Solution of Boundary Layer and Thermal Boundary Layer Equation

Asian Research Journal of Mathematics 11(4): 1-15, 2018; Article no.ARJOM.45267 ISSN: 2456-477X Solution of Boundary Layer and Thermal Boundary Layer Equation Farhana Mamtaz1, Ahammad Hossain2* and Nusrat Sharmin3 1 Department of Mathematics, California State University, Northridge, USA. 2 Department of Mathematics, Sonargaon University, Bangladesh. 3 Department of Mechanical Engineering, Sonargaon University, Bangladesh. Authors’ contributions This work was carried out in collaboration between all authors. All authors read and approved the final manuscript. Article Information DOI: 10.9734/ARJOM/2018/45267 Editor(s): (1) Dr. Krasimir Yankov Yordzhev, Associate Professor, Faculty of Mathematics and Natural Sciences, South-West University, Blagoevgrad, Bulgaria. Reviewers: (1) Hakeem Ullah, AK University, Pakistan. (2) N. Vishnu Ganesh, Ramakrishna Mission Vivekananda College, India. (3) Rahim Shah, Qurtuba University of Science and IT, Pakistan. Complete Peer review History: http://www.sciencedomain.org/review-history/27853 Short Research Article Received: 19 September 2018 Accepted: 04 December 2018 Published: 19 December 2018 _______________________________________________________________________________ Abstract We studied equation of continuity and boundary layer thickness. The Blasius and Falkner equations are studied in order to investigate the guess values in various boundary layer thicknesses, and Falknar-skan equations shows when velocity profile has a point of inflection in case of accelerated and decelerated. The solutions of the above mentioned equations are shown graphically. Finally, the thermal boundary layer equation has been derived from Navier-Stoke equation by boundary layer technique. Boundary Layer equation has been non-dimensionalised by using non-dimensional variable. The non-dimensional boundary layer equations are non-linear partial differential equations. These equations are solved by finite difference method. The effect on the velocity and temperature for the various parameters entering into the problems are separately discussed and shown graphically. We use Fortran Program for taking Data and for graphical representation we use TecPlot. Keywords: Falknar-skan; navier-Stoke; blasius equation; boundary layer; thermal boundary. _____________________________________ *Corresponding author: E-mail: [email protected], [email protected]; Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 1. Introduction It all started in 1904 at the International Mathematical Congress in Heidelberg, when Ludwig Prandtl give a lecture entitled “Über Flü ssigkeitsbewegungen bei sehr kleiner Reibung” (English: “On fluid flow with very little friction”) [1]. He explained that the viscosity of a fluid plays a role in a (very) thin layer adjacent to the surface, which he called “Uebergangsschicht” or “Grenzschicht”. Translated into English, the latter led to term boundary layer [2,3]. The boundary layer theory plays a vital role in the variety area of engineering and scientific applications [4]. With this lecture, the understanding of fluid flow was significantly increased. For instance, D’Alembert’s paradox, stating that a body placed in a potential flow does not experience a forceclearly in conflict with everyday experience-was resolved. Subsequently, it could be explained, e.g., why birds and airplanes can fly. The thus far invisible boundary layer was responsible. It plays a vital role in fluid dynamics and has become a very powerful of analysing the complex behavior of real fluids. The boundary layer flow due to a shrinking sheet is emerged as an interesting problem in fluid dynamics [5]. The shrinking sheet flows occur in some practical situations, such as, for rising shrinking balloon and it is very useful in packaging of bulk products. The drag on ships and missiles, the efficiency of compressor and turbines in jet engines, the effectiveness of air intakes for ram and turbojets and so on depend on the concept of the boundary layer and its effects on the main flow. The boundary-layer flow induced by a stretching surface has been the focus of large area during the last few decades in view of its many applications in the polymer extrusion, in a melt spinning processes, aerodynamic extrusion of plastic sheets, glass fiber production, the cooling and drying of paper and textiles, water pipes, sewer pipes, irrigation channels, blood vessels etc. The boundary-layer flow of nanofluid over a stretching sheet is a current attractive topic among the researchers [6]. With solving any equation and depending on the arguments in physical terms, the bounary layer theory is capable of explaining the difficulties encountered by ideal fluid dynamics [7]. When a real fluid (viscous fluid) flows past a stationary solid boundary, a layer of fluid which comes in contact with the boundary surface adheres to it (on account of viscosity) and condition of no slip occurs(The slip no-slip condition implies that the velocity of fluid at a solid boundary must be same as that of boundary itself). Thus, the layer of fluid which can’t slip away from the boundary surface undergoes retardation for the adjacent layer of the fluid, thereby developing a small region in the immediate vicinity of the boundary surface in which the velocity of the following fluid increases rapidly from zero at the boundary surface and approaches the velocity of main stream [2]. The layer adjacent to the boundary is known as Boundary Layer. Boundary Layer is formed whenever there is relative motion between the boundary and the fluid. Since  u  0      y  y  0 the fluid exerts a shear stress on the boundary and boundary exerts an equal and opposite force on the fluid known as the shear resistance [8]. According to boundary layer theory the extensive fluid medium around bodies moving in fluid can be divided into following two regions: (i) (ii) A thin layer adjoining the boundary called the boundary layer where the viscous shear takes place. A flow outside the boundary layer where the flow behaviour is quite like that of an ideal fluid and the potential theory is applicable [9]. Recntly, many authors are working on 3-D boundary layer flow. Three-dimensional boundary layer flow transition over rotating disks has been a subject of many studies. These studies are served as the foremost model problem for the subsequent investigations of the 3-D boundary layer flows over axisymmetric bodies of revolution [10]. For the Both case theoretically and experimentally, the case of a flow field structure of the laminar boundary layer flow over rotating spheres has been greatly. The flow visualisation studies led by the papers are related to the transition of the laminar boundary layer flow over rotating spheres and cones [11]. The stability and instability of the boundary layer flows on rotating spheres, spheroids, and disks. The theoretical studies of [12] related to the transition phenomena of the laminar boundary layer flow over various rotating geometries like disk, sphere and cone were carried out in such a way that the governing 2 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 laminar flow equations were first derived using some appropriate coordinate systems for each geometry. These laminar flow equations are actually a set of simultaneous 3-D nonlinear partial differential equations and are solved by using advanced numerical methods. Subsequently, the perturbation equations that govern the transition of the laminar boundary layer are derived for each body. The solutions of the laminar flow equations are then used in solving the related perturbation equations for each body [10]. The Falkner-Skan equation has been considered in the last 40 years due to its significance in the boundary layer theory. At the first time the solution of the Falkner-Skan equation has been studied numerically and solved this equation by the shooting method. A researcher Maksyn solved the Falkner-Skan equation by analytic approximation. After that, found its solution using finite differences, homotopy analysis and Fourier series to solve Falkner-Skan equation. An important case is the Blasius equation [11,4]. This problem was solved by Rosales and Valencia [13] using Fourier series. Mathematician Boyd found the solution of Falkner-Skan equation by numerical method. An enormous amount of research work has been invested in the study of nonlinear boundary value problems [4]. 2. Experimental Details The momentum thickness represents the vertical distance that the solid boundary must be displaced upward so that the ideal fluid has the same mass momentum as the real fluid. Expression for ∗ and Table 1. using various types of velocity profiles in the boundary layer is tabulated in Table 1. ∗ and Types of velocity distribution Linear profile, = Parabolic profile, =− Boundary layer momentum thickness, 2 6 2 15 3 8 2 Turbulent profile, = Boundary layer displacement thickness, ∗ 3 +2 Cubic profile, 3 1 = − 2 2 Sin-Cos profile, = for various types of velocity profiles in the boundary layer / 1− 2 8 Thus we can write, 2 f '''    f   f ''    0 39 280 7 72 2 − 1 2 (1) With boundary conditions, f    0, f '    0 And at  0 f '    1 as    (2) (3) 3 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 The differential equation (1) with boundary conditions (2) and (3) is known as Blasius equation [14]. Its solution is shown in Fig. 1. Fig. 1. Velocity distribution of Blasius equation The solution (Blassius solution) is tabulated as follows: Table 2. The Blassius solution y 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 U0 x ′( ) = ⁄ 0 0.1328 0.2647 0.3938 0.5168 0.6298 0.7290 0.8115 0.8767 η ′( ) 3.6 4.0 4.4 4.8 5.0 5.2 5.6 6.0 ∞ 0.9233 0.9555 0.9759 0.9878 0.9916 0.9943 0.9975 0.9990 1.0000 2.1 Mathematical formulation T T T k  2T    v u For unsteady state y ρ c p y 2 c p x t The Navier-stokes equation in X into account.  u     y  2 (4)  direction with boundary layer without body force due to gravity taken u u u 1 p  2u  u  v  g   2 t x y y  x (5) 4 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 With the corresponding initial and boundary conditions are At =0 >0 = 0, = Where = 0, , = 0 everywhere = 0, = 0, → = 0, → = 0, ∞ → ∞ (6) =0 (7) =0 →∞ x, y are Cartesian coordinate system. u, v are x, y component of flow velocity respectively local acceleration due to gravity ;  is the kinematic viscosity; thermal conductivity ; Cp  is the density of the fluid ; is the is the is the specific heat at the constant pressure [7]. Since the solutions of the governing equations (4)-(5) under the initial (6) and boundary (7) conditions will be based on a finite difference method it is required to make the said equations dimensionless [14]. we attempt to solve the governing second order nonlinear coupled dimensionless partial differential equations with the associated initial and boundary conditions. For solving a transient free convection flow with heat and mass transfer past a semi infinite plate, Callahan and Marner (1976) used the difference method [15]. From the concept of the above discussion, for simplicity the explicit finite difference method has been used to solve equations continuity and momentum equation subject to the conditions . To obtain the difference equations the region of the flow is divided into a grid of lines parallel to where X -axes is taken along the the plate of height Y   i.e. Y plate and = 100 i.e. X Y X and Y axes - axes is normal to the plate. Here we consider that varies from 0 to 100 and regard varies 0 to 20. There are m  400 and directions respectively as shown in the Fig. 2 n 400 grid = 25 as corresponding to spacing in the X and Y Fig. 2. The finite difference space grid 5 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 It is assumed that  X ,  Y are constant mesh sizes along follows, X and Y directions respectively and taken as X  0.5(0  x  200) Y  0.05(0  y  20) With the smaller time step,   0.001. 2.2 Falkner equation Let us consider the boundary layer equation of the type of potential flow for which similar solution exists. The boundary layer equation for two dimensional flow is [5] u u  2u U v u  2 v U x y x y (8) u v  v =0 y x (9) with boundary conditions u  v  0 at y  0 and u  U ( x ) at (10) y  (11) by solving we get, f   ff   (1 f 2 )  0 (12) and the boundary conditions may be re-written as f  0, f   0 at   0 f   0 as    (13) Equation (12) with boundary conditions (13) was first deduced V.M. Falkner and S. W. Skan. So this equation is known as Falknar-Skan Equation   and solution of Falknar equation is u  U e   1 1          e 2 2   1  1      e2   . 2 2    2.3 Thermal boundary layer equation Continuity equation u v  0 x y (14) 6 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 Momentum equation + + Energy equation + + = = + ( − ∞) + (16) With the corresponding initial and boundary conditions are At =0 >0 = 0, = 0, → = 0, = 0, → Where = , = 0 everywhere = 0, ∞ = 0, → =0 →∞ x, y are Cartesian coordinate system. u, v local acceleration due to gravity ; thermal conductivity ; Cp (17) (18) =0 ∞  (15) are x, y component of flow velocity respectively is the kinematic viscosity;  is the density of the fluid ; is the is the is the specific heat at the constant pressure [11,16]. Since the solutions of the governing equations (14)-(16) under the initial (17) and boundary (18) conditions will be based on a finite difference method it is required to make the said equations dimensionless. For this purpose we now introduce the following dimensionless variables; X xU 0  ,Y  yU 0  ,U  tU 2 u v T  T , V  ,   0 ,T  U0 U0 Tw  T  Using these relations we have the following derivatives are u U 0 2 U u U 03 U u U 0 2 U  ,  ,  , t   x  X y  Y 2  2u U 03  2U v U 0 V  , ,  y 2  2 Y 2 y  Y T U 0 Tw  T  T T U 0 Tw  T  T  ;  ; x  X y  Y = ( − ∞) Now we substitute the values of the above derivatives into the equations (14)-(16) and by simplifying we obtain the following nonlinear coupled partial differential equations in terms of dimensionless variables 7 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 U V  0 X Y Where, (19) + + = + + = Grashof number + 1 (20) +  Gr   g  Prandlt number  Pr  Eckert number = C p (21) Tw  T  U 03 K ( − ∞) Also the associated initial and boundary conditions become = 0 = 0, = 0, >0 = 0, = 0, = 0 = 0, = 0, = 0, = 0, =1 =0 = 0 everywhere (22) =0 (23) =0 →∞ 3. Results and Discussion 3.1 Boundary layer equation and its solution Our main purpose is to analysis the variation of stressing factor and velocity f  . That means if we change how the nature of velocity in terms of stressing factor will alter, the values of and suction parameter that is it will increase or decrease, or it will remain constant. In Figs. 3 to 4, we show that the velocity profile for different values of at various suction parameter = 3,6,9 respectively. From these figures we conclude that the velocity profile decrease with the increase of suction parameter. In Figs. 5 and 7 it is found that the velocity profile is increase with the increase of suction parameter for values of = 60,80. In Figs. 6 and 8 the velocity profile is decrease with the increase of suction parameter for values of −60, −80 In Figs. 9 to 10, it is said that the velocity profile are same for different values of = 3,6 parameter In Fig. 11 the velocity profile decreases with the increase of at =9 = (±) at the suction (+) and increases with the increase of (−) 8 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 1 1.1 0.9 1 fw=3 fw=6 fw=9 0.8 0.8 0.7 f' 0.7 0.6 f' 0.5 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.5 1 n 1.5 Fig. 3. For β=20 and 0 2 0 1 2 n Fig. 4. For β=-20 and = , , 1 3 = , , 1.5 0.9 1.4 fw=3 fw=6 fw=9 0.8 0.7 fw=3 fw=6 fw=9 1.3 1.2 1.1 0.6 1 0.5 0.9 0.4 f ' 0.8 0.7 f ' 0.3 0.2 0.6 0.1 0.5 0 0.4 -0.1 0.3 -0.2 0.2 0.1 -0.3 0 1 2 n Fig. 5. For β=60 and 0 3 0 1 n 2 Fig. 6. For β=-60 and = , , 3 = , , 1.8 1 0.8 1.6 fw=3 fw=6 fw=9 0.6 fw=3 fw=6 fw=9 1.4 1.2 0.4 f' fw=3 fw=6 fw=9 0.9 0.2 f' 1 0.8 0 0.6 -0.2 0.4 -0.4 0.2 -0.6 0 1 n Fig. 7. For β=80 and 2 = , , 3 0 0 1 n Fig. 8. For β=-80 and 2 3 = , , 9 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 Fig. 9. For different values of β (±) at = Fig. 10. For different values of β (±) at Fig. 11. For different values of β (±) at 3.2 Thermal boundary layer equation and its solution = = The main goal of the computation is to obtain the steady state solutions for the non-dimensional velocity U and temperature T for different values of Prandtl Pr , Eckert E c , and Grashof number Gr. For this purpose, computations have been carried out up to time  =80. The results of the computations, however, show little changes in the above mentioned quantities after  =50 have been reached. Thus the solution for  =80 are essentially steady state solutions. Along with the steady state solutions the solutions for the transient values of U andT are shown in Fig. 12-22 for time  =10, 50, respectively. The most important fluids are atmospheric air, water and saltwater. So the results are limited to Pr =0.71 (Prandtl number for air at 20 0 C , Pr =7.0 (Prandtl number for air at 20 0 C ), and Pr =1.0 (Prandtl number for saltwater). The values of another parameter Eckert number is chosen arbitrarily. Fig. 11 to Fig. 14 shows the velocity profile for different values of Grashof number (Gr) at time  =10,50. From these figure the velocity profile increase with the increase of Grashof number. Fig. 15 to 18 shows the velocity and temperature profile for different values of Prandtl number (Pr) at time  =10,50. From these figure the velocity and temperature profile decrease with the increase of Prandtl number. Fig. 19 to Fig. 22 shows the velocity and temperature profile for different values of Eckert number (Ec)at time  =10,50. From these figure the velocity and temperature profile increase with the increase of Eckert number(Ec). 10 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 In Fig. 11 to Fig 12, we show velocity profiles for different values of Gr at time   10 and   50 respectively. From these figures it is concluded that the velocity profile increases with the increase of Grashof number. In Fig. 13 to Fig. 14 show the temperature profiles for different values of Gr at time   10 and   50 respectively. From these figures the temperature profiles decrease with the increase of Grashof number. In Fig. 15 to Fig. 16 show the velocity profiles for different values of Pr at time   10 and   50 respectively. From these figures, we show that the velocity profile decrease with the increase of Prandtl number. 18 18 16 16 Gr=4.0 Gr=5.0 Gr=9.0 14 12 U 12 10 10 U 8 8 6 6 4 4 2 2 0 0 2 4 Y 6 8 0 10 Fig. 11. Velocity profiles for different values of Grashof number when Pr=0.71 Ec=0.03 at time = 0.9 0.6 0.6 0.5 T 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Y 4 6 Fig. 13. Temperature profile for different values of Grashof number when Pr=0.71, Ec=0.03 at time = 6 8 10 Gr=4.0 Gr=5.0 Gr=9.0 0.5 0.4 2 Y 0.8 0.7 0 4 0.9 0.7 0 2 1 Gr=4.0 Gr=5.0 Gr=9.0 0.8 0 Fig. 12. Velocity profiles for different values of Grashof number when Pr=0.71 Ec=0.03 at time = 1 T Gr=4.0 Gr=5.0 Gr=9.0 14 8 0 0 2 Y 4 6 8 Fig. 14. Temperature profile for different values of Grashof number when Pr=0.71, Ec=0.03 at time = 11 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 12 2.5 11 Pr=0.71 Pr=1.0 Pr=7.0 10 2 Pr=0.71 Pr=1.0 Pr=7.0 9 8 1.5 7 U U 6 5 1 4 3 0.5 2 1 0 0 2 4 Y 6 8 0 10 Fig. 15. Velocity profiles for different values of Prandtl number when Gr=4 Ec=0.01 at time = 4 0.9 Pr=0.71 Pr=1.0 Pr=7.0 0.8 0.7 Y 6 8 10 1 0.9 Pr=0.71 Pr=1.0 Pr=7.0 0.8 0.7 0.6 0.6 0.5 T 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 2 Fig. 16. Velocity profiles for different values of Prandtl number when Gr=4 Ec=0.01 at time = 1 T 0 0 2 Y 4 6 Fig.17. Temperature profile for different values of Prandtl number when Gr=4, Ec=0.01 at time = 8 0 0 2 Y 4 6 Fig. 18. Temperature profile for different values of Prandtl number when Gr=4, Ec=0.01 at time = In Fig. 17 to Fig. 18 show the temperature profiles for different values of Pr at time   10 and   50 respectively. From these figures the temperature profile decreases with the increase of Prandtl number. In Fig. 19 to Fig 20 show the velocity profiles for different values of Ec at time   10 and   50 respectively. From these figures it is found that the velocity profile increase with the increase of Eckert number. 12 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 In Fig. 21 to Fig 22 show the temperature profiles for different values of Ec at time   10 and   50 respectively. From these figures the temperature profiles increase with the increase of Eckert number. 8 2.5 7 2 Ec=0.01 Ec=0.02 Ec=0.03 Ec=0.01 Ec=0.02 Ec=0.03 6 5 1.5 U U 1 4 3 2 0.5 1 0 0 2 Y 4 6 8 0 10 Fig. 19. Velocity profiles for different Values of Eckert number when Gr=4 Pr=0.71 at time = 1 0.9 T 1 0.7 0.6 0.6 0.5 T 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2 4 Y 6 8 Fig. 21. Temperature profile for different values of Eckert number when Gr=4, Pr=0.71 at time = 4 Conclusion Y 6 8 10 10 Fig. 20. Velocity profiles for different values of Eckert number when Gr=4 Pr=0.71 at time = Ec=0.01 Ec=0.02 Ec=0.03 0.5 0.4 0 4 0.8 0.7 0 2 0.9 Ec=0.01 Ec=0.02 Ec=0.03 0.8 0 0 0 2 4 Y 6 8 10 Fig. 22. Temperature profile for different values of Eckert number when Gr=4, Pr=0.71 at time = The Thermal Boundary Layer Equation had been derived from Navier-stoke and concentration equation by boundary layer technique. Boundary Layer equation has been non-dimensionalised by using nondimensional variable. The non-dimensional boundary layer equations are non-linear partial differential 13 Mamtaz et al.; ARJOM, 11(4): 1-15, 2018; Article no.ARJOM.45267 equations. These equations are solved by finite difference method. Finite difference solution of heat and mass transfer flow is studied to examine the velocity and temperature. The effect on the velocity and temperature for the various important parameters entering into the problems are separately discussed into the problem with the help of graphs. Then the results in the form of velocity and temperature distribution are shown graphically. Competing Interests Authors have declared that no competing interests exist. References [1] Makanga Robert Ayiecha. August 2017, ISSN-2410-1397, The numerical solution for laminar boundary layer flow over a flat plate, university of Nairobi. [2] Swarep, Shanti. Hydrodynamics (5th edition). Krishna prakashan Media (p). Ltd. Pijush, K. K and Cohen M. Ira, Fluid Mechanics (second edition). 2005;323-330. 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McGraw-Hill Company. _______________________________________________________________________________________ © 2018 Mamtaz et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) http://www.sciencedomain.org/review-history/27853 15