 7
figure 7

Variation of skin friction \( \left( {\tau_{\lambda } } \right) \) for different value of \( Da \left( {H = 2.0, \gamma = 0.5} \right) \)

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Fig. 8
figure 8

Variation of skin friction \( \left( {\tau_{1} } \right) \) for different value of \( \gamma \left( {H = 2.0,Da = 0.1} \right) \)

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Fig. 9
figure 9

Variation of skin friction \( \left( {\tau_{\lambda } } \right) \) for different value of \( \gamma \left( {H = 2.0, Da = 0.1} \right) \)

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Figure 2 shows the temperature variation with time (t). It is evident that temperature increases with increase in time, also there is higher temperature with slow convergence when \( Pr = 7.0 \) when the fluid is heat generating and the temperature decreases towards the outer cylinder in both (a) and (b) except when \( Pr = 0.71 \) in Fig. 2a. This further confirmed the findings reported in the previous works documented in the introductory session of this work that fluid temperature attains steady state faster when the fluid is heat absorbing. Generally, temperature is found to increase with decrease in Prandtl number.

Figure 3 depicts velocity variation for different values of time (t), it is noticed that the fluid velocity increases with time (t) when the fluid is either heat generating or absorbing until the steady state is reached for both cases of (Pr). This is physically true since convection current is always higher in the case of air due to its low density in comparison to water. Moreover, the effect of the heat generation parameter on the fluid velocity is more evident with increases in time, this can be linked to the fact that both heat generating parameter and the heating at the inner cylinder enhanced convective processes in the system.

The velocity variation for different values of Darcy number (Da) and the viscosity ratio \( (\gamma ) \) is represented in Figs. 4 and 5 respectively. It is seen that there is a sharp increase in the fluid velocity when the fluid is heat generating as higher velocity profile are perceived in the annular gap when the fluid is heat generating. This can be attributed to the fact that as Darcy number (Da) increases, there is an equivalent increase in permeability of the porous materials. This further confirm the findings in the work of (Yusuf and Jha 2018). On the other hand, we see that with an upsurge in the viscosity ratio \( (\gamma ) \), a corresponding decline in the fluid velocity in the annular gap with a much more decline when the fluid is heat absorbing is observed. Thus, we conclude that fluid velocity (U) is inversely proportional to the fluid viscosity ratio \( (\gamma ) \).

Figures 6 and 7 reveal the combined effect of time (t), Darcy number (Da) and the viscosity ratio \( (\gamma ) \) on the skin friction \( \left( {\tau_{1} } \right) \) at the outer surface of the inner cylinder for both heat generating and absorbing flow. It is evident from Figs. 6 and 7 that the skin friction increases with time and Darcy number (Da). However the magnitude of skin friction \( \left( {\tau_{1} } \right) \) is seen to be higher with heat generating fluid in comparison to the heat absorbing case (see Fig. 6b). In a similar fashion, skin friction \( \left( {\tau_{\lambda } } \right) \) at the outer cylinder presented in Fig. 7a, b show a similar occurrence revealed in Fig. 6a, b respectively.

It can also be noticed that the magnitude of the skin friction is higher in the case of heat generation for the aforementioned cases. As it can be viewed clearly these figures show that skin friction is higher at the heated inner surface of the outer cylinder.

Figures 8 and 9 show the combined effect of the time (t), Prandtl number (Pr) and viscosity ratio \( (\gamma ) \) on the skin frictions \( \left( {\tau_{1} } \right) \) and \( \left( {\tau_{\lambda } } \right) \) respectively for both heat generating and heat absorbing cases. Obviously, the figures revealed that skin friction decreases with increase in both Prandtl number and viscosity ratio. However, an increase in skin friction on both walls is obtained with time. It is interesting to note that skin friction on both walls attain steady state faster in the case of the heat absorption. It is good to also note that similar to the observation in Figs. 6 and 7, skin friction is observed to be higher at the heated inner cylinder.

The numerical values of the rate of heat transfer at the outer surface of the inner cylinder and fluid mass flow rate is presented in Tables 1 and 2 respectively. It is clear in Table 1 that rate of heat transfer is inversely proportional to time for both heat generation and heat absorption. While mass flow rate is directly proportional to time for both cases. The mass flow rate is also seen to be higher in the case of air, this can be attributed to the low viscosity nature of air in comparison to water.

Table 1 Numerical values for the Nusselt number \( \left( {R = 1,\lambda = 2} \right) \)
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Table 2 Numerical values for mass flow rate \( \left( {\lambda = 2, Da = 0.1, \gamma = 1.0} \right) \)
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4 Conclusions

In this paper, a semi-analytical solution of the mathematical model accountable for the impact of heat generation/absorption on transient natural convective flow in an annulus filled with porous materials subject to isothermal and adiabatic boundaries has been considered in terms of both Bessel and modified Bessel functions of first and second kind. The effect of various flow parameter on the solution of the governing equations has been presented. The study revealed that the flow in the system can be controlled by selecting appropriate value of flow parameter as well as insulating the outer cylinder. This mechanism can be found in annular fin where heat loss is found to me maximum when fin is isothermal. It is also applicable in piping systems in engineering and boiler plant among others.