ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com RESEARCH ARTICLE OPEN ACCESS Implicit Finite Difference Solution of Boundary Layer Heat Flow over a Flat Plate Satish V Desale*, V.H.Pradhan** * R.C.Patel Institute of Technology, Shirpur, India, 45405 ** S.V.National Institute of Technology, Surat, India, 395007 ABSTRACT In this paper, boundary layer heat flow over a flat plate is discussed. Similarity transformation is employed to transform the governing partial differential equations into ordinary ones, which are then solved numerically using implicit finite difference scheme namely Keller box method. The obtained Keller box solutions, in comparison with the previously published work are performed and are found to be in a good agreement. Numerical results for the temperature distribution have been shown graphically for different values of the Prandtl number. Keywords - Boundary layer flow, Convection of heat, Keller box method, nonlinear differential equation. I. INTRODUCTION The mathematical complexity of convection heat transfer is found to the non-linearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields. The boundary layer concept, first introduced by Prandtl in 904, provides maor simplifications. This concept is based on the belief that under special conditions certain terms in the governing equations are much smaller than others and therefore can be neglected without significantly affecting the accuracy of the solution. Most boundary-layer models can be reduced to systems of nonlinear ordinary differential equations which are usually solved by numerical methods. In this paper, solution of boundary layer heat flow over a flat plate is study with the help of implicit finite difference Keller box method. The same problem is solved by M. Esameilpour et al. [] with the help of He s Homotopy perturbation method. H. Mirgolbabaei [] solved using Adomian Decomposition Method (ADM). II. GOVERNING EQUATIONS The boundary layer equations assume the following: (i) steady, incompressible flow, (ii) laminar flow, (iii) no significant gradients of pressure in the x-direction, and (iv) velocity gradients in the x- direction are small compared to velocity gradients in the y-direction. The reduced Navier Stokes equations for boundary layer over flat plate [3] Continuity Equation: u v 0 () x y Momentum Equation: u u u u v g T T () x y y Under a boundary layer assumption, the energy transport equation is also simplified. T T u u v (3) x y y Subect to the boundary conditions i) No slip u 0 ii) Impermeability v 0 iii) Wall Temperature T Tw (4) iv) Uniform flow u U v) Uniform flow v 0 vi) Uniform Temperature T T The solution to the momentum equation is independent of the energy solution. However, the solution of the energy equation is still depends on the momentum solution. The following dimensionless variables are introduced in the transformation [] y 0.5 Rex x T T ( ) T T where is non -dimensional form of the temperature and the Reynolds number is defined as ux Re v the governing equations ()-(3) can be reduced to two equations, where f is a function of the similarity variable w www.iera.com 6 P a g e
ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com f ff 0 (5) 0 f (6) where and f is related to the u velocity Pr by f u. u The corresponding boundary conditions are f(0) 0, f (0) 0, (0), f ( ), ( ) 0. III. KELLER BOX METHOD (7) Equation (5) and (6) subect to the boundary conditions (7) is solved numerically using implicit finite difference method that is known as Keller-box in combination with the Newton s linearization techniques as described by Cebeci and Bradshaw [4]. This method is completely stable and has secondorder accuracy. In this method the transformed differential equations (5)-(6) are writes in terms of first order system, for that introduce new dependent variable u,, v w such that f u u v (8) w where prime denotes the differentiation w.r.to. Equation (5) and (6) become v fv 0 (9) w fw 0 (0) with new independent boundary conditions are f(0) 0, u(0) 0, (0) u( ), ( ) 0 () Now write the finite difference approximations of the ordinary differential equations (8) for the midpoint n x, of the segment using centered difference n derivatives, this is called centering about x,. f f u u h n n n n () u u v v h n n n n w w h n n n n (3) (4) Ordinary differential equations (9) and (0) are approximated by the centering about the mid-point n x, of the rectangle. v v f f v v h v v h fv n n w w f f w w h w w h fw (5) (6) Now linearize the nonlinear system of equations ()- (6) using the Newton s quasi-linearization method [5] h f f u u r h u u v v r h w w r3 (7) ( a ) v ( a ) v ( a ) f ( a ) f r 3 4 4 ( b ) w ( b ) w ( b ) f ( b ) f r 3 4 5 The linearized difference equation of the system (7) has a block tridiagonal structure. In a vector matrix form, it can be written as www.iera.com 6 P a g e
ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com A C B A C B A C [ ] [ r] [ ] [ r] [ 3 3 3 3] [ r3] : : : : BJ AJ C J : : BJ AJ [ J] [ rj] This block tridiagonal structure can be solved using LU method explained by Na [5]. IV. RESULTS AND DISCUSSION The numerical results of boundary layer heat flow over a flat plate are shown in Table. Table gives a comparison of the obtained numerical solution with HPM solution which is found to be in good agreement. Fig.,, 3 gives graphical comparison of Keller box and HPM solution for f, f and.in Fig. 4 numerical results for the temperature distribution have been shown graphically for different values of the Prandtl number Table Keller Box solution of boundary layer heat flow over a flat plate when Pr= f f f 0 0 0 0.3309-0.3309 0. 0.00664 0.06644 0.3306 0.933586-0.330 0.4 0.0656 0.3776 0.335 0.8674-0.335 0.6 0.05974 0.98954 0.33007 0.80046-0.330 0.8 0.066 0.6473 0.3744 0.7357-0.374 0.65583 0.39803 0.3309 0.67097-0.3303. 0.37964 0.39380 0.36608 0.60699-0.366.4 0.33 0.45688 0.30788 0.5437-0.30788.6 0.4034 0.56783 0.96676 0.4837-0.9668.8 0.5954 0.574784 0.894 0.456-0.894 0.65005 0.6979 0.66758 0.37009-0.6676. 0.78 0.68335 0.48355 0.38665-0.4835.4 0.937 0.79005 0.8093 0.70995-0.809.6.07533 0.77477 0.06453 0.753-0.0645.8.3004 0.853 0.8400 0.8847-0.84 3.396834 0.846064 0.6354 0.53936-0.635 3..569 0.876 0.399 0.39-0.39 3.4.746975 0.9078 0.7865 0.098-0.787 3.6.99549 0.93347 0.098074 0.076653-0.09807 3.8.6054 0.9435 0.080 0.058865-0.080 4.30577 0.955535 0.0649 0.044465-0.064 4..498064 0.966973 0.050504 0.03307-0.0505 4.4.69386 0.975885 0.038957 0.045-0.03896 4.6.88874 0.98697 0.09468 0.07303-0.0947 4.8 3.085347 0.98780 0.0857 0.098-0.086 5 3.8330 0.99553 0.05893 0.008447-0.0589 www.iera.com 63 P a g e
ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com Table Comparisons of results obtained by using Keller Box method and HPM [] HPM f NM f Keller Box HPM NM Keller Box HPM NM Keller Box 0 0 0 0 0 0 0 0. 0.00697 0.00664 0.00664 0.0696975 0.0664077 0.066443 0.93030 0.93359 0.933586 0.4 0.07876 0.06676 0.0656 0.393444 0.3764 0.37763 0.860656 0.86736 0.8674 0.6 0.06696 0.0597 0.05974 0.08805 0.98937 0.98954 0.7989 0.80063 0.80046 0.8 0.374 0.0608 0.066 0.7788 0.647094 0.64796 0.7 0.7359 0.7357 0.7380 0.6557 0.65583 0.346538 0.3978 0.398033 0.653746 0.670 0.67097. 0.49804 0.37949 0.37964 0.435539 0.393776 0.393803 0.586446 0.6064 0.60699.4 0.339 0.398 0.33 0.4793309 0.45667 0.456879 0.50669 0.543738 0.5437.6 0.4440 0.403 0.4034 0.5430747 0.567567 0.56783 0.45695 0.48343 0.4837.8 0.5568 0.5958 0.5954 0.60489 0.574758 0.5747843 0.39577 0.454 0.456 0.68883 0.65004 0.65005 0.66097 0.697657 0.6979 0.33779 0.37034 0.37009. 0.808 0.7893 0.78 0.7649 0.68303 0.683346 0.8357 0.3869 0.38665.4 0.96987 0.99 0.937 0.76636 0.78989 0.79005 0.33677 0.708 0.70995.6.7077.07506.07533 0.83803 0.77455 0.774769 0.886 0.7545 0.753 3.46743.396808.396834 0.885438 0.8460444 0.846064 0.4567 0.43955 0.53936 3..647758.569095.569 0.9400 0.876084 0.876004 0.095999 0.398 0.39 3.4.8335.74695.746975 0.9369507 0.9076 0.907795 0.063049 0.08839 0.098 3.6.0379.9955.99549 0.954578 0.93396 0.933473 0.05548 0.06667 0.076653 3.8.7865.603.6054 0.9673977 0.948 0.9435 0.0360 0.05888 0.058865 4.45336.305746.30577 0.97606 0.95558 0.9555345 0.03789 0.0348 0.044465 4..669.49804.498064 0.98037 0.966957 0.966976 0.07976 0.033043 0.03307 4.4.849.6936.69386 0.9860369 0.9758708 0.9758854 0.03963 0.049 0.045 4.6 3.03455.88848.88874 0.989554 0.986835 0.986969 0.00446 0.0737 0.07303 4.8 3.49458 3.0853 3.085347 0.993854 0.9877895 0.987808 0.00646 0.0 0.098 5 3.478658 3.8374 3.8330 0.9999999 0.99549 0.99553 3.36E-0 0.008458 0.008447 www.iera.com 64 P a g e
ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com Figure Comparisons of results by Keller Box method, HPM and NM for f(η) Figure Comparisons of results by Keller Box method, HPM and NM for f (η) Figure 3 Comparisons of results by Keller Box method, HPM and NM for θ(η) www.iera.com 65 P a g e
ISSN : 48-96, Vol. 3, Issue 6, Nov-Dec 03, 6-66 www.iera.com V. CONCLUSION In this paper, Keller Box Method has been successfully applied to boundary layer heat transfer problem with specified boundary conditions. The obtained solutions are compared with ones from numerical method and Homotopy Perturbation Method. The excellent agreement of the Keller Box solutions and the exact solutions shows the reliability and the efficiency of the method. As prandtl number increases temperature distribution decreases which agrees well with the physical phenomena. REFERENCES [ ] M. Esmaeilpour, D.D. Gani, Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate, Physics Letters A 37, 007, 33 38. [ ] H. Mirgolbabaei, Analytical solution of forced-convective boundary-layer flow over a flat plate, Archives of civil and mechanical engineering, Vol. X (),00, 4-5 [ 3 ] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, (3/e, McGraw Hill, New York, 993) [ 4 ] Cebeci T., Bradshaw P. Physical and computational Aspects of Convective Heat Transfer, (New York, Springer 988).. [ 5 ] Na T.Y., Computational Methods in Engineering Boundary Value Problem, (New York: Academic Press 979). Figure 4 Effect of Prandtl Number on θ(η) www.iera.com 66 P a g e