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\title{Study of Viscous Dissipation on Natural Convection in a Vertical Conical Annular Porous Medium}
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             \author[1]{N. Ameer  Ahmad}

             \affil[1]{  University of Tabuk, Tabuk, Kingdom of Saudi Arabia.}

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\date{\small \em Received: 6 December 2013 Accepted: 4 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


In this paper, we study the heat transfer by Natural convection in a saturated porous medium including viscous dissipation in a vertical conical annular porous medium. Finite Element Method (FEM) has been used to solve the governing partial differential equations. Results are presented in terms of average Nusselt number (???? ???? ), streamlines and isothermal lines for various values of Rayleigh number (Ra), Cone angle (C A ), Radius ratio (R r ) and Viscous dissipation(?).Keywords: viscous dissipation (?), rayleigh number (ra), cone angle (ca) and radius ratio (rr).

\end{abstract}


\keywords{viscous dissipation(?), rayleigh number (ra), cone angle (ca) and radius ratio (rr).}

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\let\tabcellsep& 	 	 		 \par
The effect of viscous dissipation on natural convection in fluids has been studied by Gebhart \hyperref[b11]{[12]} for power law vertical wall variation. He obtained a perturbation solution in terms of a parameter which could not be expressed in terms of either the Rayleigh number or the Prandtl number, and observed its increasing effect as the Prandtl number increases. Later Gebhart and Mollendor \hyperref[b12]{[13]} obtained the similarity solution for the same problem when exponential wall temperature variation is used and a similar trend was observed. A comment was made by Fand and Brucker \hyperref[b13]{[14]} that the effect of viscous dissipation might be significant in the case of natural convection in porous medium in connection with their experimental correlation for heat transfer in external flows. The validity of the comment was tested for the Darcy model by Fand et al. \hyperref[b14]{[15]}, both experimentally and analytically while estimating the heat transfer coefficient from a horizontal cylinder embedded in a saturated porous medium. Their mathematical analysis is confined to studying the dissipation effect using a steady, energy Equation, the basis of the equation is from the analogy given by Bejan \hyperref[b15]{[16]} for the inclusion of viscous dissipation effects. The influence of viscous dissipation can be seen from the analogy given by Tucker and Dessenberger \hyperref[b16]{[17]} to model the heat transfer\par
The effect of viscous dissipation on natural convection has been studied for some different cases including the natural convection from horizontal cylinder embedded in a porous media by Fand and Brucker \hyperref[b18]{[19]} and Fand et al. \hyperref[b19]{[20]}. They reported that the viscous dissipation may not be neglected in all cases of natural convection from horizontal cylinders and further, that the inclusion of a viscous dissipation term in porous T Temperature ?T Non-Dimensional Temperature defined large Reynolds number-limit Rayleigh number. The non-similar boundary-layer equations resulting from the Forchheimer natural convection with power law wall variation were solved by Chen and Ho \hyperref[b4]{[5]}.\par
The transverse thermal dispersion effects will become important, and the analysis is dealt with at length in works by Plumb \hyperref[b5]{[6]}, Cheng \hyperref[b6]{[7]}, Hong and Tien Introduction atural convection flow and heat transfer in a saturated porous media has gained much attention during the past two decades because of its wide range of applications in packed bed reactors, porous insulation, beds of fossil fuels, nuclear waste disposal, usage of porous conical bearings in lubrication technology, geophysics and energy related engineering problems. A good review of buoyancy driven boundary layer flows in Darcian fluid is given in Nield and Bejan \hyperref[b0]{[1]}. When the Reynolds number is high enough for the Darcy flow model to breakdown, Pumb and Huenefeld N medium may lead to more accurate correlation equations. This observation has been pointed out also by Murthy and Singh \hyperref[b20]{[21]} for the natural convection flow along an isothermal vertical wall embedded in a porous medium. Recently, Nawaf H. Saeid and I.Pop \hyperref[b21]{[22]} studied the viscous dissipation effects on free convection in a porous cavity. 
\section[{II.}]{II.} 
\section[{Mathematical Formulation}]{Mathematical Formulation}\par
A vertical annular cone of inner radius r i and outer radius r 0 as depicted by schematic diagram as shown in figure (A) is considered to investigate the heat transfer behavior in the presence of viscous dissipation.\par
The co-ordinate system is chosen such that the r-axis points towards the width and z-axis towards the height of the cone respectively. Because of the annular nature, two important parameters emerges, which are Cone angle (C A ) and Radius ratio (R r ) of the annulus. They are definedas i i r i t A r r r R r r H C ? = ? = 0 0 ,\par
Where H t is the height of the cone.\par
The inner surface of the cone is maintained at isothermal temperature T h and outer surface is at ambient temperature T ? . It may be noted that, due to axisymmetry, only a section of the annulus is sufficient for analysis purpose. The horizontal surfaces of the vertical annular cone are considered adiabatic.\par
The flow inside the porous medium is assumed to obey Darcy law and there is no phase change of fluid. The properties of the fluid and porous medium are homogeneous, isotropic and constant except variation of fluid density with temperature. The fluid and porous medium are in thermal equilibrium with these assumptions, the governing equations are given by Continuity Equation:0 ) ( ) ( = ? ? + ? ? z rw r ru (2.1)\par
The velocity in r and z directions can be described by Darcy law as Velocity in horizontal directionz p K u ? ? ? = µ (2.2) Velocity in vertical direction ? ? ? ? ? ? + ? ? ? = g z p K w ? µ (2.3)\par
The permeability K of porous medium can be expressed as Bejan \hyperref[b23]{[24]} 2 3 2 ) 1 ( 180 ? ? ? = p D K (2.4)\par
The variation of density with respect to temperature can be described by Boussinesqapproximation as ? = ? ? [1 -? T (T -T ? )] (2.5) Momentum Equation: r T gK z u r w ? ? = ? ? ? ? ? ? ? (2.6) Energy equation: ) ( ) ( 1 2 2 2 2 w u C K z T r T r r r z T w r T u f p + + ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? = ? ? + ? ? ? µ ? (2.7)\par
The continuity equation (2.1) can be satisfyed by introducing the stream function? as z r u ? ? ? = ? 1 (2.8) r r w ? ? = ? 1 (2.9)\par
The cor responding dimensional boundary conditions are at r = r i , T = T w , ? = 0 (12.10a)at r = r 0 , T = T ? , ? = 0 (2.10b) (except at z = 0)\par
The new parameters arising due to cylindrical co-ordinates system areNon-dimensional Radius L r r = (2.11a) Non-dimensional Height L z z = (2.11b) Non-dimensional stream function L ? ? ? = (2.11c) Non-dimensional Temperature ( ) ( ) ? ? ? ? = T T T T T w (2.11d) Rayleigh number ?? ? TKL g Ra T ? = (2. 
\section[{11e)}]{11e)} 
\section[{Study of Viscous Dissipation on Natural Convection in a Vertical Conical Annular Porous Medium}]{Study of Viscous Dissipation on Natural Convection in a Vertical Conical Annular Porous Medium} 
\section[{Viscous dissipation parameter}]{Viscous dissipation parameter}p C TK u ? ? ? ? = (2.11f)\par
The non-dimensional equations for the heat transfer in vertical cone are Momentum equation:r T Ra r r r r z ? ? = ? ? ? ? ? ? ? ? ? ? + ? ? ? ? 1 2 2 (2.12) ume XIV Issue II Version I Year 2014 Energy equation ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 2 1 1 1 1 z r r r z T r T r r r r T z z T r r ? ? ? ? ? (2.13)\par
The corresponding non-dimensional boundary conditions are at ?r = ?r i , ?T=1 , ?? = 0 (2.14a) at r = ?r 0 , T=0 , ?? = 0 (2.14b) III. 
\section[{Solution of Governing Equations}]{Solution of Governing Equations}\par
Applying Galerkin method to momentum equation (2.12) yields:\{ \} ? ? ? d r T Ra r r r r r z N R T V e ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? = ? 1 2 2 (3.1) \{ \} dA r r T Ra r r r r r z N R T A e ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? = ? 2 1 2 2 ? ? (3.2)\par
where R e is the residue. Considering individual terms of equation (  {\ref 3}.2) [ ] [ ] r r N r N r N r T T T ? ? ? ? + ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ] [ 2 2 (3.3) Thus, [ ] r r N dA r r N r dA r N T A T A T A ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ] [ 2 2 2 2 2 (3.4)\par
The first term on right hand side of equation (3.4) can be transformed into surface by the application of Greens theorem and leads to inter-element requirement at boundaries of an element. The boundary conditions are incorporated in the force vector.\par
Let us consider that the variable to be determined in the triangular area as "T" The polynomial function for "T" can be expressed as T = ? 1 + ? 2 r + ? 3 z(3.5)\par
The variable T has the value T i , T j \& T k at the nodal position i, j \& k of the element. The r and z coordinates at these points are r i , r j , r k and z i , z j , z k respectively.\par
SinceT = N i T i + N j T j + N k T k (3.6)\par
Where N i , N j \& N k are shape functions given by A z c r b a N m m m m 2 + + = (3.7) Making use of (3.7) gives dA r N r N dA r z T N T A T A ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ?? ? ? = ? ? ? ? ? 2 2 (3.11) Since M 1 = N 1 , M 2 = N 2 , M 2 = N 3\par
Where M 1 , M 2 , and M 3 are the area ratios of the triangle and N 1 , N 2 and N 3 are the shape functions.\par
Replacing the shape functions in the above equation  {\ref (3.11)} givesdA r T T T r N M M M Ra r dA r r T Ra r N A T A ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? 2 ) ( 2 3 2 1 3 2 1 (3.12) [ ] ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? ? ? ? = 3 2 1 3 2 1 2 2 2 1 1 1 3 T T T b b b A R A Ra ? ? ? ? ? ? ? ? ? ? + + + + + + ? = 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 2 6 2 T b T b T b T b T b T b T b T b T b Ra R (3.13) Now Momentum equation leads to 0 6 2 4 2 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 2 3 2 1 2 3 3 2 3 1 3 2 2 2 2 1 3 1 2 1 2 1 2 3 3 2 3 1 3 2 2 2 2 1 3 1 2 1 2 = ? ? ? ? ? ? ? ? ? ? + + + + + + ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? T b T b T b T b T b T b T b T b T b Ra R c c c c c c c c c c c c c c c b b b b b b b b b b b b b b b A R ? ? ? (3.14) 
\section[{Which is in the form of the stiffness matrix [K s ] \{?\} = \{f\}}]{Which is in the form of the stiffness matrix [K s ] \{?\} = \{f\}}\par
Similarly application of Galerkin method to Energy equation gives\{ \} ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? 2 2 1 1 z T r T r r r r T z z T r r N R T A e ? ? dA r z r r r ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? 2 1 1 2 2 ? ? ? (3.15)\par
Considering the terms individually of the energy equation and following the same above steps. We get the stiffness matrix of energy equation as: [ ] [ ] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + + + + + ? ? ? ? ? ? ? ? ? ? ? ? + + + + + + ? 3 2 1 3 2 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 2 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 , ,\textbf{12 2 ,}A R [ ] [ ] 0 1 1 1 1 1 1 12 2 2 3 3 2 2 1 1 2 3 3 2 2 1 1 = ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? ? ? ? + + + ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? c c c b b b r A (3.16)\par
IV. 
\section[{Results and Discussion}]{Results and Discussion}\par
Results are obtained in terms of the average Nusselt number (???? ???? )at hot wall for various parameters such as Rayleigh number (Ra), Radius ratio (R r ) Cone angle (C A ) and Viscous dissipation (?) when heat is supplied to the vertical annular cone.\par
The average Nusselt number(???? ???? ), is given by   Fig.  {\ref 2 (a-c}) illustrates the streamlines and isothermal lines distribution inside the porous medium for various values of Radius ratio (R r ) at Ra = 100, C A = 75 and ? = 0.01. It is seen that the magnitude of streamlines decreases with the increase in Radius ratio (R r ). This happens due to the reason that at high Rayleigh number (Ra) with the viscous dissipation Year 2014 parameter (?), leads to more fluid movement at the hot wall of the cone. The isothermal lines tend to move towards the hot surface of the cone as the Radius ratio (R r ) increases, because the thermal boundary layer becomes thicker.? ? ? ? ? ? ? ? ? ? = ? r T u N z 0 a) b)\par
Fig.  {\ref 3 (a-c}) illustrates the streamlines and isothermal lines distribution inside the porous medium for various values of viscous dissipation parameter (?) at Ra=100, C A = 75 and R r = 1. For increasing the values of viscous dissipation parameter (?) no change of has been observed in the formation and occupation of the domain by streamlines and isothermal lines only half of the domain is covered with streamlines and isothermal lines.\par
Fig.  {\ref 4 (a-c}) shows the streamlines and isothermal lines distribution inside the porous medium of the vertical annular cone for various values of Rayleigh number (Ra) at ? = 0.003, C A = 15 and R r = 1. As the value of Rayleigh number (Ra) increases the magnitude of stream lines also increases. This is due to the reason that the increased Rayleigh number (Ra) promotes the fluid movements due to higher buoyancy force, which in term allows the convection heat transfer at lower partition of the hot wall of the vertical annular cone. increased by 41\% at Ra = 10. Whereas at Ra = 100, it is found to be 45\% with increase in Radius ratio (R r ) from 1 to 5.\par
Nusselt number ( Nu ). This figure corresponds to the values C A = 75, R r = 1. It can be seen that the effect of viscous dissipation parameter (?) is to reduce the average Nusselt number ( Nu ) at hot wall. The temperature difference near the hot wall increases with increase in viscous dissipation parameter (?). This happens due to the reason that the viscous dissipation leads to local heat generation, which increases the temperature in the porous medium. As the temperature of hot wall T W is constant, the increased temperature of porous medium reduces the temperature difference between the hot wall and the near region. Due to this reason the heat transfer from hot wall to the porous medium increases which results in increasing the average Nusselt number ( Nu ). The effect of viscous dissipation(?) is higher at the lower values of Rayleigh number (Ra) as compared to the higher values of Rayleigh number (Ra). At Ra = 10, the average Nusselt number ( Nu ) decreased by 4\% when viscous dissipation parameter (?) is increased 0 to 0.01. Whereas the corresponding reduction in the average Nusselt number ( Nu ) at Ra = 100 is found to be 18\%.\par
The effect of viscous dissipation becomes more dominant at high Rayleigh number (Ra) as compared to lower Rayleigh number (Ra). Nusselt number ( Nu ) decreases with the increase in viscous dissipation parameter (?). When there is no viscous dissipation then the average Nusselt number ( Nu ) at hot wall always increases with increase Cone angle (C A ). This happens due to reason that higher Cone angle (C A ) leads to high buoyancy force and thus Fig.  {\ref 7} demonstrates the effect of Rayleigh number (Ra) and viscous dissipation (?) on the average faster fluid movement. This faster fluid movement enhances the local friction between fluid and solid matrix thus increasing the local heat generation, which in turn reduces the average Nusselt number ( Nu ). When there is no viscous dissipation parameter (?), there is a decrease in the average Nusselt number ( Nu ). Which is found to be 26.3 \%, when Cone angle (C A ) increases from 15 to 45. At ? = 0.01, it is found that there is a decrease in the average Nusselt number ( Nu ) by 46.2\%. This shows that there is a decrease in the average Nusselt number ( Nu ) as the viscous dissipation parameter (?) increases. various values of Radius ratio (R r ). This figure is obtained for C A = 75, Ra = 100. It can be seen that the average Nusselt number ( Nu ) decreases with the increase in viscous dissipation parameter (?). When there is no viscous dissipation (?), at ? = 0, the average Nusselt number ( Nu ) at hot wall always increase with increase in Radius ratio (R r ). Whereas at ? = 0.005, the average Nusselt number ( Nu ) decreases as R r is reduced. At ? = 0.01, the average Nusselt number ( Nu ) always decreases with increase in Radius ratio (R r ). This happens due to the reason that higher Radius ratio (R r ) leads to high buoyancy force and this faster fluid movement. This faster fluid movement enhances the local friction between fluid and solid matrix thus increasing the local heat generation. Fig. \hyperref[fig_5]{10} illustrates the effect of Viscous dissipation parameter (?) on the average Nusselt number ( Nu ) for various values of Rayleigh number (Ra). This figure is obtained for C A = 75, R r =1. It can be seen that the average Nusselt number ( Nu ) decreases with the increase in viscous dissipation parameter (?). At Ra=25, the average Nusselt number ( Nu ) is linear, whereas at Ra=100, at lower viscous dissipation parameter (?), the average Nusselt number ( Nu ) increases and decreases with higher viscous dissipation parameter (?).This happens due to the reason that higher Rayleigh number (Ra) leads to high buoyancy force and thus faster fluid movement. This faster fluid movement enhances the local friction between fluid and social matrix thus increasing the local heat generation.\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1245}\includegraphics[]{image-3.png}
\caption{\label{fig_3}Figure 1 :Figure 2 :Figure 4 :Figure 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6789}\includegraphics[]{image-4.png}
\caption{\label{fig_4}Figure 6 :Figure 7 :Figure 8 :Figure 9 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10}\includegraphics[]{image-5.png}
\caption{\label{fig_5}Figure 10 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-6.png}
\caption{\label{fig_6}Fig. 5}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8}\includegraphics[]{image-7.png}
\caption{\label{fig_7}Fig. 8}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{9}\includegraphics[]{image-8.png}
\caption{\label{fig_8}Fig. 9}\end{figure}
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