Introduction

The study of fluid flow in composite tubes has caught the interest of scholars for decades because of the use of vertical composite tubes in science and engineering. The impact of thermodynamic and hydrodynamic variables on fluid flow and thermal behavior has been investigated. Viscous dissipation is the term for the internal heat production caused by viscous forces in molecular fluid-particle interactions. The mechanical energy released in this way significantly alters the thermal buoyancy and thermal behavior of a fluid in motion. Because fluid-particle interactions change fluid flow and temperature, viscous dissipation is essential in the majority of lubrication organizations. Other notable studies on the impact of viscous dissipation are the works of [1,2,3,4,5,6,7,8].

The study of heat transmission in porous media has gained importance due to its expanding practical applications. Examples include petroleum reserves, porous solids drying, thermal insulation, packed-bed catalytic reactors, and geothermal reservoirs. Several studies conducted on the impact of Darcy dissipation include [9,10,11,12,13,14,15,16].

The steady natural convection in a vertical tube partially filled with porous material was not directly studied in any of the examined papers. Because viscous dissipation is used in the lubrication industry, nuclear reactor cooling, the cooling of electric appliances, and other applications, this article was written to investigate how these governing thermo physical and flow parameters can control temperature, velocity, and energy produced in a channel. Since the momentum and energy equations in the current model are linked and nonlinear, finding a closed-form solution is difficult. Numerous additional approximate methods have also been developed to address similar issues, such as numerical solutions and perturbation techniques. The homotopy perturbation strategy has been selected as the solution approach for the current problem. This approach was chosen because it decouples the system systematically and gets over the limitation of tiny parameters imposed by the widely used perturbation method. The Homotopy perturbation approach also reveals the interactions between the parameters that are concealed in numerical methods.

Mathematical Analysis

A steady flow of viscous incompressible fluid in an infinite vertical tube is considered as seen in Fig. 1. The section of the tube near \({r}^{\prime}=0\) is occupied by clear fluid while the other section near \({r}^{\prime}=a\) is occupied by a fully saturated porous medium. The clear fluid and the porous region are separated by the interface at \({r}^{\prime}= {d}^{\prime}\). The direction of the flow is taken along the vertical axis, while the \({r}^{\prime}-\) axis is taken in the radial direction. The fluid motion is set up as a result of natural convection due to density change with temperature. On the other hand, the energy equations took into account the effect of viscous dissipation in the clear fluid region while taking also the Darcy dissipation in addition to the viscous dissipation in the porous medium section. The temperatures of the porous medium and the clear fluid are equivalent (LTE condition). The governing equations are provided below using the common Boussinesq approximation:

$$ {\upmu }\left( {\frac{{{\text{d}}^{2} {\text{u}}_{f}^{{\prime }} }}{{{\text{dr}}^{{{\prime }2}} }} + \frac{1}{{{\text{r}}^{\prime }}}\frac{{{\text{du}}_{f}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \right) + {\text{g}}{\uprho \beta }\left( {{\text{T}}^{\prime }_{{\text{f}}} - {\text{T}}_{0} } \right) = 0, $$
(1)
$$ {\upmu }_{{{\text{eff}}}} \left( {\frac{{{\text{d}}^{2} {\text{u}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{{{\prime }2}} }} + \frac{1}{{{\text{r}}^{\prime }}}\frac{{{\text{du}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \right) - \frac{{{\upmu }{\text{u}}_{{\text{p}}}^{{\prime }} }}{{\text{K}}} + {\text{g}}{\uprho \beta }\left( {{\text{T}}_{{\text{p}}}^{{\prime }} - {\text{T}}_{0} } \right) = 0, $$
(2)
$$ {\text{k}}\left( {\frac{{{\text{d}}^{2} {\text{T}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}^{{{\prime }2}} }} + \frac{1}{{{\text{r}}^{\prime }}}\frac{{{\text{dT}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \right) + {\upmu }\left( {\frac{{{\text{du}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \right)^{2} = 0, $$
(3)
$$ {\text{k}}\left( {\frac{{{\text{d}}^{2} {\text{T}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{{{\prime }2}} }} + \frac{1}{{{\text{r}}^{\prime }}}\frac{{{\text{dT}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \right) + \left( {{\upmu }_{{{\text{eff}}}} \left( {\frac{{{\text{du}}_{{\text{p}}}^{{\prime }} }}{{{\text{dy}}^{\prime }}}} \right)^{2} + \frac{{{\upmu }{\text{u}}^{{{\prime }2}} }}{{\text{K}}}} \right) = 0. $$
(4)
Fig. 1
figure 1

Schematic diagram

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The boundary and interfacial conditions in the dimensional form used in this present study are:

$$ \frac{{{\text{du}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}}} = 0,\quad \quad \frac{{{\text{dT}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}}} = 0,~~\quad \quad \quad \quad ~{\text{r}}{\prime } = 0 $$
$$ {{\text{u}}_{{\text{p}}}^{{\prime }} = 0,~\quad \quad - {\text{k}}\frac{{{\text{dT}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}{\prime }}} = {\text{h}}_{1} \left( {{\text{T}}_{1} - {\text{T}}_{{\text{p}}}^{{\prime }} \left( {\text{r}} \right)} \right),~~\quad \quad \quad \quad {\text{r}}^{\prime } = {\text{a}}~} $$
(5)
$$ {\text{r}}^{\prime } = ~{\text{d}}^{\prime }\left\{ {\begin{array}{*{20}l} {{\text{u}}_{{\text{f}}}^{{\prime }} = ~{\text{u}}_{{\text{p}}}^{{\prime }} ,~} \hfill & {{\text{T}}_{{\text{f}}}^{{\prime }} = ~{\text{T}}_{{\text{p}}}^{{\prime }} } \hfill \\ {{\upmu }_{{{\text{eff}}}} \frac{{{\text{du}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}} - ~{\upmu }\frac{{{\upalpha }{\text{u}}_{{\text{f}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}} = ~{\upmu }\frac{{{\upalpha }u_{{\text{p}}}^{{\prime }} }}{{\sqrt {\text{K}} }},} \hfill & {~\frac{{{\text{dT}}_{{\text{f}}}^{'} }}{{{\text{dr}}^{\prime }}} = ~\frac{{{\text{dT}}_{{\text{p}}}^{{\prime }} }}{{{\text{dr}}^{\prime }}}} \hfill \\ \end{array} } \right. $$
(6)

Defining a reference velocity and average Temperature

$$ {\text{u}}_{0} = \frac{{{\text{g}}\upbeta {\text{r}}^{2} \left( {{\text{T}}_{2} - {\text{T}}_{1} } \right)}}{\upvartheta }\quad {\text{and}}\quad {\text{T}}_{0} = \frac{{{\text{T}}_{1} + {\text{T}}_{2} }}{2} $$

The dimensionless parameters considered are as follows:

$$ \begin{aligned} \upgamma & = ~\frac{{\upmu _{\text{eff}} }}{\upmu },\quad \text{u}_{\text{p}} = ~\frac{{\text{u}_{\text{p}}^{\prime } }}{{\text{u}_{0} }}~,\quad \text{u}_{\text{f}} = ~\frac{{\text{u}_{\text{f}}^{\prime } }}{{\text{u}_{0} }}~,\quad \upvartheta = ~\frac{\upmu }{\rho },\quad \uptheta _{\text{p}} = ~\frac{{\text{T}_{\text{p}}^{\prime } - \text{T}_{0} }}{{\text{T}_{2} - \text{T}_{1} }}, \\ \uptheta _{\text{f}} & = ~\frac{{\text{T}_{\text{f}}^{\prime } - \text{T}_{0} }}{{\text{T}_{2} - \text{T}_{1} }},\quad \text{r} = ~\frac{{\text{r}^{\prime} }}{\text{a}},\quad \text{z} = ~\frac{{\text{d}^{\prime} }}{\text{a}}. \\ \end{aligned} $$
(7)

Using Eqs. (5–7) the temperature and velocity equations in (1–4) have been reduced to:

$$ \frac{{{\text{d}}^{2} {\text{u}}_{{\text{f}}} }}{{{\text{dr}}^{{2}} }} + \frac{1}{{\text{r}}}\frac{{{\text{du}}_{{\text{f}}} }}{{{\text{dr}}}} +\uptheta _{{\text{f}}} = 0, $$
(8)
$$\upgamma \left( {\frac{{{\text{d}}^{2} {\text{u}}_{{\text{p}}} }}{{{\text{dr}}^{2} }} + \frac{1}{{\text{r}}}\frac{{{\text{du}}_{{\text{p}}} }}{{{\text{dr}}}}} \right) - \frac{{{\text{u}}_{{\text{p}}} }}{{{\text{Da}}}} +\uptheta _{{\text{p}}} = 0, $$
(9)
$$ \frac{{{\text{d}}^{2}\uptheta _{{\text{f}}} }}{{{\text{dr}}^{2} }} + \frac{1}{{\text{r}}}\frac{{{\text{d}}\uptheta _{f} }}{{{\text{dr}}}} + {\text{Br}}\left( {\frac{{{\text{du}}_{{\text{f}}} }}{{{\text{dr}}}}} \right)^{2} = 0, $$
(10)
$$ \frac{{{\text{d}}^{2} \uptheta _{{\text{p}}} }}{{{\text{dr}}^{2} }} + \frac{1}{{\text{r}}}\frac{{{\text{d}}\uptheta _{{\text{p}}} }}{{{\text{dr}}}} + {\text{Br}}\left( {\left( {{\gamma }\frac{{{\text{du}}_{{\text{p}}} }}{{{\text{dr}}}}} \right)^{2} + \frac{{{\text{u}}_{{\text{p}}}^{2} }}{{{\text{Da}}}}} \right) = 0. $$
(11)

Subject to the following boundary conditions:

$$ \frac{{{\text{du}}_{{\text{f}}} }}{{{\text{dr}}}} = 0, \quad \quad \frac{{{\text{d}}\uptheta _{{\text{f}}} }}{{{\text{dr}}}} = 0,\quad \quad \quad {\text{at}}\quad {\text{ r}} = 0 $$
$$ {\text{u}}_{{\text{p}}} = 0,\quad \quad \frac{{{\text{d}}\uptheta _{{\text{p}}} }}{{{\text{dr}}}} = {\text{Bi}}\left( {0. 5 +\uptheta _{{\text{p}}} \left( {\text{r}} \right)} \right), \quad \quad \quad {\text{at}}\quad {\text{r}} = 1 $$
(12)
$$ {\text{r}} = {\text{d}}\left\{ {\begin{array}{*{20}l} {{\text{u}}_{{\text{f}}} = {\text{u}}_{{\text{p}}} ,} \hfill & {\uptheta _{{\text{f}}} =\uptheta _{{\text{p}}} } \hfill \\ {\upgamma \frac{{{\text{du}}_{{\text{p}}} }}{{{\text{dr}}}} - \frac{{{\text{du}}_{{\text{f}}} }}{{{\text{dr}}}} = \frac{\upalpha }{{\sqrt {{\text{Da}}} }}{\text{u}}_{{\text{f}}} ,} \hfill & {\frac{{{\text{d}} \uptheta _{{\text{f}}} }}{{{\text{dr}}}} = \frac{{{\text{d}}\uptheta _{{\text{p}}} }}{{{\text{dr}}}}} \hfill \\ \end{array} } \right. $$
(13)

where \(Da, Br and Bi\) are defined as:

$$ {\text{Da}} = ~\frac{{\text{K}}}{{{\text{H}}^{2} }},\quad {\text{Br}} = ~\frac{{\upmu {\text{u}}_{0}^{2} }}{{{\text{k}}\left( {{\text{T}}_{2} - {\text{T}}_{1} } \right)}}\quad {\text{and}}\quad {\text{Bi}}_{1} = \frac{{{\text{h}}_{1} {\text{r}}^{\prime }}}{{\text{k}}}. $$

Result and Discussion

In this study, the impact of various flow parameters on the thermodynamics and flow generation in the composite tube is discussed. The Brinkman number, Biot number, and the porosity of the porous media are the major variables whose impacts are being studied. The inquiry is conducted using the line graphs in Figs. 2, 3, 4, 5, which were solved using the Homotopy perturbation method. When using water and air as the working fluids at 1 atm pressure, the following values are used: \(\upmu =1.002 \times {10}^{-3}\mathrm{ kg}/\mathrm{ms}\) at \({20 }^{0}\mathrm{C}\), \(\upbeta = {2.09 \times {10}^{-4} }^{0}{\mathrm{C}}^{-1},\) \(\mathrm{g}=9.8 {\mathrm{m}}^{2}/\mathrm{s}\), \(\mathrm{k}=0.5861\mathrm{ W}/\mathrm{mk},\) \(\upnu =1.003 \times {10}^{-6}{\mathrm{m}}^{2}/\mathrm{s}\), \({\mathrm{u}}_{0}=5.0\mathrm{ m}/\mathrm{s}\) and \(\mathrm{H}=1.0\mathrm{ m}\) and for air at 1 atm pressure we have \(\upmu =1.806 \times {10}^{-5}\mathrm{ kg}/\mathrm{ms}\) at \({20 }^{0}\mathrm{C}\), \(\upbeta = {\frac{1}{293} }^{0}{\mathrm{C}}^{-1},\) \(\mathrm{g}=9.8 {\mathrm{m}}^{2}/\mathrm{s}\), \(\mathrm{k}=0.024\mathrm{ W}/\mathrm{mk},\) \(\upnu =1.5 \times {10}^{-5}{\mathrm{m}}^{2}/\mathrm{s}\), \({\mathrm{u}}_{0}=5.0\mathrm{ m}/\mathrm{s}\) and \(\mathrm{H}=1.0\mathrm{ m}\) are taken into account. Table 1 shows several computed values \(\mathrm{Br}\) for both air and water at various temperatures.

Fig. 2
figure 2

Effects of \(Br\) on temperature and velocity profile

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Fig. 3
figure 3

Effects of \(Bi\) on temperature and velocity profile

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Fig. 4
figure 4

Effects of \(Da\) on temperature and velocity profile

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Fig. 5
figure 5

Effects of \(d\) on Temperature and velocity profile

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Table 1 Shows the Brinkman number at different temperatures
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Figures 2 displays the effect of Brinkman number on the temperature and velocity with \(\mathrm{Da}=0.1,\mathrm{ d}=0.8,\mathrm{ \upalpha }=0.5,\mathrm{ Bi}=2.0\mathrm{ and \upgamma }=1.0\). It is observed that with increase in the Brinkman number results in to increase in the temperature profile within the composite tube. Due to viscous dissipation, there is an increase in heat generation which results in to increase in the convection current and consequently results in to increase in the temperature. With the increase in temperature as shown in Fig. 2a there's a corresponding increase in the velocity. This is a result of the effect of heat generation due to the viscous dissipation term bringing about an increase in the convection current which leads to a rise in the buoyancy force in the tube and finally an increase in the velocity. The impact of the Brinkman number is more on water than air due to high dynamic viscosity and heat conductivity. Furthermore with gradual increase in the Brinkman number the effect on both the temperature and velocity is more pronounced.

Figures 3 displays the effect of Biot number on the temperature and velocity with fixed \(\mathrm{Da}=0.1,\mathrm{ d}=0.8,\mathrm{ \upalpha }=0.5, {\mathrm{Br}}_{(\mathrm{W})}=100, {\mathrm{Br}}_{(\mathrm{A})}=93\mathrm{ and \upgamma }=1.0\). Since Biot number is the ratio of internal conducive resistance of the composite tube to external convective resistance. With increase in the Biot number results to increase in the internal conductive resistance and a decrease in the external conductive resistance. Increase in the internal conductive resistance leads to a decrease in the buoyancy force and as a result the fluid thickness and this leads to decrease in the velocity of the fluid. It is also observed that the temperature decrease with increase in the Biot number.

Figures 4 displays the effect of Darcy number on the temperature and velocity with fixed values of \(\mathrm{d}=0.8,\mathrm{ \upalpha }=0.5, {\mathrm{Br}}_{(\mathrm{W})}=100, {\mathrm{Br}}_{(\mathrm{A})}=93,\mathrm{ Bi}=2.0\mathrm{ and \upgamma }=1.0\). Due to the increase in the value of the Darcy number, there is a decrease in the permeability of the porous region making the region like a clear fluid material. As indicated in Fig. 4a, with an increase in the Darcy number there is a correspondence decrease in the temperature. This is true since the increase in the Darcy number results in to decrease in the permeability of the fluid material due to a decrease in the collusion of the particles which leads to a decrease in the heat generation within the tube and hence a decrease in the temperature profile. The velocity increases with an increase in the Darcy number, this is a result of a decrease in the permeability of the fluid which brings about a reduction in the buoyancy forces within the tube, and this leads to an upsurge in the velocity.

Figures 5 displays the effect of the thickness of the porous material on the temperature and velocity with fixed values of \(\mathrm{Da}=0.1,\mathrm{ \upalpha }=0.5, {\mathrm{Br}}_{(\mathrm{W})}=100, {\mathrm{Br}}_{(\mathrm{A})}=93,\mathrm{ Bi}=2.0\mathrm{ and \upgamma }=1.0\). As seen in Fig. 5b increase in the thickness of the porous material leads to an upsurge in the temperature. This is a result of the fact that reducing the clear fluid region in addition to the effect of the permeability of the porous region leads to an increase in the temperature. There is a decrease in the velocity due to an increase in the thickness of the porous material which leads to a decrease in the buoyancy force.

Table 2 shows the rate of heat transfer for both water and air being considered as the working fluids. It can be observed that an increase in the Brinkman number leads to an increase in the rate of heat transfer for both air and water. Also, an increase in the thickness of the porous materials leads to a decrease in the rate of heat transfer. This is true since growing the thickness of the porous region decreases the rate at which heat is been transferred into the tube. Furthermore, the rate of heat transfer decreases as the permeability of the porous region is been increased.

Table 2 Rate of heat transfer
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Conclusion

The current study examined the steady natural convection flow in a vertical composite tube partially filled with porous material under the effects of Darcy and viscous dissipation. The solutions of the governing equations was carried out using the approximate solution technique (HPM). The main results were summarized as follows:

  1. 1.

    An increase in the Brinkman number upsurges the temperature and velocity profiles of both the working fluids.

  2. 2.

    An increase in the internal conductive resistance leads to a decrease in the buoyancy force and as a result the fluid thickness and this leads to decrease in the velocity of the fluid.

  3. 3.

    An increase in the thickness of the porous material can be used to increase the velocity of the fluid in the composite vertical tube.

  4. 4.

    An increase in the permeability of the material leads to increase in the velocity of the fluid.