Archive of SID
Journal of Applied Fluid Mechanics, Vol. 10, No. 4, pp. 1071-1077, 2017. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.73.241.27544
New Perspectives on the Laminar Boundary Layer Physics in a Polarized Pressure Field with Temperature Gradient: an Analytical Approximation to Blasius Equation
M. Moeini1 and M. R. Chamani2†
1 Department of Civil Engineering, Amirkabir University of Technology, Tehran, Iran 2 Department of Civil Engineering, Isfahan University of Technology, Isfahan, Iran
†Corresponding Author Email: [email protected]
(Received December 26, 2016; accepted February 8, 2017)
ABSTRACT
This study proposes a semi-analytic approximation to the laminar boundary layer growth in a polarized pressure field with temperature gradient represented by the joint Blasius-energy equation. We illuminate that f () is a probability density function (PDF) approximated by an amended Gaussian PDF with zero mean and standard deviation 2.18 . This implies a diffusive structure for the molecular momentum conversion as well as the energy flux in the boundary layer. A new limit for the boundary layer edge is also presented. Results suggest an augmented boundary layer when compared to accepted values in the literature. We also reproduce the inverse proportionality of the free stream velocity to the diffusion of both momentum and energy. Keywords: Blasius laminar flow; Semi-analytic approximation; Boundary layer thickness; Momentum diffusion; Energy diffusion.
NOMENCLATURE
a constant determined to be v crosswise velocity 9.24sqrt (\ nu / U 0) density of fluid c p fluid specific heat f () = uU/ 0 c =erfaU 2 0 0 2 thermal diffusivity Db uniform-pressure momentum constant kinematic viscosity dimensionless temperature defined in Dt uniform-pressure thermal constant Eq. (6) k thermal conductivity * * Stefan–Boltzmann constant k mean absorption coefficient yU0 / x k = 3NR/(3NR+4) 0 **3Radiation parameter N R kk./4 T , amended Gaussian PDF defined in p piezometric pressure Eq. (9) Pr modified Prandtl number =0 (Mean of the Gaussian Pr Prandtl number distribution) free stream velocity U 0 , Gaussian PDF with mean and T fluid temperature standard deviation temperature of the ambient fluid T =2.18 (Standard deviation of the Gaussian distribution) Tw temperature of the wall u streamwise velocity boundary layer thickness
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1. INTRODUCTION analysis (Lari and Moeini, 2015) and the joined differential transform method with the Padè The thorough investigation of inviscid flows led to approximants (Rashidi et al., 2013 and Thiagarajan the increased awareness and development of the and Senthilkumar, 2013). An important outcome boundary layer concept (Schlichting, 1979). This was investigation into more complex physical notion, which was largely evolved by the work of problems. For example, the boundary layer Prandtl, introduced various levels of simplification development was examined under the influence of of the governing equations of fluid flow (Hirsch, magnetic response, with results showing the 2007). A direct consequence was the feasibility of possibility of devaluating the skin friction in an simulation of industrial-scaled applications which orthogonal magnetic field configuration used to be recalcitrant by the full Navier-Stokes (Thiagarajan and Senthilkumar, 2013; and Vyas and equations (Chen et al., 1984; Raptis et al., 2004; Srivastavat, 2012). In addition, several works Rashidi and Erfani, 2011; Lu and Law, 2014). arrived at a simulation for the behavior of non- Newtonian and thixotropic fluids in the boundary One of the most spectacular examples of boundary layers (Sadeqi et al., 2011; Pan et al., 2016 and layers was studied by Blasius almost a century ago. Ashraf et al., 2016) which have been unknown until It was based on the low-speed laminar flow passing quite recently. over a flat plate in a polarized pressure field (Blasius, As a sequel to this corpus, our goal here is to suggest a semi-analytic approximation analogous to 1950). Along with experimental visualizations, results in Ahmad and Al-Barakati (2009), Yun Blasius exploited the Prandtl equations in (2010), Savas (2012) and Bataller (2008) with the conjunction with assumptions about the absence of aim of providing more detailed analysis of the pressure gradient and similarity of velocity profiles laminar boundary layer physics in a polarized to elaborate the so-called Blasius equation. With the pressure field with temperature gradient. The reason help of recursive power series, Blasius (1950) behind the appeal to this approximation, which solved it with satisfying levels of accuracy. makes it preferable to the comparable works, lies in Subsequently, Howarth (1938) performed the its simplicity; In fact, we will justify that through Runge-Kutta method to provide an enhanced this closed-form approximation, the important numerical scheme. properties of boundary layer thickness/diffusion can Due to the engineering significance and be investigated with little computational expense. mathematical value of Blasius equation, there has appeared an extensive body of literature 2. GOVERNING EQUATIONS surrounding it. For instance, Healey (2008) investigated the effect of a controlled point-source The Prandtl equation (Bird et al., 2001, p. 135] disturbance in the Blasius laminar regime. By means of a wave-envelope steepening mechanics, uu1 dpu 2 uv (1) they illustrated the extent to which the flow xy dxy2 becomes nonlinear at low amplitudes. Kuo (2004) presented the solution to the momentum and energy lays the theoretical foundation to identify the equations of Blasius flow in the presence of external flow characteristics. Here, u is the temperature gradient. Zuccher et al. (2006) streamwise velocity, v the crosswise velocity, p the suggested a numerical scheme to solve the whole piezometric pressure and the kinematic viscosity. boundary layer equations with the aim of In the case of a uniform pressure field through the determining the maximum energy growth in a boundary layer, Prandtl equation reduces to the surging instable Blasius flow. Abdallah and Blasius equation (Bird et al. 2001, p. 138] Zeghmati (2011) examined the relative tendency to heat and mass transfer through the boundary layer u f when exposed to buoyancy gradients in the vicinity U of a cylindrical surface. Their results verified the ff 20 f 0 (2) U dominance of mass transfer over energy scattering y 0 when the ratio of Prandtl to Schmidt number x witnessed a significant plummet. Considering the no-slip condition on the plate (u = v On the mathematical side, major advances in = 0 at y = 0) and an asymptotically-reached numerical/analytical techniques have been made to potential flow outside the boundary layer (u = U0 at the need for an enhanced quantification of fluid y = ), Eq. (2) will get subject to motion problems. Such spectrum is very broad ranging from the revelation of homotopy analysis ff001 f (3) 00 (Liao, 1992; Liao, 1999; Bég et al., 2012; Malvandi et al., 2014; Hassan and Rashidi, 2014 and In addition, thermal radiation in the Blasius regime Makukula and Motsa, 2014), differential transforms is typically formulated as (Bataller, 2008) (Rashidi et al., 2013 and Ganji et al., 2016) and Adomian decompositions (Adomian, 1994; Wang, TT 2 T 2004; Aski et al., 2014 and Akpan, 2015) to a uv (4) xyky 2 combination of these and analogous techniques 0 (e.g., coupled integral transform and functional
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where T is the temperature, k / c the negative branch of the Gaussian PDF is cut and the p positive values are doubled to keep the thermal diffusivity, k the thermal conductivity, integral ˆ(,)=1(analogous with the integral the fluid specific heat, k0 = 3NR/(3NR+4) where c p **3 N kk.4/ T is the radiation parameter, in Eq. (8) and with the definition of a probability R density function). As a result of a qualitative * k the mean absorption coefficient and * the interpolation, it can be seen that the option 0 Stefan–Boltzmann constant. This PDE can be and 2.18 results in a close consistency between transformed to (Bataller, 2008) the two PDFs (see Fig. (1)). Therefore,
Pr k0 ˆ () f () ()=0 (5) f()=(0,) (10) 2
where Pr is the Prandtl number and the dimensionless temprature defined as TT = w (6) TT w
where Tw is the temperature of the wall and T the temperature of the ambient fluid. The boundary conditions for Eq. (5) are =0; =1 (7) ||=0 =
resulting from the temperature equilibrium at the interface of solid-fluid boundary (T = Tw at y = 0) and the prevailing uniform ambient temperature at the far-field. Fig. 1. Close consistency between the density
function f ''\sigma=2.18() and the Amended 3. SEMI-ANALYTICAL Gaussian PDF in Eq. (9). APPROXIMATION
The present approximation is based on the This can be validated by solving ODE (10) to obtain hypothesis that a dimensionless quantity taking the closed-form approximation values in the interval [0, 1] can be regarded as a 2 cumulative distribution function (CDF). By 22 2 comparing the CDF against numerical schemes, we fe ()= erf propose a closed-form solution to the ODE in this 2 section, and portray some of its outcomes in section (11) (4). On the grounds of the important property (Iacono and Boyd, 2015)
0 fd fdf f 01 (8) f ″() can be thought of a probability density function (PDF) (Roussas, 2003, p. 34). Vias and Srivastava (2012) and Iacono and Boyd (2015) justified that f″() attains an exponential decline when linearly grows. This provides us with a criterion to the need for a proximate estimation of the PDF. For this purpose, we define an Amended Gaussian function as Fig. 2. Comparison of the suggested approximation f=2.18() with the comparable 20 , schemes in Howarth (1938), Ahmad and Al- , (9) Barakati (2009) and Savas (2012). Note that the 00 results are coincident. 2 222 where , e is the PDF which is fully concordant with the exact solution of of a Gaussian variable with mean and standard Blasius equation (e,g., Howarth, 1938 (numerical), deviation (Roussas, 2003). The function (9) is Ahmad and Al-Barakati, 2009 and Savas, 2012 called the Amended Gaussian PDF since the (semi-analytical)) as shown in Fig. (2).
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Fig. 3. Visualization of the dimensionless vertical and mainstream velocities near the flat plate (lying in 6 2 the xz plane) in the Blasius regime with U0 = 0.1 m/s and = 10 m /s.
Eqs. (11) and (1) enable us to provide a closed-form calculations on the criterion ||=VU, where ||V is representation of the velocity vector as 0 the magnitude of velocity field. The aim here, therefore, is to solve 0.5Uy0 uxy(, )= U0 erf; vxy (, )= 2 x 2 (12) U y2 Uy22erf0 0.25 U 2 0022 erf ( ) 2x x 2 (14) 2 2 U0 2 2 Indeed, these formulas are of practical use in terms erf (y ) = U0 2x 22 of performing separate mathematical/programming operators, as opposed to the typical coupled relationship existing in the literature [e.g., Hirsh, where both sides of equation ||=VU0 has been 2007, p. 619] squared. Solving the quadratic Eq. (14) leads to the relation 1 (,)=uv Uf00,() U x f f (13) U 2 erfy 0 = 2 2x (15) An immediate fruit of this approximation is the 0.542x 4442xy 4 2 0.5 22y velocity field visualization, as seen in Fig. (3), with the advantage of eliminating extra computational 42xy 0.25 2 expense. The plots correspond to U0 = 0.1 m/s and 6 2 = 10 m /s. From the graphs we observe that the Since the boundary-layer thickness increases as the crosswise velocity plays a marginal role inside the square root of the streamwise coordinate (Bird et boundary layer (bellow the dashed-curve) compared with the streamwise component. In fact, computing al., 2001, p. 137), we set yax= where ’a ’ is a the boundary layer edge is another outcome of this constant to be determined. As a result, the right- approximation that will be discussed in subsection hand side of Eq. (15) is simplified (4.1). to erf aU 2 , which is constant for any 0 2 values of parameters involved. For the reason of 4. SOLUTION OUTCOMES convenience, we denote c =erfaU 2 . 0 0 2 4.1. Determination of Boundary Layer Equation (15) then can be written as Locus 42 8 4 6 4 4 Our goal here is to identify the boundary layer edge 0.0625ac00 0.25 ac 0.25 a by the Eq. (12). Although there is no orthodox 22 8 2 6 2 8 2 4 measurement of the viscous region edge due to its xacac0.500 0.25 a a (16) asymptotic attenuation (Schlichting, 1979), it is 228 8 generally accepted to be the locus of points at which xc0 =0 the streamwise velocity attains 99% of the potential velocity (Schlichting, 1979, p. 140). Equation (16) holds when the coefficients of variables xn (n {0, 1, 2}) vanish. The Here, we suggest a new measure u = 0.999980U0, simultaneous solution of this system indicates that with the reason behind whose appeal becoming c0 = 1 and = 1.41 (note that among the three, only apparent in later paragraphs, and base the two equations are independent). The constant
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then can be calculated from the definition of c0. At of the viscous zone. In Fig. 3, we have plotted the the same time, it is noted that these values introduce boundary layer parabola by the dashed-line. a slight deviation from the expected value = 2.18 4.2. Momentum and Energy Diffusion corresponding to f 2.18 (). This issue can be addressed by providing a smooth solution to the Our goal here is to infer the momentum and energy system (16). By setting = 2.18, Eq. (16) is conservation equations in the diffusion modes from simplified into the previous approximation. A direct calculation using Eq. (11) in conjunction with the chain rule 42 4 4 shows that 31.881ac00 26.833 ac 5.646 a x uu2 = Db (19) 22 2 22 2 255.048ac0 107.334ac0 127.524aa 22.585 x y
22 2 xc510.0960 510.096 = 0 where Db = /(2U0) represents the effect of mollecular mometum flux in the laminar visous (17) zone. Since this equation is valid for the absence of It is worth to note that erf(x) is a monotone pressure gradient, we call it the uniform-pressure increasing function with a rapid rate of convergence momentum constant. Equation (19) describes a to unity for large enough arguments (limx erf(x) stationary diffusion process in the 2-dimensional =1). A computer algebra package shows that erf (3) spatial domain (the reader can compare with the 1< 5105. If an acceptable accuracy is satisfied typical time-dependent heat diffusion in Logan by 0.005% error at this stage (we shall show that (2004)). Here, the constant of diffusion (Db) this amount of error results in the negligible 2% depends on the viscosity and the upstream velocity; error in later equations), we are able to set c0 = the more the fluid is viscous and its flow retarded, 0.999980. One should bear in mind that c0 is the the more the momentum flux transpires in the dimensionless velocity u/U0 combined with the control volume. This leads us to a significant assumption that boundary layer locus is a parabola. inference that the transport of momentum in the This demonstrates that the intuitive criterion u = laminar boundary layer is governed by a diffusion 0.99U0 in the boundary layer literature (Schlichting, process. 1979; Bird et al., 2011) has been changed to u = Furthermore, applying the chain rule to Eq. (6) 0.999980U0 in this work. leads to 0 With this attribution, the coefficients of x and x TyT become of orders O(a2) and O(a4) respectively, being =0.5 (20) x xy negligible on the level of reality in which 0.1 m/s < U0 < 100 6 2 m/s and = 10 m/s. Therefore, the predominant Solving this equation for y/x and substituting to the 2 source of error is the coefficient of x , with a nearly vertical and mainstream velocity components in Eq. 2% error (acceptably small). This verifies that the (12) shows that assumption of parabolic growth of y versus x still remains valid for = 2.18. Txuxy(, ) 2 vxy(, )= U0 . (21) Ty U 2 The final outcome, therefore, is the determination of 0 boundary layer locus as Finally, combination of Eq. (21) and Eq. (4) results in the differential equation x y = 9.24895 (18) 2 U0 TT = Dt (22) x y2 21 where a has been derived by 2erf() Uc00. 2 It should be mentioned that this result is in where Dt = 2kU00is the uniform-pressure agreement with the results in Schlichting (1979) in thermal constant since it represents how heat terms of dimension and relation among the diffuses between the plate and the physical infinity parameters involved. However, the value suggested in the paucity of pressure gradient. Indeed, the in Eq. (18) is nearly twice higher than Schlichting increased radiation parameter and free stream (1979, p. 140) ( ()=5x xU0 ). The distinction velocity would be impediments to the diffusion of heat, while the thermal diffusivity is proportionally mainly arisen from the fact that in Schlichting correlated, predicting a linearly grown flux as a (1979) u = 0.99U0 is accepted as a premise without consequence of a steady improvement in the any recourse to mathematical analysis. Although thermal diffusivity value. setting u = 0.99U0 here results in an approximately the same factor as that in Schlichting (calculated to These results conform well with Hirsch (2007, p. be yxU=5.6 0 ), the current outcomes, being on 96) demonstrating that the Navier-Stokes equations, as a consequence of boundary layer approximations, the grounds of merely three will reduce to relationships very close to standard assunptions ||=VU0 , yx and u = 0.999980U0, parabolic second order PDEs. However, to the are believed to provide more reasonable estimations authors' knowledge, this study is the first one that explicitly presents the momentum/energy laws in
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the Blasius regime in the form of diffusion viscous region. It was shown that, due to the equations. Interestingly, by dividing the uniform- amended criteria suggested in this work, the pressure momentum constant by the uniform- boundary layer edge expands to include almost 85% pressure thermal constant, we attain an indication more volume in the vicinity of the flat plate. We for the boundary layer relative readiness to transport also introduced a uniform-pressure thermal momentum/energy. This ratio introduces a modified constant and a uniform-pressure momentum Prandtl number constant for the two suggested energy and momentum time-independent equations. Db Pr k 0 (23) Dt REFERENCES which is analogous with the definition of Prandlt Abdallah, M. S. and B. Zeghmati (2011). Natural number Pr for a typical flow system (Bird et convection heat and mass transfer in the boundary layer along a vertical cylinder with al., 2001, p. 268). However, the modified form of opposing buoyancies, Journal of Applied Fluid Prandtl number includes the intrinsic effect of Mechanics4(4), 15-21. radiation parameter as well. It is noted that for large values of radiation parameter, the Pr asymptotes to Adomian, G. (1994). Solving frontier problems of Pr . physics: The decomposition method, Springer, Boston, USA. Finally, we support the validation of some of our results by means of comparison against higher level Ahmad, F. and W. H. Al-Barakati (2009). An approximations seen in Bataller (2008). If we approximate analytic solution of the Blasius substitute Eq. (11) for f () in Eq. (5), there appears problem, Communications in Nonlinear a linear second-order ODE having as the only Science and Numerical Simulation 14(4), unknown involved. This can represent a remarkable 1021–1024. simplification in the solving process since the Akpan, I. P. (2015). Adomian decomposition calculation of becomes decoupled from the approach to the solution of the Burger’s calculation of f (). Table (1) shows the solution of equation, American Journal of Computational this ODE, representing the value of (0) in the Mathematics 5(3), 329-335. current work and Bataller (2008), which the former Ashraf, M. B., T. Hayat and H. Alsulami (2016). has been obtained by typical numerical methods in Mixed convection Falkner-Skan wedge flow of ordinary differential equations. A good agreement is an Oldroyd-B fluid in presence of thermal observed between the results. This also suggests radiation, Journal of Applied Fluid Mechanics that the problem of thermal radiation would be 9(4), 1753-1762. more tractable, in terms of investigating its physics, if we benefit from this or similar semi-analytical Aski, F. S., S. J. Nasirkhani, E. Mohammadian and approximations. Asgari, A. (2014). Application of Adomian decomposition method for micropolar flow in a porous channel, Propulsion and Power Table 1 Comparison of the solutions to Eq. (5) Research 3(1), 15-21. obtained in this work and in Bataller (2008) with K0 = 1 Bataller, R. C. (2008). Radiation effects for the Blasius and Sakiadis flows with a convective (0) Pr surface boundary condition, Applied Bataller (2008) Current Study Mathematics and Computation 206(2), 832– 840. 0.1 - 0.14035 Bég, O. A., M. M. Rashidi, T. A. Bég and M. Asadi 0.4 - 0.24094 (2012). Homotopy analysis of transient magneto-bio-fluid dynamics of micropolar 0.7 0.29268 0.29607 squeeze film in a porous medium: a model for 5 0.57669 0.59023 magneto-bio-rheological lubrication, Journal of Mechanics in Medicine and Biology 12(3), 10 0.72814 0.74735 1250051.11250051.21. 50 1.24729 1.28518 Bird, R., W. Stewart and E. Lightfoot (2001). 100 1.57183 1.62100 Transport phenomena, 2nd edition, New York. USA.
Blasius, H. (1950). The boundary layers in fluid 5. CONCLUSIONS with little friction, Zeitschrift fur Mathematikund Physik 56, 1–37. In this study, we investigated the basic properties of Chen, T., B. Armaly and M. Ali (1984). Natural laminar flow regime near a flat plate governed by convection-radiation interaction in boundary the Blasius ODE. By providing a semi-analytical layer flow over horizontal surfaces, AIAA approximation, we calculated the thickness of Journal. 22 (12), 1797–1803.
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M. Moeini and M. R. Chamani / JAFM, Vol. 10, No. 4, pp. 1071-1077, 2017.
Ganji, H. F., M. Jouya, S. A. Mirhosseini-Amiri and Malvandi, A., F. Hedayati and M. R. H. Nobari D. D. Ganji (2016). Traveling wave solution (2014). An HAM analysis of stagnation-point by differential transformation method and flow of a nanofluid over a porous stretching reduced differential transformation method, sheet with heat generation, Journal of Applied Alexandria Engineering Journal 55(3), 2985- Fluid Mechanics 7(1), 135-145. 2994. Pan, M., L. Zheng, F. Liu and X. Zhang (2016). Lie Hassan, H. and M. M. Rashidi (2014). An analytic group analysis and similarity solution for solution of micropolar flow in a porous fractional Blasius flow, Communications in channel with mass injection using homotopy Nonlinear Science and Numerical Simulation analysis method, International Journal of 37, 90–101. Numerical Methods for Heat & Fluid Flow 24(2), 419-437. Raptis, A., C. Perdikis and H. S. Takhar (2004). Effect of thermal radiation on MHD flow, Healey, J. J. (2008). Wave-envelope steepening in Applied Mathematics and Computation 153(3), the Blasius boundary-layer, European Journal 645–649. of Mechanics-B/Fluids 19(6), 871–888. Rashidi, M. M. and E. Erfani (2011). A new Hirsch, C. (2007). Numerical Computation of analytical study of MHD stagnation-point flow Internal and External Flows: The in porous media with heat transfer, Computers Fundamentals of Computational Fluid and Fluids 40(1), 172–178. Dynamics, Elsevier Science, Burlington, USA. Rashidi, M. M., T. Hayat, T. Keimanesh and H. Howarth, L. (1938). On the solution of the laminar Yousefian (2013). A study on heat transfer in a boundary layer equations, Proceedings of the second-grade fluid through a porous medium Royal Society of London, Series A, with the modified differential transform Mathematical, Physical and Engineering method, Heat Transfer—Asian Research 42(1), Sciences 164, 547–579. 31-45. Iacono, R. and J. P. Boyd (2015). Simple analytic Roussas, G. (2003). An introduction to probability approximations for the Blasius problem, and statistical inference, Elsevier Science. Physica D: Nonlinear Phenomena 310, 72–78. Sadeqi, S., N. Khabazi and K. Sadeghy (2011). Kuo, B. L. (2004). Thermal boundary-layer “Blasius flow of thixotropic fluids: A problems in a semi-infinite flat plate by the numerical study”, Communications in differential transformation method, Applied Nonlinear Science and Numerical Simulation Mathematics and Computation 150(2), 303– 16(2), 711-721. 320. Savas, O. (2012). An approximate compact Lari, K. S. and M. Moeini (2015). A single-pole analytical expression for the Blasius velocity approximation to interfacial mass transfer in profile, Communications in Nonlinear Science porous media augmented with bulk and Numerical Simulation 17(10), 3772–3775. reactions, Transport in Porous Media 109(3), 781-797. Schlichting, H. (1979). Boundary-layer theory, McGraw-Hill, New York, USA. Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear Thiagarajan, M. and K. Senthilkumar (2013). DTM- problems, PhD Thesis, Shanghai Jiao Tong Pade approximants for MHD flow with University, Shanghai, China. suction/blowing, Journal of Applied Fluid Mechanics 6(4), 537-543. Liao, S. J. (1999). A uniformly valid analytic solution of two-dimensional viscous flow over Vyas, P. and N. Srivastavat (2012). On dissipative a semi-infinite flat plate, Journal of Fluid radiative MHD boundary layer flow in a Mechanics 385, 101–128. porous medium over a non isothermal stretching sheet, Journal of Applied Fluid Logan, J. (2004). Applied partial differential Mechanics 5(4), 23-31. equations, Springer Undergraduate Texts in Mathematics and Technology, Springer, New Wang, L. (2004), A new algorithm for solving York, USA. classical Blasius equation, Applied Mathematics and Computation 157(1), 1-9. Lu, Z. and C. K. Law (2014). An iterative solution of the Blasius flow with surface gasification, Yun, B.I. (2010). Intuitive approach to the International Journal of Heat and Mass approximate analytical solution for the Blasius Transfer 69, 223–229. problem, Applied Mathematics and Computation 215(10), 3489–3494. Makukula, Z. G. and S. S. Motsa (2014). Spectral homotopy analysis method for PDEs that Zuccher, S., A. Bottaro and P. Luchini (2006). model the unsteady Von Karman swirling Algebraic growth in a Blasius boundary layer: flow, Journal of Applied Fluid Mechanics Nonlinear optimal disturbances, European 7(4), 711-718. Journal of Mechanics-B/Fluids 25(1), 1–17.
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