MHD Heat and Mass Transfer Forced Convection Flow Along a Stretching Sheet with Heat Generation, Radiation and Viscous Dissipation

Transcription

1 Dhaka Univ. J. Sci. 59(): -8, (July) MHD Heat and Mass Transer Forced onvection Flo Along a Stretching Sheet ith Heat Generation, Radiation and Viscous Dissiation M. A. Samad, Sajid Ahmed and Md. Motaleb Hossain Deartment o Mathematics, Dhaka University, Dhaka-, Bangladesh. Institute o Natural Sciences, United International University, Dhaka-9, Bangladesh Received on Acceted or Publication on.. Abstract The resent study comrises o steady to dimensional magnetohydrodynamic heat and mass transer orced convection lo along a vertical stretching sheet in the resence o magnetic ield ith radiation and viscous dissiation. The roblem has been analyzed by alying Nachtsheim-Sigert shooting iteration technique ith sith order Runge-Kutta integration scheme. The nonlinear artial dierential equations governing the lo ield occurring in the roblem have been transormed to dimensionless nonlinear ordinary dierential equations by introducing suitably selected similarity variables. The ensuing equations are simultaneously solved by alying numerical iteration scheme or velocity, temerature and concentration ields. The results are dislayed grahically in the orm o velocity, temerature and concentration roiles. The corresonding skin-riction coeicient, sselt number and Sherood number are dislayed in tabular orm. Several imortant arameters such as the Prandtl number (Pr), radiation arameter ( N), magnetic ield arameter ( M), heat source arameter (Q), Schmidt number (Sc), suction arameter ( ) and Eckert number (Ec) are conronted. The eects o these arameters on the velocity, temerature and concentration roiles are investigated. Key Words: MHD, Forced onvection, Stretching sheet, Heat Generation, Radiation. I. Introduction Out o to tyes o convective heat and mass transer, orced convection is the one here the velocity o the lo dominates over the other arameters. Some kind o eternal orces are emloyed here. The cooling system in a car engine is an eamle o orced convection. Thermal radiation eects may lay an imortant role in controlling heat transer in industry here the quality o the inal roduct deends on the heat controlling actors to some etent. High temerature lasmas, cooling o nuclear reactors, liquid metal luids, and oer generation systems are some imortant alications o radiative heat transer rom a vertical all to conductive gray luids. Thermal radiation eect on orced and ree convection have been studied in recent years etensively, as the magnetohydrodynamic [MHD] lo and heat transer roblems have become more imortant in many engineering and industrial alications. Hossain and Takhar [] studied the radiation eects using the Rosseland diusion aroimation [] (Seigel and Hoell, 97) that leads to non similar boundary layer equation governing the mied convection lo o an otically dense viscous incomressible luid ast heated vertical late ith a ree uniorm velocity and surace temerature. Shateyi et al. [3] studied the thermal radiation and buoyancy eects on heat and mass transer over a semi-ininite stretching surace ith suction and bloing. Ali et al. [4] studied the radiation eect on natural convection lo over a vertical surace in a gray gas. Folloing Ali et al. Mansour [5] studied the interaction o mied convection ith thermal radiation in laminar boundary layer lo over a horizontal, continuous moving sheet ith suction / injection. Elbashbeshy et al. [6] studied the determination o the eect o radiation on orced convection lo o a microolar luid over a horizontal late. Aydin et al. [7] investigated the mied convection heat transer about a ermeable vertical late in the resence o magneto and thermal radiation eects. ess [8] studied to determine the inluence o radiation heat transer uon the orced convection sselt number. Samad and Rahman [9] investigated thermal radiation interaction ith unsteady MHD lo ast a vertical orous late immersed in a orous medium. Samad and Karim [] studied thermal radiation interaction ith unsteady MHD ree convection lo through a vertical lat late ith time deendent suction in the resence o magnetic ield. As in the case o stretching sheets, hen [] studied laminar mied convection adjacent to vertical continuously stretching sheet. hiam [] investigated magnetohydrodynamic heat transer over non-isothermal stretching sheet. Po et al. [3] studied radiation eect on the lo near the stagnation oint o a stretching sheet. Abo-eldahab [4] studied lo and heat transer in a microolar luid ast a stretching surace embedded in a non-darcian orous medium ith uniorm ree stream. Samad et al. [5] investigated magnetohydrodynamic heat and mass transer orced convection lo along a stretching sheet ith heat generation/ absortion. T. S. Khaleque and Samad [6] studied the eects o radiation, heat generation and viscous dissiation on MHD ree convection lo along a stretching sheet. In the resent study thermal radiation interaction on MHD orced convection lo in the resence o a uniorm magnetic ield over a vertical stretching sheet ith heat generation and viscous dissiation has been investigated. II. Mathematical Formulation We consider a steady to-dimensional magnetohydrody namic heat and mass transer lo o a viscous

2 incomressible luid along a vertical stretching sheet ith constant heat generation/absortion ith radiation. M. A. Samad, Sajid Ahmed and Md. Motaleb Hossain Rosseland aroimation and alying the Taylor series eansion the energy equation (3) takes the orm, u = D g u v T T u v y Q ( T T T u y y 3 T 6 ) 3 k y T (5) v () u = D O Fig.: Flo coniguration and oordinate system. We take the -ais along the sheet and y-ais erendicular to it. To equal and oosite orces are introduced along the -ais so that the sheet is stretched keeing the origin ied. A uniorm magnetic ield o strength B is imosed along the y-ais. A radiation deending on temerature is alied on the stretching sheet. The governing equations reresenting the roosed lo ield are: ontinuity Equation: u v y Momentum Equation: u u u B u v sg ( T T ) u () y y Energy Equation: T oncentration Equation: T T T T u u v y y y Q q r ( TT ) y u v D y m y We take s = to vanish the second term o the right hand side o () or the orced convection. Ater using the T B B y () (3) (4) Here, the second term in the right hand side reers to the viscous dissiation term. The boundary conditions as a hole are u D, u, v v, T T, T T, at y, as y The olloing similarity variables are introduced to obtain a similarity solution corresonding to the above roblem. (6) D y (7) D () T T () T T (8) (9) () () Here is the stream unction, is the dimensionless distance normal to the sheet, is the dimensionless stream unction, is the dimensionless luid temerature and is the dimensionless concentration. Since u y and v e have the velocity comonents rom equation (4) given by u D ( ) () v D ( ) Here rime denotes the derivatives ith resect to. No substituting the similarity variables rom equations (7) to () into the equations (), (5) and (4) e have the system as, ( ) M () 3N Pr 3N Pr Q 3N 4 3N 4 3N Pr Ec ( ) 3N 4 (3) Sc (4)

3 MHD Heat and Mass Transer Forced onvection Flo along a Stretching 3 The transormed boundary conditions are:,,, at (5),, as Here, B M is the magnetic ield arameter, Pr D is the Prandtl number, Q Q is the heat source/ sink D arameter, v N D k 3 4 T is the radiation arameter and, is the suction arameter. III. merical omutations The nonlinear ordinary dierential equations (8), (9) and () under the boundary conditions () are solved numerically or various values o the arameters occurring in the roblem. Here e use the standard initial-value solver shooting method namely Nachtsheim-Sigert (965) [7] shooting iteration technique (g uessing the missing value) together ith the sith order Runge-Kutta-Butcher initial value solver. ' = 5., Pr =., N =., M =., Q =., Sc =., Ec = Fig. (a). Velocity roiles or various ste sizes = 5., Pr =., N =., M =., Q =., Sc =., Ec =. A ste size o =. has been used together ith 6 accuracy o in all the cases. The value o ma as selected in accordance ith the values o each grou o arameters Pr,, M, Q, N and Ec to satisy the accuracy requirement. The code veriying grahs or dierent ste sizes are shon in the Fig. (a) and Fig. (b). Here, e see that or ste sizes, =.,.5,. the velocity and the temerature roiles are in ine agreement among them. IV. Results and Discussion We use the dimensionless velocity, temerature, and concentration roiles to resent the results obtained in the numerical comutations. merical comutations have been carried out or various values o the arameters entering into the roblem in comliance ith the dierent hysical conditions. These arameters are, the Prandtl number (Pr), suction arameter ( ), magnetic ield arameter ( M), radiation arameter ( N), Schmidt number ( Sc), heat source arameter (Q) and Eckert number (Ec). Signiicant eect o Prandtl number variation is observed in the temerature ield shon in the Fig. 3. The temerature roiles increase u to a value o about. 5 and then it start to decrease ith the increase o the Prandtl number. The thermal boundary layer decreases ith the increase o the Prandtl number. We see that or air that is, or Pr =.7 the thermal boundary layer is thicker than that or the other values o Pr. With the increase o the Prandtl number the thermal boundary layer thickness start to decrease. For Pr = 7. the temerature rises the most and then raidly decrease. The result is in good agreement ith that given by Samad et al. [5]. Fig. 4 shos the eect o radiation arameter on the velocity, temerature and concentration roiles. The velocity roiles in the Fig. 4 (a) do not sho any signiicant eect = 5., Ec = 5., M =., Q =., N =., Sc =. Pr =.7,.,., 5., Fig. (b). Temerature roiles or various ste sizes Fig. 3. Temerature roiles or various values o Prandtl numbers.

4 4 This is because o the act that in the orced convection the velocity is relatively large Thereore, no signiicant eects in the velocity roiles are observed. The momentum boundary layer thickness does not change due to variations in the radiation arameters. The temerature roiles have shar eects as shon in the Fig. 4(b). The temerature roiles cross each other near about.. The temerature at irst rises and then decreases don. The thermal boundary layer thickness also changes due to the change in the radiation arameter. For N =. the thermal boundary layer thickness is very high. As the radiation arameter value increases the thermal boundary layer thickness also decreases. M. A. Samad, Sajid Ahmed and Md. Motaleb Hossain ' =., Pr = 7., N =., Q =., Sc =.6, Ec =. M =.,., 3., 4., Fig. 5(a). Velocity roiles or values o magnetic ield arameter..8 = 5., Pr = 7., M =., Q =., Sc =., Ec = 5. =., Pr = 7., N =., Q =., Sc =.6, Ec =. '.6 N =.,.5,.,., M =.,., 3., 4., Fig. 4(a). Velocity roiles or various values o radiation arameter = 5., Pr = 7., M =., Q =., Sc =., Ec = 5. N =.,.5,.,., Fig. 5(b). Temerature roiles or values o magnetic ield arameter..8.6 =., Pr = 7., N =., Q =., Sc =.6, Ec =. M =.,., 3., 4., Fig. 4(b). Temerature roiles or various values o radiation arameter. In the Fig. 5 it is shon that all the dimensionless velocity, temerature, and concentration roiles have eects or the variation in the magnetic ield arameter M. The velocity roiles are shon in the Fig. 5(a). We see that the velocity roiles decrease ith the increase in the magnetic ield arameter values. This haens due to the retarding eect o the magnetic ield in the lo ield. The momentum boundary layer thickness decreases ith the increase o the magnetic ield arameter values. This indicates e can eectively use the magnetic ield to control the boundary layer. The temerature roiles are shon in the Fig. 5(b) Fig. 5(c). oncentration roiles or various values o magnetic ield arameter. The temerature roiles at irst mark an increase and then decrease ith the increase o the magnetic ield arameter. The thermal boundary layer thickness also changes a small amount due to the increase in the magnetic ield arameter. The concentration roiles are shon in Fig. 5(c). They sho a slight increase ith the increase o the magnetic ield arameter. This occurs because the magnetic eect on the luid articles increases the luid concentrations. The results are analogous in Samad et al. [5]. A small dierence is observed in the temerature roiles. This is

5 MHD Heat and Mass Transer Forced onvection Flo along a Stretching 5 due to the changed values o the various arameters and the resence o additional arameters as radiation and viscous dissiation. The concentration roiles sho a shar change or the change in the Schmidt number ( Sc). This is shon in the Fig. 6. With the increase in the Schmidt number the concentration roiles mark a major decrease. The concentration boundary layer thickness also changes ith the increase in the Schmidt number (Samad et al. [6]). ' Pr = 7., N =., M =., Q =., Sc =.6, Ec = 3. =.,., 3., 4., = 5., Pr = 7., N =., M =.5, Q =., Ec =. Sc =.,.,.4,.6,..5.5 Fig. 8(a). Velocity roiles or various values o suction arameter Pr = 7., N =., M =., Q =., Sc =.6, Ec = 3. =.,., 3., 4., Fig. 6. oncentration roiles or various values o Schmidt number = 5., Pr = 7., N =.5, M =., Sc =.6, Ec = 3. Q =.,., 4., 6., Fig. 8(b). Temerature roiles or various values o suction arameter..8.6 Pr = 7., N =., M =., Q =., Sc =.6, Ec = 3. =.,., 3., 4., Fig. 7. Temerature roiles or values o heat source arameter. The temerature roiles in the Fig. 7 shos eects ith the change in the heat source arameter. The temerature increases at irst as the heat source arameter increases. Then it decreases gradually. This is evident as because the increase in the heat source arameter ill obviously increase the temerature. The thermal boundary layer thickness changes signiicantly in Samad et al. [5], but in the resent ork this change is insigniicant oing to the resence o viscous dissiation in the lo ield..5.5 Fig. 8(c). oncentration roiles or various values o suction arameter. In the Fig. 8 the eect o suction arameter is shon. Here in this case, the dimensionless velocity roiles in the Fig. 8(a) shos a decrease ith the increase in the suction arameter. This is because the increase in the suction arameter contributes more to remove the boundary layer materials rom the lo ield. Also or this reason the boundary layer thickness decreases ith the increase o the suction arameter. The temerature roiles are shon in the Fig. 8(b). Here e see that the temerature rises raidly hen suction arameter has a smaller value, that is,.. As the suction

6 6 arameter value increases the temerature starts to decrease evidently. The thermal boundary layer decreases a large amount ith the increase o the suction arameter. The concentration roiles are ehibited in the Fig. 8(c). As evident they also sho a shar decrease ith the increase o the suction arameters. The results sho a resemblance ith Samad et al [5]. Finally the variation o Eckert number is shon in the Fig The temerature roiles sho a shar increase ith the increase o the Eckert number. This haens due to the buoyancy eects on the lo ield. The thermal boundary layer thickness changes a small amount = 5., Pr = 7., N =., M =., Q =., Sc =. Ec =.,.,., 3., Fig. 9. Temerature roiles or various values o Eckert number. V. Skin-Friction oeicient, sselt mber and Sherood mber The skin riction coeicient ( ), local sselt number ( ) and the local Sherood numbers ( Sh) are signiicant in the engineering ield. These arameters reer to the all shear stress, local all heat transer rate and all mass transer rate resectively. It is observed that the skin-riction coeicient, sselt number, and Sherood number are roortional to ( ), (), and () resectively. Tables comrising o the roortional values o,, and Sh corresonding to the revious grahs are shon in the olloing tables. The numerical values are in ecellent agreement comared to Samad et al. ()[5]. The local heat transer rate i. e., shos some change caused by the additional arameters introduced in the energy equation (3). Table. (a). Values o Pr. comared to ublished results or Pr Samad et al. () Present M. A. Samad, Sajid Ahmed and Md. Motaleb Hossain Table. (b). Values o Sh comared to ublished results or Pr. Pr Samad et al. () Present Table. (c). Values o Pr. comared to ublished results or Pr Samad et al. () Present Table. (a). Values o M. comared to ublished results or M Samad et al. () Present Table. (b). Values o Sh comared to ublished results or M. M Samad et al. () Present Table. (c). Values o results or M. comared to ublished M Samad et al. () Present

7 MHD Heat and Mass Transer Forced onvection Flo along a Stretching 7 Table. 3(a). Values o or. comared to ublished results Samad et al. () Present Table. 3(b). Values o Sh comared to ublished results or. Samad et al. () Present Table. 3(c). Values o results or.. comared to ublished Samad et al. () Present Table. 4(a). Values o or Sc. comared to ublished results Sc Samad et al. () Present Table. 4(b). Values o Sh comared to ublished results or Sc. Sc Samad et al. () Present Table. 4(c). Values o results or Sc. comared to ublished Sc Samad et al. () Present Table. 5(a): Values o results or Q. comared to ublished Q Samad et al. () Present Table. 5(b): Values o Sh comared to ublished results or Q. Q Samad et al. () Present Table. 5(c): Values o results or Q. comared to ublished Q Samad et al. () Present Table. 6:, and Sh or dierent values o N. N Sh

8 8 Table. 7:, Ec and Sh or dierent values o Ec VI. onclusion The roblem has dealt ith the to dimensional heat and mass transer orced convection lo o an MHD luid along a vertical stretching sheet in the resence o magnetic ield ith heat generation, radiation and viscous dissiation. From the results obtained in the resent analysis and comaring ith the results by Samad et al. [5] the olloing conclusions can be made:. In the orced convection the velocity being large, the Prandtl number has no eective dominance over velocity and concentration. Nevertheless on the temerature it has a signiicant eect. With viscous dissiation taken into consideration, local heat transer rate decreases ith the increase o the Prandtl number.. The radiation arameter has a large eect on the temerature roiles, and hence can be used eectively to control the temerature o the lo ield. 3. The magnetic arameter has signiicant eect on the velocity, temerature and concentration roiles. The local skin-riction coeicient, heat transer rate and mass transer rate decrease ith the increase o the magnetic arameter. So, magnetic ield can eectively be used to control the lo ield. 4. As aarent the temerature increases ithin the boundary layer ith the increase o the heat source arameter. 5. For the increase o the suction arameter the velocity, temerature and concentration signiicantly decrease and as a consequence making it one o the most essential arameter in the boundary layer control theory. 6. The Schmidt number has large eects on the concentration roiles dominating the mass transer rate o the lo. 7. The eect o Eckert number is clearly visible on the lo attern. With viscous dissiation in eect the temerature rises in the vicinity o the all Hossain, M. A., H. S. Takhar, 996. Radiation Eect on Mied onvection along a Vertical Plate ith Uniorm Surace Temerature. Heat and Mass Trans. 3(4), Sh M. A. Samad, Sajid Ahmed and Md. Motaleb Hossain. Seigel, R., J. R. Hoell, 97. Thermal Radiation Heat Transer. McGra-Hill, Ne York. 3. Shateyi, S., M. Petersen, 8. Thermal Radiation and Buoyancy Eects on Heat and Mass Transer over a Semi- Ininite Stretching Surace ith Suction and Bloing. J. Al. Math. 8 (8),. 4. Ali, M. M., T. S. hen, and B. F. Armaly, 984. Natural onvection Radiation Interaction In Boundary Layer Flo over Horizontal Suraces. AIAA Journal, (), Mansour, M. A, 99. Radiative and Free onvection Eects on the Oscillatory Flo ast a Vertical Plate. Astrohysics Sace Sci. 66, Elbashbeshy, E. M. A., M. A. A. Bazid,. Eect o Radiation on Forced onvection Flo o a Microolar Fluid over a Horizontal Plate. an. J. Phys. 78, Aydin, O., A. Kaya, 8. Radiation Eect On MHD Mied onvection Flo About A Permeable Vertical Plate. Heat Mass Trans. 45, ess, R. D., 966. The Eect o Radiation uon Forced onvection Heat Transer, Al. Sci. Res. Sec A., Samad, M. A., M. M. Rahman, 6. Thermal Radiation Interaction ith Unsteady MHD Flo ast a Vertical Porous Medium. J. Nav. Arch. Mar. Eng. 3, Samad, M. A., M. E. Karim, 9. Thermal Radiation Interaction ith Unsteady MHD Flo ast a Vertical Flat Plate ith Time Deendent Suction. Dhaka Univ. J. Sci. 57(), hen,. H., 998. Laminar Mied onvection Adjacent to Vertical, ontinuously Stretching Sheet. Heat and Mass Tran. 33, hiam, T.., 997. Magnetohydrodynamic Heat Transer over a Non-Isothermal Stretching Sheet. Acta Mechanica., Po. S. R., T. Grosan, I. Po, 4. Radiation Eect on the Flo near the Stagnation Point o a Stretching Sheet. The hnische Mechanik, 5, Abo-Eldahab, E. M., 5. Flo And Heat Transer in a Microolar Fluid ast a Stretching Surace Embedded in a Non-Darcian Porous Medium ith Uniorm Free Stream. Al. Math. om. 6, Samad, M. A., M. Mohebujjaman, M. Mustak Mia, M. A. Rahman,. Magnetohydrodynamic heat and Mass Transer Forced onvection Flo along a Stretching Sheet ith Heat Generaion/ Absortion. Dhaka Univ. J. Sci. 58(), Khaleque, T. S., M. A. Samad,. Eects o Radiation, Heat Generation and Viscous Dissiation on MHD Free onvection Flo along a Stretching Sheet. Res. J. Al. Sci. Eng. Technol., (3), Nachtsheim, P. R., P. Sigert, 965. Satisaction o the Asymtotic Boundary onditions in merical Solution o the System o Non-Linear Equations o Boundary Layer Tye. NASA TND-34.