Analytical Solution to the Boundary Layer Slip Flow and Heat Transfer over a Flat Plate using the Switching Differential Transform Method

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1 Journal o Applied Fluid Mechanics, Vol., o., pp. 33-, 9. Available online at ISS , EISS DOI:.8869/acadpub.jam Analytical Solution to the Boundary Layer Slip Flow Heat ranser over a Flat Plate using the Switching Dierential ransorm Method H. H. Mehne M. Esmaeili Aerospace Research Institute, ehran, , Iran Department o Mechanical Engineering, aculty o Engineering, Kharazmi University, ehran, 579-9, Iran Corresponding Author hmehne@ari.ac.ir (Received February 8, 8; accepted August, 8) ABSRAC In this paper, the two-dimensional steady boundary layer low heat transer over a lat plate with slip velocity temperature jump conditions at the walls were analyzed. Using similarity transorms, the governing equations were reduced to a system o ordinary dierential equations. Semi-analytical solutions to the resulting boundary value problem were obtained using the dierential transorm method (DM). In order to cover the asymptotic boundary conditions, a method o switching curves was proposed. In this switching approach, the traditional solution to the DM, which is valid or inite horizons, was ollowed by another path that was also an analytical solution to the problem. he main preerence o the resulting closed orm solution with respect to numerical solution is the possibility o parametric studies. he method was veriied using some available numerical data, the results showed that our proposed method had reasonable eiciency accuracy. Keywords: Shape optimization; Optimization; Heat transer; Approimation. OMECLAURE C p Kn l Pr Re g luid heat capacity transormed velocity luid thermal conductivity Knudsen number characteristic length Prtl number Reynolds number temperature gas temperature adjacent to the wall w wall temperature ree-stream temperature u stream-wise velocity component u gas velocity adjacent to the wall g U ree-stream velocity u w wall velocity v y M luid inematic viscosity velocity component vertical to stream directions component o the coordinate system component o the coordinate system tangential momentum coeicient non-dimensional velocity parameter non-dimensional thermal slip parameter speciic heat ratio similarity variable transormed temperature ree path thermal accommodation coeicient luid density stream unction. IODUCIO Boundary layer low theory has been o interest to many researchers in luid mechanics, its various applications appear in many real world situations. Boundary layer theory was initially developed by some pioneering publications (Blasius 97, Falner San 93, Prtl 95, Saiadis 96), comprehensive reviews o this theory related topics are given by Calisch

2 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. Sammartino(), Garratt (99), Kachanov (99), Oleini Samohin (999), Schlichting et al. (995), ani (977), Weinan (). Among the dierent aspects o the theory, the problem o laminar boundary layers heat transer low on a semi-ininite lat plate, called Blasius boundary layer low, have been broadly studied (e.g. see Aziz 9, Bataller 8, Bhattacharyya, Cortell 8), where the latter can be used as abenchmar low or the validation o dierent numerical methods (Cortell 5 Wang ). During recent years, with the rapid development o micro- nano-measuring science technologies, it has been ound that there are many signiicant dierences between luid low at macroscopic scales that at micro/nano scales, e.g. the wall-slip phenomena (Kumaran Pop ). Rareied gas lows with slip boundary conditions are oten countered in micro-scale devices low-pressure situations (Gad-el-Ha 999). he eects o slip conditions are very important in technological applications or some luids that ehibit wall slip, e.g. when polishing artiicial heart valves their internal cavities. he irst reerence that proposed boundary layer with slip condition is Beavers Joseph (967), where the eect o the boundary layer was replaced by the slip velocity. In the wor o Martin Boyd (), the slip low condition was added to the Blasius problem in order to study microelectromechanical system (MEMS) scale lows. Among more recent studies (Aziz et al.5, Aziz et al., Cai 5 Parida et al. 5), Aziz et al.5 considered the convective boundary layer low o a power-law luid on a porous plate with suction/injection. hey investigated the eects o dierent physical parameters, such as the powerlaw inde, slip, permeability on the luid low heat transer characteristics. In the present paper, the Blasius boundary layer heat transer low with slip boundary conditions were studied. Applying the dierential transorm method (DM) led to closed orms o the solutions to this problem. Introduced by Zhou (986), the DM was originally an analytical method or solving initial value problems. he method was subsequently etended adopted to solve various mathematical problems such as boundary value problems, partial dierential equations integral equations (Usmanet al. 7, Sepasgozar et al. 7, Sheiholeslami Ganji 5, Mosayebidorcheh et al. 7, Vimala Omega 6, Mosayebidorcheh 5 hiagarajan Senthilumar 3). A recent application o this method appears, or eample, in Usman et al. (7), where the DM was applied to study the two unsteady phases o a nano luid low heat transer between moving parallel plates in the presence o a magnetic ield. In the wor o Sepasgozar et al. (7), the analytical solutions o the momentum heat transer equations o a non- ewtonian luid lowing in an aisymmetric channel with a porous wall were obtained via the DM. his method was also used successully by Sheiholeslami Ganji (5) to solve the problem o nano luid hydro thermals in the presence o a variable magnetic ield. A hybrid versiono the DM the inite dierence method was also developed to solve low heat transer equations by Mosayebidorcheh et al. (7). In the problems considered in the current paper, there are two boundary conditions at ininity. As the validity o the regular DM is restricted to inite horizons, the related solution diverges over this range, thereore the asymptotic behavior o the solution is not attained. he traditional method used to cover asymptotic conditions is to represent the polynomial solution with a rational unction, i.e. a Padé approimation (Rashidi ). However, this approach the consequent solutions depend on the degree o the polynomials used in the numerator denominator. hereore, determining the best itting values o these parameters is based on trial error. In the approach o this paper, the switching o the DM result to a solution that satisied the inal boundary condition, joint restriction, was introduced. In comparison with the Padé approimation, the proposed method was not based on trial error, but instead it provided a systematic procedure to ind the optimal location o switching. hereore, the novelty o our method lies in introducing this technique. In order to chec the validity accuracy o the method, we applied it to dierent eamples with without the slip condition. Comparison o our results with those published in the literature conirmed the eiciency o our proposed approach.. MAHEMAICAL FORMULAIO he steady two-dimensional laminar viscous luid low heat transer over a lat plate were considered in this study. he governing equations or the conservation o mass, momentum energy, based on the boundary layer approimation, can be epressed as: u v y u v y y u u u u v y C y p () () (3) where is the temperature, are the velocity components in the stream-wise vertical directions, respectively, v is the luid s inematic viscosity, is the luid s thermal conductivity, C is the luid s heat capacity, is the luid s p density. Considering the no-slip constant temperature conditions on the wall, the ree-stream condition outside o the boundary layer, the proper boundary conditions o the boundary layer low over a lat plat are given by: u v 3

3 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. u,, v,,, w (),,,,, u U v (5) w Where is the wall s temperature, U are the ree-stream s velocity temperature, respectively. For lows with is the Knudsen number that is deined as the mean ree path ( ) divided by the characteristic length ( ), the no-slip constant temperature conditions are valid. In the slip low regime,. Kn., ollowing slip boundary conditions can be used (Cai 5): Kn / l u g g M u uw M y l 3 g w y w Pr y y where the w g Kn., where (6) (7) subscripts represent the wall the adjacent gas, respectively. In addition, M are the tangential momentum thermal accommodation coeicients, respectively, Pr is the Prtl number, is the speciic heat ratio. he second term in Eq. (6) contains the stream-wise temperature gradient, also called the thermal creep, which is neglected in this study. By introducing the ollowing similarity transormations: y U U, w (8) (9) the variables are transormed rom, y to,. Here the stream unction deined as: u, v y is a similarity variable is () he velocity components also are based on the stream unction as: U u U, v () By substituting Eqs. (9) () into the momentum energy equations (Eqs. () (3)), the ollowing system o ordinary dierential equations (ODE) is obtained:, () Pr, (3) he boundary equations are also transormed into the ollowing orm as: '',,, M Kn M Re /,, Kn Pr Re / () (5) where Kn Re are, respectively, the Knudsen Reynolds numbers based on, are the non-dimensional velocity thermal slip parameters, respectively. Using the viscosity deinition based on the inetic theory o gases (Cai 5 Howarth 938), we ind: 3 8R (5) Based on the inetic gas estimation o the viscosity, the slip boundary conditions can be also epressed as ollows: ' '' ' / M 3 M M / 3 M Pr 3. MEHOD OF SOLUIO 3. he raditional DM (6) (7) Herein, we review some basic deinitions o the dierential transorm, which were initially introduced by Zhou (986): Deinition : I y t is an analytical unction in domain, then it can be dierentiated continuously with respect to t. Let t be ied, then the dierential transorm o this unction at is deined as a t sequence Y, Y, Y,, in which: 3

4 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. d y Y,,,,... dt tt (8) It is clear thaty, which is called the spectrum o o y t y t, is the -th coeicient in the aylor series at t t. hereore, i D denotes the dierential transorm, then the transorm o Y t t y t Y! : yt () D Y Y Y,,, is the inverse o (9) Some useul important properties o the dierential transorms are summarized in able. In this table, it is assumed that X st, R, S are the dierential transorms o t, respectively., r t able Some properties o the dierential transorm Original unction ransormed orm t r t s t X R S t r t, X R r t s t X R S l t l dr t X R t t dt n d r t t dt n X t r d t t t n n! R n! R X, X n t e X t sin t t cos t t X X, i n, otherwise sin!!! cos When solving a two-point boundary equation with the DM, the irst step is to tae the dierential transorm o both sides o the equation. I the, original equation is: h t L y Where () L. is thedierential operator, then the equivalent equation in transormed orm is: D h t D L y () Ater perorming the dierential transorm, this equation has an algebraic orm lie: Y K H K,,,, Where Y transorms K o y. H K, h () are the dierential. respectively. Equation () is also called the equivalent equation in the -domain, which is usually a recursive K iteration. I Eq. () is solved or Y K, then the solution o Eq. () will be calculated using Eq. (9). When the initial conditions, or eample when L is o the second degree, aregiven as:, y t y y t y hen the initial values or solving Eq. () are: Y y, Y y When Eq. () has boundary conditions lie: y t y y t y, hen the method starts with: Y y, Y Where applying is a parameter that will be ound by y t y at the end to the resulting solution o the orm Eq. (9). In the present problem o this paper, some inial conditions have an asymptotic orm as: Some reerences propose using the Padé approimation to approimate the resulting polynomial o Eq. (9) as a rational unction, in order to ind solutions that will have the appropriate asymptotic behavior. 3.. he Switching DM In the proceeding section, we introduced a dierent approach based on switching to a second curve to satisy the inal conditions. Lets tae the dierential transorm rom both sides o Eqs. ()- (3). hen, the corresponding equations based on the properties inable in K- space are: 3

5 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. F 3 3 l l F l F l l,,, Pr Θ where F l F l l l,, Θ, (3) () denote the dierential transorms o, respectively. o start the above iterations, the ollowing initial conditions are employed, which are equivalent to the initial conditions o Eqs. () (5), which are also in K-space: F, F F, (5) Θ Θ, (6) With these initial conditions, the resulting coeicients obtained rom Eqs. (3) () are unctions o Pr,, that are given two unnown coeicients F Θ. hese two coeicients are taen as parameters that control the inal conditions o Eqs. () (5) by deining: F, (7) t Θ, (8) Let us now consider that the resulting solutions to.. truncated in F, which are ininite series, are terms as ollows:, Θ, where (9) (3) satisies both Eq. () approimately, its initial condition. In order to satisy the inal condition at ininity, we propose that the velocity proile is a constant value at a suitable point as ollows:,, (3) hen, it satisies the inal condition trivially. ow, are two parameters which have to be ound in such a way that at the switch point, the value o the irst derivative is the second derivatives must be as small as possible, that is optimal values o. In order to ind the, the ollowing optimization problem needs to be solved numerically: '' min (3) Similarly, the switched orm o the temperature is assumed as:,, Unlie the inal course curve or (33) in Eq. (3), which is a constant straight line, in the case o, some analytical eort is needed. Due to the switching o at to a constant value o, where, has the orm a or, a is a nown constant. hereore, Eq. (3) changes to the ollowing ormat: Pr a, (3) or equivalently: d d Pr a, (35) he above linear irst-order equation can be solved or : d Pr a Ln hereore: d Ln Pr a d, Pr ep ep Pr (36) (37) By integrating the above equation, the inal orm o 3

6 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. is ound as: where Er Pr ep Pr Pr Er Pr Er Er is the error unction deined as: s ds ep (38) (39) In order that the proposed solution to Eq. (33) is continuous, it should satisied at established when in Eq. (39), that is: Pr ep Pr Pr Er Pr Er, which is are used () his solution should also satisy the inal condition. It is clear that the proposed solution has a parameter a, which we will use as a controlling parameter or its asymptote. Since the limit o the error unction at ininity is, i we tae the limit rom as, we have: lim Pr ep Pr Er Pr () ow, it is enough to solve the ollowing equation or :,, L t Pr ep Pr Er Pr () But, as it has been seen beore, is estimated rom Eq. (3), in order to ind its optimal value, Eq. () must be zero. et, we can combine Eqs. (3) () by minimizing the ollowing objective unction:,, L,, t I t (3) By solving Eq. (3), the switch point o Eqs. (3) (33), the required parameters that satisy the inal conditions are obtained.. RESUS In this section, our proposed method was applied to three cases to demonstrate its accuracy eiciency. Case : In the simplest case, the ollowing Blasius problem is considered, where the velocity slip is zero:,,,, () he corresponding equation in K-space is the ollowing single recursive relation: F 3 3 l l F l F l l,, 3,, he initial conditions are also translated as: (5) F, F, (6) It is also assumed that F is a ree parameter that will be obtained in such a way that the inal condition is satisied. In this eample, the solution is epressed up to 7 th power o as ollows: 3

7 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, (7) his solution gives a amily o approimate solutions o the Blasius problem that satisies the Blasius equation initial conditions in a region o. ow, the proposed solution that satisies the inal condition has a s witched orm as: 7,, (8) hereore, are two parameters which have to be ound in such a way that at the switch point, the value o the irst derivative is the second derivatives must be as small as possible, that. In order to ind is the optimal values o, the ollowing optimization problem is solved numerically: min (9) he optimal values o the decision variables are he results are depicted in Figs.. In these igures, the computed results o its irst second derivatives are compared with the numerical results o Howarth (938). As can be seen, very good agreement is observed between the results obtained with the switching DM the numerical results reported by Howarth (938). Another parameter that measures the quality o the solution is the absolute value o the dierence o the two sides o the Blasius equation with the approimate solution put into the right h side o Error. A Eq. (). hat is, smaller Error means a better approimation. In Fig. 3, the corresponding errors are depicted or dierent number o terms. () present method um. o Howarth (938) Fig.. he proile o or case. he solid line indicates presents our dierential transorm method (DM) results, while the circles show the numerical results o Howarth (938). '(), ''() present method um. o Howarth (938) '() ''() Fig.. he proiles o ( ) ( ) or case. he solid lines present our dierential transorm method (DM) results, the circles indicate the numerical results o Howarth (938). Error =5 =8 = = = Fig. 3. Errors o the irst part o the solution in case. It can be concluded rom Fig. 3 that the validity o polynomial part o the solution changes with the 3

8 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. order o the polynomial. he decreasing error rom shows that higher orders lead to better solutions o the dierential equations. It is also noted that the error will increase ater a near zero course, where the length o this avorable region increases with the polynomial s order. As the solution obtained with the proposed method has changed rom a polynomial to a straight line, this grows o error lost its importance, 5 to at 7 while in the remaining, no errors are achieved. It should also be noted that in case with, the maimum error beyond is hereore, the solution obtained with our proposed method satisies the Blasius equation with reasonable accuracy in its polynomial part, when the error starts to increase, we switched to its straight line orm, which is an eact solution to Eq. (). Case : In this case, the Blasius problem with was considered. he boundary value problem is:,,,, (5) he corresponding equation in K-space is similar to thato case, but the second initial condition has changed to: F, F F, So, with (5) F as the unnown parameter, the general orm o the semi-analytical solution up to the 8 th power o is ound as: (5) Computations or case were perormed or dierent values o. Figure shows the normalized velocity proiles within the boundary layer or,,,3, 5. he velocity proile o represents the classical solution tothe Blasius low in which the no-slip condition is imposed at the wall. As shown in Fig., the normalized slip velocity increases with an increase in. As an eample, or, the values o are compared with the numerical result o Martin Boyed () obtained with the shooting method, where reasonable agreement is observed. In Figs. 5 6 two important parameters,, which represent the normalized slip velocity wall shear stress are shown, respectively. he accuracy o our results is conirmed by comparing them with the results o Martin Boyed (). It is noted that when the Knudsen number is large enough, approaches ininity, the normalized slip velocity the wall shear stress approaches zero, respectively. his condition corresponds tocomplete slip low (i.e. hundred percent slip at the wall). '().5.5 um. o Martin Boyed () Fig.. ormalized velocity proiles u ( ) orcase.he solid lines indicate U our dierential transorm method (DM) results or dierent values o δ, while the circles show the numerical results o Martin Boyed () or δ he percentage o riction reduction FR=% '' was also obtained or three dierent values ov.,.5,.8, as shown in Fig. 7. he results show that the riction is large at the initial portion o the plate, but then it reduces the rate proportional to the value o. Case 3. In this case, boundary layer low heat transer over a lat plate with velocity slip thermal jump conditions are considered. So, the boundary value problem is epressed by Eqs. ()- (5). v

9 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. '() present method um. o Martin Boyed () 3 5 u Fig. 5. Slip velocity ( ) or U wall various values o δ or case.he solid lines give ourdierential transorm method (DM) results, the circles indicate the numerical results o Martin Boyed (). ''()..3.. present method um. o Martin Boyed () 3 5 Fig. 6. Wall shear stress / / / U w / or various values o δ or case. he solid lines indicate thedierential transorm method (DM) results, the circles indicate the numerical results o Martin Boyed (). FR v =. v =.5 v = / Fig. 7. he percentage o riction reduction in terms o / or various values o v or case. For eample, with eight approimate terms, the solution has the ollowing closed orm: 3 Prt 8 t t Prt 8 5 Pr t 6 t 5 Prt Pr t Prt Pr 8 3 t Prt Pr t Pr t 5 Prt Pr 8 8 t Prt Pr 3 t Prt Pr t Pr t 5 Prt Pr 8 Pr t 3 7 Pr t 3 6 (53) 3 Prt (5) 8 t Prt t 5 Prt 5 Pr t Pr 3 8 where Pr, t 7 are given constants are parameters that control the solutions in order to match the inal conditions., t are ound by minimizing Eq. (3). As an eample to chec the accuracy o our method, we solved the problem or Pr.5,... he results or are as ollows: ,.75 (55) Due to rounding the coeicients to our decimal 6 places, the conidences o to are zero. hen, the complete solution is:, , 3.8 Moreover, when inding results are obtained: (56) the ollowing

10 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. t.33, a Er , 3.8, 3.8 (57) In Figs. 8 9, the velocity temperature proiles or case 3 are compared with the results o Bhattacharyya et al. (), respectively. Reasonable agreement between the switching DM the shooting method is clearly detected. he highest deviation o the normalized velocity rom the numerical results o Bhattacharyya et al. () is observed where the normalized velocity approaches (i.e. close to the boundary layer s edge). '() present method um. o Bhattacharyya et al. () 6 Fig. 8. he velocity proile o case 3 compared with the results o Bhattacharyya et al. (). () present method um. o Bhattacharyya et al. () 6 8 Fig. 9. he temperature proile orcase3 compared with the results o Bhattacharyya et al. (). Moreover, to study the impact o thermal slip on heat transer rom the plate, the proiles o ' or values o.,.3,.7,.3,. are depicted in Figs., respectively. hese igures show that as the thermal slip increases, the magnitude o the temperature its gradient monotonically decrease, which leads to weaening o the heat transer rom the plate. () Fig.. emperature proiles or case 3 against various values o the thermal slip. '() Fig.. emperature gradient proiles or case 3 versus various values o the thermal slip. he problem has been also solved or..,., 3., 8.. In Fig., the temperature proiles o the various values o the velocity slip parameters are shown. As epected, the results indicate that when the slip parameter increases, the temperature reduces. Figure 3 demonstrates the temperature gradient proiles or the dierent velocity slip parameters. It is observed that, in an agreement with the results o Bhattacharyya et al. (), the temperature gradient decreases with beore, it increases ater this point.

11 H. H. Mehne M. Esmaeili / JAFM, Vol., o., pp. 33-, 9. () Fig.. emperature proiles or case3 versus various values o the velocity slip = = = 3 = Fig. 3. emperature gradient proiles or case 3 against various values o the velocity slip. 5. COCLUSIOS he switching DM method was proposed in this paper to solve problems with asymptotic boundary conditions. he method was successully applied to the boundary layer slip low with heat transer over a lat plate. he results revealed the accuracy o our method or dierent cases, which were comparable with other available numerical methods. he advantage o the proposed method with respect to numerical methods lies in the act that with DM, a semi-analytical solution will be obtained rather than a discrete one. he presented switching version o DM did not contain the problem o choosing the degree o the numerator denominator polynomials that appeared in the Padé approimation. REFERECES Aziz, A. (9). A similarity solution or laminar thermal boundary layer over a lat plate with a convective surace boundary condition. Communications in onlinear Science umerical Simulation (), Aziz, A., J. I. Siddique. Aziz (). Steady boundary layer slip low along with heat mass transer over a lat porous plate embedded in a porous medium. PLoS OE 9(): e5. Aziz, A., Y. Ali,. Aziz J. I. Siddique (5). Heat ranser Analysis or Stationary Boundary Layer Slip Flow o a Power-Law Fluid in a Darcy Porous Medium with Plate Suction/Injection. PLoS OE (9): e Bataller, R. C. (8). Radiation eects or the Blasius Saiadis lows with a convective surace boundary condition. Applied Mathematics Computation 6(), Beavers, G. S. D. D. Joseph (967). Boundary conditions at a naturally permeable wall. Journal o luid mechanics 3(), Bhattacharyya, K., S. Muhopadhyay G. Laye (). MHD boundary layer slip low heat transer over a lat plate. Chinese Physics Letters 8(), 7. Blasius, H. (97).Grenzschichten in Flüssigeiten mit leiner Reibung: Druc von BG eubner. Calisch, R. M. Sammartino (). Eistence singularities or the Prtl boundary layer equations. ZAMM Journal o Applied Mathematics Mechanics/Zeitschrit ür Angewte Mathemati und Mechani 8(-), Cai, C. (5). ear continuum boundary layer lows at a lat plate. heoretical Applied Mechanics Letters 5(3), Cortell, R. (5). umerical solutions o the classical Blasius lat-plate problem. Applied Mathematics Computation 7(), Cortell, R. (8). A numerical tacling on Saiadis low with thermal radiation. Chinese Physics Letters 5(), 3. Falner, V. S. W. San (93). Some approimate solutions o the boundary-layer equations. he London, Edinburgh, Dublin Philosophical Magazine Journal o Science, Gad-el-Ha, M. (999). he luid mechanics o microdevices-the Freeman scholar lecture. Journal o Fluids Engineering (), Garratt, J. (99). he internal boundary layer a review. Boundary-Layer Meteorology 5(- ), 7-3. Howarth, L. (938). On the solution o the laminar boundary layer equations. Proceedings o the Royal Society o London A: Mathematical, Physical EngineeringSciences 6, Kachanov, Y. S. (99). Physical mechanisms o laminar-boundary-layer transition. Annual 3

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