The effects of suction and injection on a moving flat plate in a parallel stream with prescribed surface heat flux

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1 Noriah Bachok Anuar Ishak The eects o suction inection on a moving lat plate in a parallel stream ith prescribed surace heat lux NORFIFAH BACHOK & ANUAR ISHAK Department o Mathematics Faculty o Science Universiti Putra Malaysia 3 UPM Serdang Selangor MALAYSIA School o Mathematical Sciences Universiti Kebangsaan Malaysia 36 UKM Bangi Selangor MALAYSIA noriah@math.upm.edu.my Abstract: - The eect o surace mass lux on a moving lat plate in a moving luid ith prescribed surace heat lux is studied. The governing partial dierential boundary layer equations are irst transormed into ordinary dierential equations beore being solved numerically by a inite dierence method. The eatures o the lo heat transer characteristics or dierent values o the governing parameters are analyzed discussed. It is ound that dual solutions exist hen the plate the ree stream move in the opposite directions. The results indicate that the range o knon dual solutions increases ith suction decreases ith inection the rate o heat transer increases ith increasing heat lux exponent parameter. Key-Words: - Heat transer Moving plate Moving luid Boundary layer Heat lux Suction/inection Dual solutions Introduction The classical boundary layer lo past a lat plate or the Blasius problem has attracted considerable interest o many researchers since introduced by Blasius []. Blasius considered the boundary layer lo on a ixed lat plate ithout considering the heat transer aspects. The study o the lo heat transer in an electrically conducting luid has many practical applications in manuacturing process in industry. The thermal luid lo problem have been extensively studied numerically theoretically as ell as experimentally (see [-]). The Blasius equation has never yielded to exact analytical solution Blasius himsel gave matching inner outer series solutions. Dierent rom Blasius [] Sakiadis [5] considered the boundary-layer lo on a moving plate in a quiescent ambient luid. He ound exactly the same equation as Blasius but the boundary conditions are dierent. Ishak et al. [6] extended the classical problems o Blasius [] Sakiadis [5] by considering a lat plate moving in the same or opposite directions to a parallel ree stream all ith constant velocities. Similar problems ith various boundary conditions in dierent situations have been considered by Klemp Acrivos [7] Merkin [8] Abdelhaez [9] Hussaini et al. [] Azal et al. [] Bianchi Viskanta [] Lin Huang [3] Chen [] Azal [5] Abraham Sparro [6] Sparro Abraham [7] Weidman et al. [8]. Hoever the existence o dual solutions as reported only in the papers by Merkin [8] Azal et al. [] Azal [5] Weidman et al. [8]. Examples o practical applications include the aerodynamic extrusion o plastic sheets the cooling o an ininite metallic plate in a cooling bath the boundary layers along material hling conveyers along a liquid ilm in condensation processes glass bloing continuous casting spinning o ibers etc. It is ell knon that the skin riction along a continuous moving lat surace in a quiescent luid (Sakiadis problem) is about 3% higher than those along a static lat plate in a moving luid (Blasius problem). As discussed by Abdelhaez [9] these to problems are physically dierent can not be mathematically transormed into one another. This act indicates that e cannot regard the velocity dierence U U as a relative velocity in the sense o Galilei. Excellent descriptions o the problem o laminar luid lo hich results rom the simultaneous motions o a ree stream its bounding surace in the same direction have been in detail examined by Abraham Sparro [6] Sparro Abraham [7] using the relative-velocity model hich uses magnitude o the relative velocity in conunction ith the drag ormula or the case in hich only one o the participating media is in motion. They ound that the results o exact solutions demonstrate that this model is laed under predicts the drag orce ISSN: Issue 3 Volume 5 uly

2 Noriah Bachok Anuar Ishak thus the use o the relative-velocity model can lead to gross errors in the drag orce. The extent o the error increases as the to participating velocities approach each other in magnitude. The solution depends not only on the velocity dierence U U but also on the velocity ratio U / U. Folloing Azal et al. [] in this paper e deine the reerence velocity U as U U + U thus a single set o boundary conditions is ormed irrespective o hether U > U or U < U. The eects o suction inection on a moving lat plate opposite to the ree stream in a poer la luid into or out o the origin at uniorm speed in the same or opposite direction to the ree stream ere studied by Weidman et al. [8] Zheng et al. [9] Ishak et al. [] respectively. Continuous surace heat transer problems have many practical applications in industrial manuacturing processes. Such processes are hot rolling ire draing glass iber production. Problems ith variable surace heat lux has been introduced in many other studies [-]. The purpose o this investigation is to study the eects o suction inection on a moving lat plate in a parallel stream ith variable surace heat lux. Problem Formulation Consider a to-dimensional boundary layer lo on a ixed or continuously moving permeable lat surace immersed in a viscous incompressible luid o constant temperature T. It is assumed that the plate is n subected to a variable surace heat lux q ( x) ax here a n are constants x is the distance rom the slit here the plate is issued moves in the same or opposite direction to the ree stream both ith constant velocities U U respectively. Under these assumptions the boundary layer equations are u v + x y u u u u + v ν x y y T T T u + v α x y y () () (3) subect to the boundary conditions T q u U v at y y k u U T T as y here u v are the velocity components in the x- y- directions respectively ν T q k α V are respectively the kinematic viscosity temperature o the luid in the boundary layer surace heat lux thermal conductivity thermal diusivity the mass transer velocity through the surace o the plate. In order to solve Eqs. () (3) subect to the boundary conditions () e introduce the olloing similarity transormation: ( η) U η y vx ψ ( vxu) ( ) k T T U θ( η) q vx () (5) here U is the composite velocity deined as U U + U (Azal et al. []). Further ψ is the stream unction deined as u ψ y v ψ x hich identically satisies Eq. (). Using (5) e obtain ( η) u U vu v ( η ) x here primes denote dierentiation ith respect to η. In order that similarity solutions o Eqs. () (3) exist e take νu V( x) x here ( ) (6) (7) is a non-dimensional constant hich determines the transpiration rate at the surace ith > or suction < or inection corresponds to an impermeable plate. By employing the similarity variables (5) Eqs. () (3) reduce to the olloing ordinary dierential equations: ISSN: Issue 3 Volume 5 uly

3 Noriah Bachok Anuar Ishak + (8) ( n ) Pr θ (9) C Nu x Re Re x x θ ( ) ( ) () The boundary conditions () no become ( ) ( ) λ θ ( ) ( ) ( ) as η λ θ η η here λ is the velocity ratio parameter deined by () here Re Ux x v is the local Reynolds number. The nonlinear ordinary dierential equations (8) (9) subected to () are solved numerically by a initedierence scheme knon as the Keller-box method hich is very amiliar to the present authors (see Bachok et al. [5] Bachok Ishak [6] Ishak et al. [7 8]). U λ. () U ith λ > λ < correspond to the plate moving in the same opposite directions to the ree stream respectively hile λ represents a ixed plate. We the lo notice that or an impermeable plate ( ) problem under consideration reduces to that considered by Blasius [] hen λ to that o Sakiadis [5] hen λ. The physical quantities o interest are the skin riction coeicient C the local Nusselt number Nu hich are deined as τ C ρ U xq Nu x k( T T) here the all shear stress τ the all heat lux are given by x () q 3 Solution Procedure 3. Finite-dierence method To solve the transormed dierential Eqs. (8) (9) subected to the boundary conditions () Eqs. (8) (9) are irst converted into a system o ive irst-order equations the dierence equations are then expressed using central dierences. For this purpose e η q η introduce ne dependent variables p ( ) ( ) s ( η ) θ( η) ( η) ritten as t so that Eqs. (8) (9) can be p (5) p q (6) s t (7) q + q (8) t + t ( n+ ) ps. (9) Pr u τ µ y y T q k y y (3) In terms o the ne dependent variables the boundary conditions () are given by ( ) ( ) λ ( ) ( ) ( ) p t p η λ s η as η () ith µ k being the dynamic viscosity thermal conductivity respectively. Using the similarity variables (5) e obtain We no consider the segment η η ith / η as the midpoint hich is deined as belo: η η η + h η η () here h is the η spacing... is a ISSN: Issue 3 Volume 5 uly

4 Noriah Bachok Anuar Ishak sequence number that indicates the coordinate location. The inite-dierence approximations to the ordinary dierential equations (5)-(9) are ritten or the midpoint η / o the segment η η. This procedure gives p s q Pr h p h s h q h t t h p q t p + q + t t p q / ( q) / / / ( t) ( n+ )( ps). / Rearranging o expressions ()-(6) gives ( p + p ) / () (3) () (5) (6) h (7) p p h ( q + q ) (8) s s h ( t + t ) (9) ( + )( q + q ) q q + h (3) 8 ( t t ) + h( + )( t + t ) Pr 8 h( n+ )( p + p )( s + s ). (3) Equations (7)-(3) are imposed or 3... the transormed boundary layer thickness η is to be suiciently large so that it is beyond the edge o the boundary layer. The boundary conditions are p t λ p λ s. (3) 3. Neton s method To linearize the nonlinear system (7)-(3) e use Neton s method by introducing the olloing expressions: ( k+ ) ( k) ( k) ( k+ ) ( k) ( k) + δ p p + δ p ( k+ ) ( k) ( k) ( k+ ) ( k) ( k) q q + δ q s s + δ s ( k+ ) ( k) ( k) t t + δ t (33) here k... We then insert the let-h side expressions in place o p q s t into Eqs. k (7)-(3) drop the terms that are quadratic in δ ( k ) ( k ) ( k ) ( k ) δ p δ q δ s δ t. This procedure yields the olloing linear system (the superscript k is dropped or simplicity): h δ δ ( δ p + δ p ) ( r ) / (3) h δ p δ p ( δ q + δ q ) ( r ) ' (35) h δ s δ s ( δ t + δ t ) ( r 3) / (36) ( ) ( ) ( 3) ( ) δ ( ) a δ q + a δ q + a δ + a r / ( ) ( ) ( ) ( ) ( b5) δ p + ( b6) δ p + ( b7) δ s + ( b ) δ s ( r ) b δ t + b δ t + b δ + b δ + 3 here 8 5 / a + h / a a a3 hq / a a 3 b + Pr h / b b b3 Pr h t / b b 3 b5 Pr n h s / b6 b 5 b7 Pr n h p / b8 b 7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( + ) ( ) ( ) ( ) ( + ) ( ) ( ) ( ) (37) (38) (39) ISSN: Issue 3 Volume 5 uly

5 Noriah Bachok Anuar Ishak ( ) ( ) ( ) r + + h p / / r p + p + h q / / r s + s + h t 3 / / r q q h q / / 5 / + / h( n+ )( ps ). / ( ) ( ) ( ) ( r ) ( t t ) Pr h ( t) The boundary conditions (3) become δ δ p δ t δ p δ s () () [ δ ] [ δ ] [ δ ] [ δ ] [ r] [ r ] δ r. [ r ] [ r ] The elements o the matrices are as ollos: [ A] h h h ( a) ( a3) ( a) ( b8) ( b3) ( b) (3) hich ust express the requirement or the boundary conditions to remain constant during the iteration process. 3.3 Block-elimination method The linearized dierence equations (3)-(38) can be solved by the block-elimination method as outlined by Na [9] Cebeci Bradsha [3] since the system has block-tridiagonal structure. Commonly the blocktridiagonal structure consists o variables or constants but here an interesting eature can be observed that it consists o block matrices. In a matrix-vector orm Eqs. (3)-(38) can be ritten as here Aδr () [ A] [ C] [ B ] [ A ] [ C ] A [ B ] [ A ] [ C ] [ B] [ A ] h h A h ( a3) ( a) ( b6) ( b8) ( b3) ( b) B h h ( a) ( a) ( b) ( b) h C ( b5) ( b7) () (5) (6) ISSN: Issue 3 Volume 5 uly

6 Noriah Bachok Anuar Ishak [ δ ] δ q δ s δ p δ s δ δ δ δ q δ q δ t δ t r ( r ) ( r ) ( r ) ( r ) ( r ) / / 3 / / 5 / (7) (8) To solve Eq. () e assume that A is nonsingular it can be actorized as here ALU (9) [ α] [ B ] [ α ] L [ α ] [ B] [ α ] [ I] [ Γ ] [ I] U [ I] [ Γ ] [ I] here [ I ] is a 5 5 identity matrix hile [ α i] [ Γ i] are 5 5 matrices in hich elements are determined by the olloing equations: [ ] [ ] α (5) i A [ A][ ] [ C ] [ α ] [ ] [ ] Γ (5) A B Γ 3... i [ α ] Γ 3 i (5) C.... (53) Substituting Eq. (9) into Eq. () e obtain I e deine LU δ r. (5) U δ W (55) Eq. (5) becomes here W LWr (56) [ W] [ W ] W [ W ] [ W] are 5 column matrices. The elements o W can be determined rom Eq. (55) by the olloing relations: [ ][ ] [ ] α W r (57) α W r B W. (58) When the elements o W have been ound Eq. (55) gives the solution or δ in hich the elements are ound rom the olloing relations: [ ] [ ] δ W (59) δ W Γ δ +. (6) Once the elements o δ are ound Eqs. (3)-(38) can k+ th iteration in Eq. (33). These be used to ind the ( ) calculations are repeated until the convergence criterion is satisied. In laminar boundary layer calculation the q is commonly used as all shear stress parameter ( ) the convergence criterion (Cebeci Bradsha [3]). ISSN: Issue 3 Volume 5 uly

7 Noriah Bachok Anuar Ishak This is probably because in boundary layer calculations it is ound that the greatest error usually appears in the all shear stress parameter. Thus this convergence criterion is used in the present study. Calculations are stopped hen ( k) δ q < (6) here is a small prescribed value. In this study. is used hich gives about our decimal places accuracy or most o the predicted quantities as suggested by Bachok et al. [5] Ali et al. [3]. The present method has a second-order accuracy unconditionally stable is easy to be programmed thus making it highly attractive or production use. The only disadvantage is the large amount o once--orall algebra needed to rite the dierence equations to set up their solutions. Results Discussion The step size η in η the position o the edge o the boundary-layer η have to be adusted or dierent values o the parameters to maintain the necessary accuracy. In this study the values o η beteen.. ere used depending on the values o the parameters considered in order that the numerical values obtained are independent o η chosen at least to our decimal places. Hoever a uniorm grid o η.as ound to be satisactory or a 5 convergence criterion o hich gives accuracy to our decimal places in nearly all cases. On the other h the boundary-layer thickness η beteen 5 as chosen here the ininity boundary condition is achieved. To assess the accuracy o the present method comparison ith the previously reported data available in the open literature is made. The variations o the skin riction coeicient ( ) ith λ are shon in Fig. hile the corresponding local Nusselt number θ ( ) are shon in Fig. or some values o n. It is seen that the solution is unique hen λ hile dual solutions are ound to exist hen λ < i.e. hen the plate the ree stream move in the opposite directions. The values o ( ) are positive hen λ <. 5 they become negative hen the value o λ exceeds.5 or all values o the suction/inection parameter. Figs. sho that or a particular value o the solution exists up to certain critical value o λ(say λ c ). Beyond this value the boundary-layer approximations breakdon thus the numerical solution cannot be obtained. The boundary-layer separated rom the surace at λ λc here λc denotes a critical value o λ. Based on our computations λ c or.. respectively. This value o λ c is in agreement ith those reported by Merkin [8] Azal et al. [] Weidman et al. [8] Ishak et al. [6 ] Hussaini et al. []. From this observation it can be concluded that > delays the boundary-layer separation suction ( ) hile inection ( ) < accelerates it. In contrast to the classical boundary-layer theory the separation occurs hen the skin riction coeicient ( ) > not at the point o vanishing all shear stress. The samples o velocity temperature proiles or some values o parameters are presented in Figs. 3 5 respectively. These proiles satisy the ar ield boundary conditions () asymptotically hich support the numerical results besides supporting the dual nature o the solutions presented in Figs.. ( ) (Blasius) (.5 ) (Sakiadis) λ Fig. Variation o the skin riction coeicient ( ) ith λ or various values o. ISSN: Issue 3 Volume 5 uly

8 Noriah Bachok Anuar Ishak.8.6 Pr n. n.5 3 First solution Second solution λ /θ ( ). θ (η ) - n λ Fig. Variation o the local Nusselt number ( ) θ ith λ or various values o n hen Pr η Fig. Temperature proiles ( η) θ or various values o n hen λ (η ) θ (η ) -.. n λ -.3 n.5. - λ First solution Second solution η Fig. 3 Velocity proiles ( η) or various values o hen λ η Fig. 5 Temperature proiles θ ( η) or various values o n hen λ. 3. ISSN: Issue 3 Volume 5 uly

9 Noriah Bachok Anuar Ishak 5 Conclusion The development o the boundary layer on a ixed or moving surace parallel to a uniorm ree stream ith variable surace heat lux has been investigated. The classical Blasius Sakiadis problems are to particular cases o the present problem. Discussions or the eects o the heat lux exponent parameter suction or inection parameter the velocity ratio parameter λ on the skin riction coeicient ( ) the local Nusselt number θ ( ) or Pr have been done. From the present investigation it may be concluded that: Dual solutions are obtained hen the plate the external stream move in the opposite λ< the solution exists up to directions ( ) a critical point λ c ( < ). Suction ( ) separation hile inection ( ) > delays the boundary layer < accelerates it. Larger value o the heat lux exponent parameter increases the rate o heat transer. Acknoledgement The inancial support received in the orm o a research grant (FRGS) rom Ministry o Higher Education Malaysia is grateully acknoledged. Reerences: [] Blasius H Grenzschichten in Flüssigkeiten mit kleiner Reibung Z Math Phys Vol pp [] Bacharoudis E Vrachopoulos MG Koukou MK Filios AE Numerical investigation o the buoyancy-induced lo ield heat transer inside solar chimneys WSEAS Transantions on Heat Mass Transer Vol. 6 pp [3] Abdell K Magpantay F Approximate analytic solutions or mixed orced convection heat transer rom an unsteady no-uniorm lo past a rotating cylinder WSEAS Transantions on Heat Mass Transer Vol. 7 pp [] Al-Azab T Unsteady mixed convection heat mass transer past an ininite porous plate ith thermophoresis eect WSEAS Transantions on Heat Mass Transer Vol. 9 pp [5] Sakiadis BC Boundary-layer behaviour on continuous solid suraces: I. Boundary-layer equations or to-dimensional axisymmetric lo AIChE Vol pp [6] Ishak A Nazar R Pop I Flo heat transer characteristics on a moving lat plate in a parallel stream ith constant surace heat lux Heat Mass Trans Vol. 5 9 pp [7] Klemp B Acrivos A A method or integrating the boundary-layer equations through a region o reverse lo Fluid Mech Vol pp [8] Merkin H On dual solutions occurring in mixed convection in a porous medium Eng Math Vol. 985 pp [9] Abdelhaez TA Skin riction heat transer on a continuous lat surace moving in a parallel ree stream Int. Heat Mass Trans Vol pp [] Hussaini MY Lakin WD Nachman A On similarity solutions o a boundary layer problem ith an upstream moving all SIAM Appl Math Vol pp [] Azal N Badaruddin A Elgarvi AA Momentum transport on a continuous lat surace moving in a parallel stream Int. Heat Mass Trans Vol pp [] Bianchi MVA Viskanta R Momentum heat transer on a continuous lat surace moving in a parallel counterlo ree stream Heat Mass Trans Vol pp [3] Lin HT Huang SF Flo heat transer o plane suraces moving in parallel reversely to the ree stream Heat Mass Trans Vol pp [] Chen CH Heat transer characteristics o a nonisothermal surace moving parallel to a ree stream Acta Mech Vol. pp [5] Azal N Momentum transer on poer la stretching plate ith ree stream pressure gradient Int. Eng Sci Vol. 3 pp [6] Abraham P Sparro EM Friction drag resulting rom the simultaneous imposed motions o a reestream its bounding surace Int. Heat Fluid Flo Vol. 6 5 pp [7] Sparro EM Abraham P Universal solutions or the streamise variation o the temperature o a moving sheet in the presence o a moving luid Int. Heat Mass Trans Vol. 8 5 pp [8] Weidman PD Kubitschek DG Davis AM The eect o transpiration on sel-similar boundary layer lo over moving suraces Int Eng Sci Vol. 6 pp [9] Zheng LC Zhang XX He C Biurcation behavior in the reverse-lo boundary layer ith special ISSN: Issue 3 Volume 5 uly

10 Noriah Bachok Anuar Ishak inection or suction Chin. Phys Lett Vol. 3 pp [] Ishak A Nazar R Pop I Boundary layer on a moving all ith suction inection Chin Phys Lett Vol. No.8 7 pp [] Elbashbeshy EMA Heat transer over a stretching surace ith variable surace heat lux. Phys. D: Appl. Phys. Vol pp [] Elbashbeshy EMA Bazid MA The mixed convection along a vertical plate ith variable surace heat lux embedded in porous medium Appl Math Comp Vol. 5 pp [3] Ishak A Nazar R Pop I Heat transer over a stretching surace ith variable heat lux in micropolar luids Phys. Lett A Vol pp [] Rahman MM Sultana T Radiative heat transer lo o micropolar luid ith variable heat lux in a porous medium Nonlinear Analysis: Modelling Control Vol. 3 8 pp [5] Bachok N Ishak A Pop I Mixed convection boundary layer lo near the stagnation point on a vertical surace embedded in a porous medium ith anisotropy eect Trans. Porous Med Vol. 8 pp characteristic on a moving lat plate in a parallel stream ith prescribed surace heat lux Proceedings o the 9 th WSEAS Int. Con on Appl o Comp Eng pp [7] Ishak A Nazar R Bachok N Pop I MHD mixed convection lo near the stagnation-point on a vertical permeable surace Physica A Vol. 389 pp. -6. [8] Ishak A Nazar R Bachok N Pop I Melting heat transer in steady laminar lo over a moving surace Heat Mass Transer Vol. 6 pp [8] Na TY Computational methods in engineering boundary value problems Academic Press Ne York 979. [9] Cebeci T Bradsha P Physical computational aspects o convective heat transer Springer Ne York 988. [3] Cebeci T Bradsha P Momentum transer in boundary layers Hemisphere Washington 977. [3] Ali FM Nazar R Ariin NM MHD viscous lo heat transer due to a permeable shrinking sheet ith prescribed surace heat lux Proceedings o the 9 th WSEAS Int. Con on Appl o Comp Eng pp [6] Bachok N Ishak A Flo heat transer ISSN: Issue 3 Volume 5 uly