SIMILARITY SOLUTIONS OF THE THREE DIMENSIONAL BOUNDARY LAYER EQUATIONS OF A CLASS OF GENERAL NON- NEWTONIAN FLUIDS

Transcription

1 SIMILARITY SOLUTIONS OF THE THREE DIMENSIONAL BOUNDARY LAYER EQUATIONS OF A CLASS OF GENERAL NON- NEWTONIAN FLUIDS Department of Mathematics, Veer Narmad South Gujarat University Surat India mtimol@mail.com Received 28 May 2011; accepted 16 October 2011 ABSTRACT Similarity analysis is made of the three dimensional incompressible laminar boundary layer flow of eneral class of non-newtonian fluids. This work is an extension of previous analysis by Na and Hansen (1967), where the similarity solution of laminar three dimensional boundary layer equations of Power-law fluids was investiated. For the present flow situation, it is observed that the similarly solutions exist only for the case of flow over wede. Further, it is also observed that for the more eneral case of the boundary layer flow of non-newtonian fluids over anybody shapes yields non similar solutions. Present similarity equations are well areed with those available in literature. Keywords: Non-Newtonian fluids, Power-law fluid, Reiner-Philippoff fluid, Non-similarity solution, Similarity solution,group of Transformations. 1 INTRODUCTION The classical theory of Newtonian fluid depends upon the hypothesis of linear relationship between stress tensor and strain tensor, rate of strain tensor and even rate of stress tensor. The fluids which do not follow such a linear relationship are called non-newtonian fluid. Non- Newtonian fluids are enerally divided in to two cateories like viscoinelastic fluids and viscoelastic fluids. The common feature of viscoinelastic fluids is that when at the rest they are isotropic and homoeneous and when they are subjected to a shear the resultant stress depends only on the rate of shear. However, such types of fluids show diverse behavior in response to applied stress. Numbers of rheoloical models have been proposed to explain such a diverse behavior. Some of this models are; Power-law fluids, Sisko fluids, Ellis fluids, Prandtl fluids Williamson fluids, Sutterby fluids, Reiner-Rivlin fluids, Binham plastic, Eyrin fluids, Powell Eyrin fluids, Reiner-Philippoff etc. To investiate the non-newtonian effects, the class of solutions known as similarity solutions place an important role. This is because that is the only class of the exact solutions for the overnin equations which are usually non-linear partial differential equations (PDEs) of the boundary layer type. Further this also serves as a reference to check approximate solutions. The subject of boundary layer flows of non-newtonian fluids has been a topic of an investiation from a lon time as it has an application in various industries and in day to day

2 78 life. It is well known that similarity solutions for the PDEs overnin the flow of Newtonian and non-newtonian fluids exist only for limited classes of main stream velocities at the ede of the boundary layer. For example, for two dimensional laminar boundary layer flow of Newtonian fluids, similarity solutions are limited to the well known Falkner-Skan solutions Rajaopal et al (1983). Most of the eneralization of the Falkner-Skan solutions and approximate solutions in the literature are limited to the power law fluids; this is because they are mathematically the easiest to be treated amon most of the non-newtonian fluids. In past, some special work on the topic was due to Acrivos et al ((1960), (1965)),Schowalter (1960),Bizzell et al (1962),Hayasi (1965), Kapur et al (1963), Lee and Ames (1966), Hansen and Na (1968), Walters et al (1964), Denn (1967), Seth (1974), Rajaopal et al ((1983), (1984)), Banks et al (1986). Recently, the topic developed much rapidly may be availability of fast computin devices and software facilities. Patil and Timol (2011) have investiated Similarity solutions of nonlinear partial differential equations overnin the motion of threedimensional unsteady incompressible laminar boundary layer flow of non-newtonian powerlaw fluid past a flat plate by two parameter roup of transformations method in its most eneral form.anjali Devi and Julie Andrews (2011) have discussed problem of incompressible, viscous, forced convective laminar boundary layer flow of copper water and alumina water nanofluid over a flat plate. An exact solution of the problem of oscillatory flow of a fluid and heat transfer alon a porous oscillatin channel in presence of an external manetic field has been discussed in detail by Adhikary and Misra (2011). They have considered the flow throuh a channel in which the fluid is injected on one boundary of the channel with a constant velocity, while it is sucked off at the other boundary with the same velocity. The two boundaries are considered to be in close contact with two plates placed parallel to each other. While Parmar and Timol (2011) have derived similarity transformations for the system of partial differential equations overnin the boundary layer equations for coupled heat and mass transfer natural convection of a viscous, incompressible and electrically conductin flow of non-newtonian Power-law fluids over a vertical permeable cone surface saturated porous medium in the presence of uniform transverse manetic field and the thermal radiation effects usin one parameter deductive roup-theoretic transformation technique. Even thouh considerable proress has been made in our understandin of the flow phenomena, more work is still needed to understand the effect of various parameters involvin different non-newtonian models and the formulation of accurate method of analysis for any body shape of enineerin sinificance. The theoretical research on the topic however is hampered because of complex nature of the equations describin the flows. Moreover, the nature of non-linear shearin stress and rate of a strain relationship of the various models poses additional difficulties. In the analysis of boundary layer problem, the class of solution known as similarity solutions places a vital role because it is the only class of exact solution for the boundary layer equations. For two-dimensional flow of Newtonian fluids, it is well-known that similarity solutions exist for the class of bodies known as the Falkner-Skan problem, which includes many practical eometry. On the other hand, for non-newtonian fluids, the non-linear relation between shearin stress and the rate of strain causes further restrictions on the class of problems this can be solved by similarity transformations.it is interestin to note that this nonlinear relationship can be mathematically expressed as a functional relationship between stress tensor and rate of deformation tensor and for different non-newtonian fluids this

3 Similarity Solutions Of The Three Dimensional Boundary Layer Equations 79 relationship may be implicit or explicit functional relationship. For the two dimensional case, such problem was investiated by Hansen and Na (1968) and they have drawn the conclusion that for boundary layer flows of non-newtonian fluids of any model, similarity solutions exist only for the flow passed a wede. There are certain aspects of three-dimensional boundary layers. In two-dimensional flow, since the boundary layer is restrictin to more in the direction of the outer flow due to any sufficiently stron opposin pressure radient. These ultimately results in separation of the flow from the surface and may trier important readjustment of the outer flows. In the threedimensional boundary layer, on the other hand, the flow remains in freedom of choosin the minimum difficult path and stron unfavorable pressure radient do not necessarily lead to detachment of flow from the surface but demonstrate themselves by drastic chanes in flow directions. Consequently, not only is the boundary layer separation modified in the threedimensional flow but also it carries different implications (reardin over all effects) from those which applied in two-dimensional case. Even thouh very useful information can be revealed to the various physical parameters on the boundary layer characteristics from the similarity solution, it is of limited enineerin value since for practical purposes bodies other than wede will most likely be encountered. This required a eneral formulation and solution technique which can solve any problem of boundary layer flows of non-newtonian fluids such as the Reiner Philippoff fluid treated in this paper a topic which seems to have been nelected in the literature. In this paper, we will therefore look beyond the similarity solution of the problem by considerin shapes other than a wede. A formulation is iven in which the boundary layer equations are transformed to a form which are suitable for solution by some exact or numerical solution techniques. The formulation is made into such a eneral form that boundary layer flows of any shape can be treated by enterin the expression of the main stream velocity into a eneral function p(x) and q(x) these are also known as body shape functions. Similarity equations will be presented in this paper for two examples, namely, the similarity solution of the flow over a wede and the non-similar solution of the flow over a semi-infinite flat plate with mainstream parallel to the plate. The second example is known as Blasius solution, which for the case of two-dimensional flows of Newtonian fluids is in well areement. Deviations from similarity solutions as shown in the present paper where non-newtonian fluids are treated therefore show clearly the effects of the various parameters involved in the model. There are two reasons for studyin this particular non-newtonian fluid model, namely Reiner- Philippoff fluid model. First, this model correctly represents a class of non-newtonian fluids and yet there seems to be luck of reported literature on the boundary layer flow of such fluids. Second, the present analysis introduces a method of formulation and solution which can be applied to the boundary layer flow of any non-newtonian fluid over any body shape in which the velocity radient is expressed explicitly as a function of the shearin stress. 2 PROBLEMFORMULATION The formulation of empirical relations for different non-newtonian fluid and its evaluation in terms of known variables is indeed very difficult task. Recently Patel and Timol (2010) have

4 80 derived empirical relationship for three-dimensional boundary layer equations of different non-newtonian fluids models. A number of industrially important fluids such as molten plastics, polymers, pulps and foods exhibit non-newtonian fluid behavior (Nakayama et al. 1988). Due to the rowin use of these non-newtonian materials, in various manufacturin and processin industries, considerable efforts have been directed towards understandin their flow characteristics. Many of the inelastic non-newtonian fluids encountered in chemical enineerin processes are assumed to follow the so-called "power-law model". This is because of the mathematical simplicity of the power-law fluids in which the shear stress varies accordin to a power function of the strain rate (Metzner et al. 1965). This model is purely phenomenoloical; however, it is useful in that approximately describes a reat number of real non-newtonian fluids. This model behaves properly under tensor deformation. Use of this model alone assumes that the fluid is purely viscous. But there are certain limitations of this model. For example, it is deduced from empirical relationship and it indicates an infinite effective viscosity for low shear rate, thus limitin its rane of applicability. For present study we consider Reiner-Philippoff non-newtonian fluid, mainly for two reasons: First, thismodel correctly represents a class of non-newtonian fluids and yet there seems to be a lackof reported literature on the boundary layer flow of such fluids. Second, the present analysis introduces a method of formulation and solution which can be applied to the boundary layer flow of any non-newtonian fluid over anybody shape in which the velocity radient isexpressed explicitly as a function of the shearin stress. The overnin differential equations for the three dimensional boundary layer flow of a Reiner Philippoff non-newtonian fluid can be written as[refer Na and Hansen (1967),Kalthia and Timol (1986)]. (1) { } (2) { } (3) Where are shearin stresses parallel to Y-direction and actin alon X and Z direction respectively. Followin Schowalter (1960), under the boundary layer assumptions, the only two non-vanishin components that are related explicitly to the velocity radient are iven by (4) ( ) ( ) ( ) ( ) (5) The boundary conditions are:

5 Similarity Solutions Of The Three Dimensional Boundary Layer Equations 81 y = 0: u(x) = 0; v (x) = 0;w(x) = 0 (6) (7) Let us introduce the followin dimensionless quantities (8) And a stream function, such that (9) Equations (1)-(7) become ( ) (10) ( ) (11) ( ) ( ) ( ) ( ) (12) (13) Subject to the boundary conditions: (14) (15) Equations (10)-(15) represent a system of non-linear partial differential equations, the solution of which is quite difficult. One major simplification can be achieved by usin the similarity transformation where the system of non-linear partial differential equations is reduced to a system of ordinary differential equations. Such transformations are of the limited to some special forms of the mainstream velocities. For the case of boundary layer flows of eneral non-newtonian fluids, it was proved by Timol and Kalthia (1986) that similarity solutions exist only if the mainstream velocities are iven by (16) (17)

6 82 This corresponds to the three-dimensional boundary layers flow past a wede. For other bodyshape, the flow is not self similar and a transformation will be introduced in this paper to reduce equations (10)-(15) to a form which can be solved by some suitable numerical technique. 3GROUP THEORETIC ANALYSIS Similarity analysis by the roup theoretic method is based on concepts derived from the theory of continuous transformation roup. This method was first introduced by Birkhoff (1950) and Moran (1952) and later on lot of contribution was made in these techniques by various research workers [Bluman and Cole (1974), Bluman and Kumai (1989) and Bluman and Anco (2002).] For the present problem here we select followin one parameter linear roup of transformationsg: { Where are constants and A is a parameter of roup transformation. We now seek relations amon the such that the basic equation will be invariant under this roup of transformation. This can be achieved by substitutin the transformation into equation (10)-(15). Thus, we obtain ( ) ( ) (18) And ( ) ( ) (19) ( ( ) ) (20) (21) (22) (23) From equations (10)-(15), it is seen that if the basic equation are to be invariant under theg roup of transformation, the powers of A in each term should be equal. Thus, invariance of equations (18) to (23) under the roup G ives followin relations amon.

7 Similarity Solutions Of The Three Dimensional Boundary Layer Equations 83 (24) Solvin equations (24) we et (25) The next step in this method is to find the so-c ll b ol t i ri t r ro p of transformation. Absolute invariants are function havin the same form before and after the transformation (26) Therefore, these functions are absolute invariants under this roup of transformation. We therefore obtain the transformed independent and dependent variables are: (27) Substitutin for independent and dependent variable in equations (10)-(15) expressions found from equation (27). We expect to obtain a set of equations which are ordinary differential equation or very close to ordinary differential equations; specifically we obtain similarity solution flow over a wede, ( ) (28) (29) ( ( ) (30) ) (31) Subject to the boundary conditions: (32) (33)

8 Deduction For two-dimensional case, the flow will be independent of Z-direction and hence in this case, equations (29) and (31) will vanish (as ). Hence equations (28) and (30) alon with boundary conditions (32)- (33) will be reduced to those derived by Sirohi et al (1984) and Na (1994) which they have independently solved numerically. 4 NON-SIMILARITY SOLUTIONS For the eneral case in which the boundary layer over anybody shape is to be analyzed eneral transformations are introduced as follows: ( ) ( ) (34) Under these transformations, equations (10)-(15) become, { p } p { ( ) } { } (35) { } { } { } (36) Subject to the same boundary conditions iven by equations (32)-(33). Where ( ) (37) ( ) (38) p (39) (40) Here p are known as body shape functions and it is easy to verify that for

9 Similarity Solutions Of The Three Dimensional Boundary Layer Equations 85 p Now for these values of p above set of equations will reduce to those ordinary differential equations iven by (28) and (29). This also confirms that similarity solutions of present flow problem exist only for the flow past wede. On the other hand for the body other than wede, where are real constants. For the above mentioned case, the related bodyshape functions will be: p ; ( ) And for this case, non-similarity equation (35)-(38) will become { } { ( ) } { } (41) { } { } { } (42) Where correspondin equations (37) and (38) willreduce [ ] (43) ( ) (44) With the same boundary conditions are iven by equations (39)-(40). 4.1 Deduction For two-dimensional case, the flow will be independent of Z-direction and hence in this case, equations (42) and (44) will vanish (as ). Hence equations (41) and (43) alon with boundary conditions (32) - (33) will be reduced to those derived by Na (1994) which they have already solved numerically by Kellar-Box method. 5 CONCLUSION The similarity analysis of the three dimensional boundary layer flow of non-newtonian Reiner Philippoff fluids past external surface is derived. It is observed that similarity solutions exist for only the flow past at wede.for the flow past any otherbody shape, the same formulation can be used and only chane is in two functions namely p and equations (37) and (38) by substitutin into these functions the mainstream velocity for that particular eometry. The present analysis provides useful in formulation for the boundary layer flow not only for Reiner Philippoff fluids but for the other fluids too which are studied by Hansen and Na (1968), Sirohi et al (1984), Timol and Kalthia (1986), Patel and Timol (2009) and recently, Surati and Timol (2010).

10 86 REFERENCES Acrivos A, Shah MJ and Peterson EE (1960). Momentum and heat transfer in laminar boundary layer flows of non-newtonian fluids past external surfaces. AIChE J. 6, p.312. Acrivos A, Shah MJ and Peterson EE (1965).on the solution of the two-dimensional boundary layer flow equations for a non-newtonian power-law fluid. C and M. Enn Sci. 20, p.101. Adhikary SD and Misra JC (2011). Unsteady two-dimensional hydromanetic flow and heat transfer of a fluid, International Journal of Applied Mechanics and Mathematics (IJAMM), 7 (4): pp Anjali Devi and Julie Andrews (2011). Laminar boundary layer flow of nanofluid over a flat plate, International Journal of Applied Mechanics and Mathematics (IJAMM), 7 (6): pp Banks WHH and Zatunka MB (1986). Eien solutions in boundary layer flow adjacent to a stretchin sheet IMA J. Appl. Mathematics, 36, pp Beard DW and Walters K (1964). Elastico-viscous boundary layer flows-i Two-dimensional flow near a stanation point. Proc. Camb. Phil. Sot. 60, pp Birkhoff G (1950). Hydrodynamics, Princeton Univ. Press, Princeton, New Jersey, Ch. V. Bixzell GD and AlatteryJC (1962). Non-Newtonian boundary layer flow Chem. Enn Sci. 17, p Bluman G and Anco SC (2002).Symmetry and Interation Methods for Differential Equations, Appl Math.Sci. No. 154, Spriner-Verla, New York, Bluman GW and Cole JD (1974).Similarity methods for Differential Equations, Spriner- Verla, New York, Bluman GW and Kumai S (1989). Symmetries and Differential Equations, Applied Mathematical Sciences, No. 81, Spriner- Verla, New York. Denn MM (1967). Boundary layer flows of a class of elastic fluids, Chem. Emma Sci. 22, pp Hansen AG and Na TY (1968). Similarity solutions of laminar, incompressible boundary layer equations of non-newtonian fluids, Trans. ASME, J Basic Enn.pp Hayasi N (1965). Similarity of two-dimensional and axisymmetric boundary layer flows of non-newtonian fluids. J. Fluid Mech 23, pp Kapur JN and Srivstava RC (1963).Similar solutions of the boundary layer equations for Power-law fluids. ZAMP 14, p.383.

11 Similarity Solutions Of The Three Dimensional Boundary Layer Equations 87 Lee SY and Ames WF(1966). Similar solutions for non-newtonian fluids, AIChE J. 12, p.700. Metzner AB (1965). Heat Transfer in non-newtonian fluid, Adv. Heat Transfer. Vol. 2, pp Moran AJA (1952). Redaction by One of the Number of Independent Variables in Some Systems of Partial Differential Equations, Quart, Appl. Math.2, p.250. Na TY (1994). Boundary layer flow of Reiner Philippoff fluids Int. J. Non-Linear Mechanics, Vol. 29 No. 6, pp Na TY and Hansen AG (1967).Similarity solutions of a class of laminar, three-dimensional boundary layer equations of non-newtonian fluid. Int. J. Non-Linear Mech., 2, pp Na TY(1979). Computational Methods in Enineerin Boundary value Problems, Academic Press, New York. Nakayama A and Koyama H (1988). An analysis for friction and heat transfer characteristics of power-law non-newtonian fluid flows past bodies of arbitrary eometrical confiuration, Warme-und Stoffubertraun, Vol. 22, pp Parmar Hiral and Timol MG (2011). Convection flow of non-newtonian power-law fluid over a vertical cone throuh porous medium, International Journal of Applied Mechanics and Mathematics (IJAMM), 7 (2): pp Patel Manisha and Timol MG (2009). Numerical treatment of Powell Eyrin fluid flow usin method of satisfaction of asymptotic boundary conditions (MSABC), Applied Numerical Mathematics (Elsevier), Vol 59, pp Patel Manisha and Timol MG (2010).The eneral stress strain relationship for some different visco-inelastic non-newtonian fluids.international Journal of Applied Mechanics and Mathematics (IJAMM) 6 (12), pp Patil V and Timol MG (2011). On the class of Three-dimensional unsteady incompressible boundary layer equations of non-newtonian Power-law fluids, International Journal of Applied Mechanics and Mathematics (IJAMM), 7 (7): pp Rajaopal KR, Na TY and Gupta AS (1983).Falkner-Skan flow of second-order fluid, Int. J. Non-Linear Mech. pp Rajaopal KR, Na TY and Gupta AS (1984).Flow of a viscoelastic fluid over a stretchin sheet, Rheol. Acta 23, pp Schowalter WR (1960). The application of boundary layer theory to power law pseudo-plastic fluids: similar solutions. AIChE J. 6, p.24.

12 88 Seth RW (1974).Solution of visco- elastic boundary layer equations by orthoonal collection. J. Enineerin Math. 8, pp Sirohi Vijyalaxmi, Timol MG and Kalthia NL (1984).Numerical treatments of Powell- Eyrin fluid flow past a 90 0 Wede, Re J. Enery Heat and Mass Trans. Vol6, No.3, pp Surati Hema and Timol MG (2010).Numerical Study of Forced Convection Wede Flow of Some Non-Newtonian Fluids, International Journal of Applied Mechanics and Mathematics (IJAMM), 6 (18), pp Timol MG and Kalthia NL (1986).Similarity solution of three-dimensional boundary layer equations of non-newtonian fluids.int. J. Non-Linear Mech. V. 21, pp