An Optimal Analysis of Flow and Heat Transfer over a Slender Permeable Elastic Sheet with Variable Fluid Properties

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1 IOSR Journal o Engineering (IOSRJEN) ISSN (e): 5-3, ISSN (p): Vol. 8, Issue 6 (June. 8), V (VII) PP An Optial Analysis o Flow and Heat Transer over a Slender Pereable Elastic Sheet with Variable Fluid Properties Hanuesh Vaidya Departent o Matheatics, SSA Governent First Grade College (Autonoous),Ballari-583, Karnataka, India Abstract: The present article exaines the inluence o variable liquid properties on low and heat transer over a pereable stretching sheet with variable thickness. The transored syste o coupled non-linear ordinary dierential equations is solved analytically via optial hootopy analysis ethod (OHAM). Nuerical results are analyzed graphically. Wall thickness paraeter exhibits dual nature or low and heat transer patterns when it takes the value greater than or less than. The skin riction and the wall teperature gradient are exained or inluential paraeters in this consideration. Keywords: Variable thickness; Pereability; Skin riction; Nusselt nuber; OHAM Date o Subission: Date o acceptance: I. INTRODUCTION Fluid lows through pereable edia are o enthusias or soe ields o engineering and natural sciences, or exaple, oil recuperation, soil echanics, aterial adsorption on solids, iltration, and polyer property estiations. In these designing and applied research ields, the low through a pereable ediu is norally treated by a basic relationship broadly reerred to in a sued up rae as the Kozeny or Darcy law. In view o these application Abel et al. [] exained the ipact o periablity on the low and heat transer o a non-newtonian liquid over a non-isotheral stretching sheet. Pal and Mondal [] applied Soret and Duour eects, cheical reaction and theral radiation on low ield over a porous stretching sheet. Stretching/shrinking porous sheet geoetry is considered by Rosali et al. [3] and recorded the enhanceent in the skin riction coeicient and the local Nusselt nuber or increasing pereability paraeter. Recently, Bhatti et al. [4] continued the work o Re. [3] by considering shrinking porous sheet. One o the iportant acts that all these researchers have concentrated on the pioneering work o Crane [5] and oreover nuerous r e- searchers have exained the nature o luid low by considering the geoetry proposed by Crane[5](See Shehzad et al.[6], Prasad et al.[7], Vajravelu et al.[8], Hayat et al.[9], Zeeshan et al.[]) All the above researchers have explored the nature o low and heat transer o a Newtonian/non Newtonian luid by considering linear/ nonlinear stretching sheet. However, there is one ore special type o nonlinear stretching o the sheet recorded in the literature, naely, nonlinear stretching with variable thickness. Here, the boundary conditions are dierent ro conventional nonlinear stretching sheet probles such as u x U x b at y = A x b w. For all practical purposes deoring substances like needles and nozzles were the base or variable sheet thickness. A stretching sheet with a variable thickness can be ore close to the situation in practical applications. In the year Fang et al. [] coined the word variable thickness and analysed the low pattern nuerically. Many researchers extended the work o Fang et al. [] with the addition o heat and ass transer ( Khader and Megahed [], Prasad. et al.[3], Salahuddin et al. [4], Prasad et al.[5-7]). The object o present analysis is to predict the ipact o variable liquid properties on the low and heat transer o luid towards pereable stretching sheet o variable thickness. The relevant probles are orulated. Convergent series solutions o governing equations are constructed by optial hootopy analysis ethod (OHAM) ([8]-[]). Graphical results are used to elaborate the ipacts o involved paraeters. II. MATHEMATICAL FORMULATION Consider a steady two-diensional viscous incopressible luid low past a pereable stretching sheet with variable thickness. The origin is located at the slit, through which the sheet (see Fig. ) is drawn in the luid. The Flow caused due to nonlinear stretching sheet is restricted in doain y>. Stretching velocity o the pereable sheet is U ( x) = U x b where U and b are constants ( is the velocity exponent pa- w 49 Page

2 raeter). The proble stateents in the absence o pressure gradient are: u v, () x y u u u T u v ( T) u, () x y y y K T T T cp u v k T. (3) x y y y Fig. Physical description o the proble. where, (u,v) are the luid velocity coponents in the strea wise and cross-strea directions, respectively. The subscript denotes partial dierentiation with respect to the independent variables is the constant luid density, T is the coeicient o viscosity and K is the pereability o the porous e- T is considered to vary as an inverse unction o teperature (see or details diu. Here in this work Prasad et al. [8]). The appropriate boundary conditions or the proble are C r u x, y Uw U x b, v x, y, T x, y Tw x b at y Ax b, l u x, y, T x, y T as y. (4) Let the diensionless siilarity variable be U y ( x b), (5) ( x, y) ( ) U x b, T T Tw T (6) where x, y identically satisies the continuity Eq. (), the velocity coponents can be written as u Uw ( ) and v U ( x b) ( ) ( ). (7) Using above, Eqs. ()- (3) and (4) reduces to 5 Page

3 K, r ( ) ( r) r Pr, and the corresponding boundary conditions are,,,,. () The non-diensional paraeters, naely, variable thickness,the luid viscosity paraeter r, the porous paraeter Pr are deined as K and the Prandtl nuber U Tr T = A, r, K Kb, Pr =. T T T T w w The value or is deterined by the viscosity o the luid and = = A order to acilitate the coputation, we deine Now the equations becoe K, r ( ) ( r) r Pr, and the corresponding boundary conditions are,,,, U (9) (8) is the plate surace. In ( ) = and ( ) = ( ) = ( ). (3) where the prie denotes the dierentiation with respect to. With reerence to variable transoration, the integration doain will be ixed ro to. The shear stress and the wall teperature gradient respectively becoe = () and ( )= (). The values o engineering interest are the local skin riction x and the local Nusselt nuber uy y A xb x bty y A xb w Nu x deined as C ) Re, x x U Nu x ) Rex, T T w where Re x= Uw( x b) is the local Reynolds nuber. III. SEMI-ANALYTICAL NUMERICAL SOLUTION METHOD The governing equations are highly nonlinear, coupled ODEs with variable coeicients. We use the optial hootopy analysis ethod (OHAM) to obtain appropriate analytic solutions to equations () and () with associated boundary conditions (3). The OHAM has been successully applied to a wide variety o nonlinear probles (see [8-]). We choose the auxiliary linear operators as, () () C (4) 5 Page

4 L d d d d 3, 3 L d d, and L d d. (5) Initial approxiations satisying the boundary conditions (3) are ound to be () e, () e, and () e. Let us consider the so-called zero-th order deoration equations ( q) L ˆ( ; ) ˆ ˆ q ( ) qh ( ) N ( ; q), ( ; q), ( q) L ˆ( ; ) ˆ ˆ q ( ) qh( ) N ( ; q), ( ; q), (7) ( q) L ˆ( ; ) ˆ ˆ q ( ) qh( ) N ( ; q), ( ; q). (8) Here [,] (6) q is an ebedding paraeter, while, and are the convergence control paraeters. With these approxiations, we ay evaluate the residual error and iniize it over the paraeters, and in order to obtain the optial value o, and giving the least possible residual error. To do so, one ay use the integral o squared residual errors, however this is very coputationally deanding. To get around this, we use the averaged squared residual errors, deined by [M] [M] k ) E ( ) N ( ), (, M n k M k En ( ) N ( ), (, M [M] k [M] k ), [M]( k ) M k En ( ) N ( ), (, M [M] k [M] k ), [M]( k ) M k where k k / M, k,,,... M. For dierent order approxiations, the CPU tie required or obtaining the approxiate solutions will vary. Table lists the values o individual average residual errors by considering the optial values o.389,.35,.6977, which have been obtained by iniizing the averaged residual errors at the th order approxiation. Results are validated by coparing the present results with the results available in the literature (See Table.). IV. RESULT AND DISCUSSION In order to understand the atheatical odel, we present the nuerical results graphically or the horizontal velocity proile and the teperature proile with or dierent values o and paraeters, K, r, Pr, and r in Figs. to 6. The skin riction () and the wall teperature gradient () are tabulated in Table 3. Fig. (a) and Fig. (b) illustrates the eect o on or increasing values o. It shows that the velocity decreases with an increase in the value o. This iplies that the oentu boundary layer thickness becoes thinner as increases. Fig. 3(a) portrays the velocity distribution or dierent values o K. It indicates that the porous paraeter opposes the transport phenoena. This is due to the act that the variation in K leads to the variation o the Lorentz orce which in turn produces ore resistance to the transport phenoena. It is clearly seen ro the graph that the oentu boundary layer thickness decreases as K increases, and hence there is an increase (in absolute sense) in the velocity gradient () at the surace (see Table 3). Fig. 3(b) shows the eect o r on. It is observed that the velocity decreases with increasing r. Also, as r approaches to zero the boundary layer thickness is squeezed and velocity distribution asyptotically tends to zero. This is due to the act that or a given luid, when r is saller, higher is the teperature dierence between the wall and the abient luid. The results clearly reveal that r is the indicator o the variable viscosity with teperature which has a substantial eect on the velocity coponent 5 Page

5 and hence on the skin riction. We shall now turn our attention to the inluence o various paraeters on the teperature ield.the eect o on is exhibited in Fig. 4(a) and Fig. 4(b). The eect o increasing values o is to increase the teperature distribution. This is in conoration with the act that an increase in leads to an increase in the theral boundary layer thickness which is also true even or non-zero values o K. As explained above, increase in K increases the teperature as shown in Fig. 5(a). Moreover, the rate o cooling o the end product is iportant in several anuacturing processes such as etal and polyer extrusion, which will conir the quality o the end product. Fig. 5(b) explains eect o on. Fro the graphical representation it is seen that the eect o increasing value o r r is to enhance the teperature. That is, an increase in r results in an increase in the theral boundary layer thickness. The eect o Pr on can be ound ro Fig. 6(a). The igure deonstrates that an increase in Pr (eans decrease in the theral conductivity k ) leads to a decrease in the teperature. Hence the theral boundary layer thickness decreases as Pr increases. This is because luid with a higher values o Pr posses a large heat capacity and hence intensiies the heat transer. Thereore, cooling o the heated sheet can be iproved by choosing a coolant with a large Pr. Fig. 6(b) displays the eects o on. Fluid teperature is ound to increase with increasing values o which leads to a all in the rate o heat transer. That is, the assuption o teperature dependent theral conductivity suggests a reduction in the agnitude o the transverse velocity by a quantity k( T) y which can be seen in Eq. (.3).Thereore, the rate o cooling is uch aster or the coolant aterial having sall theral conductivity paraeter. Fig. 6(c) elucidates the eect o r on in the boundary layer. An increase in the value o r leads to decrease in and this is because when r >, heat lows ro the stretching sheet into the abient ediu and, when r <, the teperature gradient is positive and heat lows into the stretching sheet ro the abient ediu. An interesting observation ro the above results is that the velocity and teperature distributions depend heavily on the paraeters and. It is noticed that the velocity at any point near the plate decreases onotonically as increases or <. Also it is obvious ro the igures that the thickness o the boundary layer becoes thinner or higher values o when <, but the reverse is true or (see Fig. (a) and Fig. (b)). This is due to the induced ass transer. This oentu transer accelerates the luid particle at the downward region. This kind o signiicant change in the velocity can also be seen or positive values o, the stretching sheet case. For higher values o, theral boundary layer becoes thinner or < when copared with (See Fig. 4(a) and Fig. 4(b)). The eects o the physical paraeters on the skin riction () and the Nusselt nuber () are presented in Table 3. It is noticed that the eect o increasing values o the paraeters, K, r, is to decrease () and to increase (). The eect o increasing values o r and the Pr is to decrease (). Further it is observed that an increase in leads to a decrease in () as well as () or <, whereas an opposite trend is observed as >. V. CONCLUSIONS The iportant indings are as ollows: The diensionless velocity and teperature distributions at any point near the plate decrease when < and the thickness o the boundary layer becoes thinner when < and a reverse is true or. In the presence o teperature dependent thero-physical properties, the eect o increasing porous paraeter is to decrease the velocity ield. However, quite opposite is true with the theral boundary layer. The non-diensional heat transer rate reduces or increasing. The eect o the Pr is to decrease the theral boundary layer thickness and the wall teperature gradient. The eect o is to enhance the teperature in the low region and is reversed in the case o the r. O all the paraeters, the variable thero-physical property paraeters have strong eects on the drag, heat transer characteristics, the horizontal velocity ield and the teperature ield. ACKNOWLEDGMENTS: The authors appreciate the coents o the reviewers which led to deinite iproveent in the paper. 53 Page

6 REFERENCES []. M.S. Abel, S.K. Khan, K.V. Prasad, Study o visco-elastic liquid low and heat transer over a stretching sheet with variable viscosity, Int. J. Nonlinear Mech. 37 () () []. D. Pal, H. Mondal, Inluence o cheical reaction and theral radiation on ixed convection heat and ass transer over a stretching sheet in Darcian porous ediu with Soret and Duour eects, Energy Conv. Manage. 6 () 8. [3]. H. Rosali, A. Ishak, I. Pop, Micropolar luid low towards a stretching/shrinking sheet in a porous ediu with suction, Int. Coun. Heat Mass Transer 39 () [4]. M.M.Bhatti, A.Shahid, M.M.Rashidi, Nuerical siulation o Fluid low over a shrinking porous sheetsuccessive linearization ethod, Alexandria Engineering Journal, Vol. 55(),(6)5-56. [5]. Crane, L.J., Flow past a stretching plate, Zeitschrit ür angewandte Matheatik und Physik,, pp (97). [6]. Shehzad, S.A., Hayat, T. and Alasidi, A., Inluence o convective heat and ass conditions in MHD low o nanoluid, Bulletin o the polish acadey o sciences technical sciences, 63, No..(5). [7]. Prasad, K. V, Vajravelu, K, Shivakuara, I. S, Vaidya Hanuesh, Basha, Neeluer. Z., Flow and Heat Transer o a Casson Nanoluid Over a Nonlinear Stretching Sheet, Journal o Nanoluids, Vol.5, 6, pp (). [8]. K. Vajravelu. K. V. Prasad, Hanuesh Vaidya, Neeluer Z. Basha, Chiu-On Ng, Mixed Convective Flow o a Casson Fluid over a Vertical Stretching Sheet, Int. J. Appl. Coput. Math (7) 3: [9]. T. Hayat, M. Zubair, M. Ayub, M. Waqas, A. Alsaedi, Stagnation point low towards nonlinear stre t- ching surace with Cattaneo Christov heat lux, Eur. Phy. J. Plus 3 (6) 355. []. Zeeshan, A., Shehzad, N., Ellahi, R., Analysis o activation energy in Couette-Poiseuille low o nanoluid in the presence o cheical reaction and convective boundary conditions, Results in Physics, 8, pp. 5-5(8). []. Fang, T., Zhang, J., Zhong, Y., Boundary layer low over a stretching sheet with variable thickness, Applied Matheatics and Coputation, 8, pp (). []. Khader, M. M., Megahed, A.M., Boundary Layer Flow Due to a Stretching Sheet with a Variable Thickness and slip Velocity, Journal o Applied Mechanics and Technical Physics, 56, pp 4 47 (5). [3]. Prasad. K. V, Vajravelu. K, Hanuesh Vaidya. Hall eect on MHD low and heat transer over a stretching sheet with variable thickness, International Journal or Coputational Methods in Engineering Science and Mechanics, DOI:.8/ ,6. [4]. T. Salahuddin, M.Y. Malik, Ari Hussain, S. Bilal, M. Awais, MHD low o Cattanneo-Christov heat lux odel or Williason luid over a stretching sheet with variable thickness: Using nuerical approach, Journal o Magnetis and Magnetic Materials, Vol. 4, 6, Pages [5]. K.V. Prasad, K. Vajravelu, Hanuesh Vaidya, Robert A. Van Gorder, MHD low and heat transer in a nanoluid over a slender elastic sheet with variable thickness, Results in Physics 7 (7) [6]. Prasad, K.V., Vaidya, H., Vajravelu, K. and Rashidi, M., Eects o Variable Fluid Properties on MHD Flow and Heat Transer over a Stretching Sheet with Variable Thickness, Journal o Mechanics, 33(4), pp. 5-5(7). [7]. K. V. Prasad, Hanuesh Vaidya, K. Vajravelu, and V. Raanjini, Analytical Study o Cattaneo-Christov Heat Flux Model or Williason-Nanoluid Flow Over a Slender Elastic Sheet with Variable Thickness, J. Nanoluids 7, (8). [8]. Liao S. A new branch o solutions o boundary-layer lows over a pereable stretching plate, Int. J. o Non-Line. Mech., Vol. 4, no. 6, pp 89-83, 7. [9]. Liao, S., An optial hootopy-analysis approach or strongly nonlinear dierential equations, Coun Nonlinear Sci Nuer Siul, 5(8), pp.3 6(). []. Van Gorder, R.A. (Optial hootopy analysis and contro o error or iplicitly deined ully nonlinear dierential equations, Nuer Algor (8) Page

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10 Table : Individual average residual error as a unction o the nuber o iterations. CPU tie required to calculate the solution is also listed. Paraeter values are ixed at Pr.9, Le.,., r,,, K. We have optial convergence control paraeter values o 3.389,.35, n n n n CPU tie (Sec) 58 Page

11 Table. Coparison o skin riction or dierent values o and with, K.. r 4 Khader and Megahed [] when Present Khader and Present Fang et Fang et Work by Megahed [] Work al.[] al.[]. OHAM when. OHAM / / by Table 3. Variation o skin riction and wall-teperature gradient or dierent values o physical paraeters. Pr r r K. () 4 () () () () () -/ / / / Page

12 Hanuesh Vaidya "An Optial Analysis o Flow and Heat Transer over a Slender Pereable Elastic Sheet with Variable Fluid Properties "IOSR Journal o Engineering (IOSRJEN), vol. 8, no. 6, 8, pp Page