Energetic Balance for the Flow Induced by a Constantly Accelerating Plate in a Second Grade Fluid

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Engineering 466-47 doi:.46/eng..66 Published Online June (http://www.scirp.

1 Engineering doi:.46/eng..66 Published Online June ( Energetic Balance for the Flow Induced by a Constantly Accelerating Plate in a Second Grade Fluid Abstract Corina Fetecau Masood Khan Constantin Fetecau Department of Theoretical Mechanics Technical University Iasi Romania Department of Mathematics Quaid-i-Azam University Islamabad Pakistan Department of Mathematics Technical University Iasi Romania cfetecau@yahoo.de {mkhan_ fetecau_constantin}@yahoo.com Received February 4 ; revised March ; accepted March 5 Dissipation power due to the shear stress at the wall and the boundary layer thickness corresponding to the unsteady flow of a second grade fluid due to a constantly accelerating plate are established in exact and approximate forms. The changing of the etic energy with time is also determined from the energetic balance. Exact expressions of the same entities for ewtonian fluids are recovered as limiting cases of general results. Keywords: Second Grade Fluid Dissipation Energetic Balance Power Kinetic Energy Constantly Accelerating Plate. Introduction In recent years the interest for flows of non-ewtonian fluids has considerably increased while the energetic aspects regarding these motions are scarcely met in the literature. To the best of our knowledge the first results of this d seem to be those of Bühler and Zierep [] concerning the Rayleigh-Stokes problem for ewtonian fluids. These results have been recently extended to second grade and Maxwell fluids by Zierep and Fetecau []. In these works the authors also determine the boundary layer thickness and use it in order to obtain series solutions for the velocity field and the shear stress. The aim of this note is to provide the energetic study for the unsteady flow of a second grade fluid driven by the transversal motion of a constantly accelerating plate and to compare the obtained results with those corresponding to the flow induced by a suddenly moved plate (Stokes first problem). Of special interest is the energetic balance of the three terms: Changing of the etic energy with time Dissipation and the Power due to the shear stress at the wall. The last term describes the input energy that is necessary to keep the medium running. A decisive question is whether these terms are larger or smaller than in the ewtonian case.. Statement of the Problem The flows to be here considered have the velocity field [-] v v() yt uyt () i () where i is the unit vector along the x-direction of the Cartesian coordinate system x y and z. For such motions the constraint of incompressibility is automatically satisfied while the governing equations in the absence of body forces and a pressure gradient in the flow direction are [4] tuyt () ( t) yuyt () () yt ( t) yuyt () () where is the dynamic viscosity is a material constant / is the ematic viscosity of the fluid ( being its constant density) / and Sxy is the non-trivial shear stress. The appropriate initial and boundary conditions corresponding to the motion over an infinite constantly accelerating plate are given by [4-6] uy () y; u() t () At t ; u() y t u() y t as y y where A is a constant. The energetic balance for an infinite volume of rectangular cross-section with x [ ] and z [ ] as it results from [-] is given by the identity d E L (4) dt Copyright SciRes. EG

2 C. FETECAU ET AL. 467 where E is the etic energy and L u( t) ( t) uyt ( ) uw() t w() t ( y) t dy y are the power of the shear stress at the wall and the dissipation. The boundary layer thickness as it results from [7] is given by u( y t) dy u( t) (6) and represents the thickness of the fluid layer moved with the plate by friction. One measure of the boundary layer thickness is the distance from the wall where the velocity of the fluid differs by % from the external velocity.. Exact Results Let us consider an incompressible second grade fluid at rest over an infinitely extended flat plate situated in the ( x z) plane. After time t the plate is subjected to a transversal motion in its plane of velocity u( t) At in the x-direction. Owing to the shear the fluid over the plate is gradually moved its velocity being of the form (). The governing equation is given by Equation () while the corresponding initial and boundary conditions are of the form (). The velocity field u( y t) and the adequate shear stress ( yt ) corresponding to this problem as it results from [8] (see Equations () and (4)) are given by (see also [5] Equation (5) with Vt () At or [6] Equation (7) for the velocity field only) uyt ( ) A t sin( y) At exp d (5) (7) ( yt ) A t cos( y) exp d. (8) Introducing (7) and (8) into (5) and (6) we find for L and the next exact expressions L At t (9) exp d 4A t cos( y) exp d () t cos( y) exp d dy t sin( y) () exp d dy. t By mag and thereby into Equations (7)-() we attain to the similar expressions u ( y t) L At t sin( A y ) e d ( yt ) A t cos( y) e d At t e d () 4 A t cos( y) e d dy () t sin( y ) e t d dy (4) corresponding to a ewtonian fluid performing the same motion. The integrals in ()-(4) can be evaluated to give the classical solutions y u( y t) 4Ati Erfc ( y t) t y A ti Erfc t (5) for velocity and shear stress and the simple expressions [9 Equations () and (4)] 8( ) 4 t t t L A t A t for L d E dt (7 4 ) t A t (6) and de / dt. Into Equation (5) Copyright SciRes. EG

3 468 C. FETECAU ET AL. n i Erfc( x) n i Erfc( ) d with x n...; i Erfc( x) Erfc( x) are the integrals of the complementary error function Erfc( ). 4. Asymptotic approximations for ( α/ νt)<< Let us now consider the case when the material module is small enough so that the dimensionless memory variable /( t) is much less than one. Simple computations show that the next approximations 4 t t t exp e exp (7) t 4 e O( ) t t exp t e ( t ) O( ) (8) are valid for each and t greater zero. Introducing Equations (7) and (8) into (9)-() and having in mind Equation (6) we can easy prove that At t L L t e d O( ) (9) t 4A e t cos( y) d t e cos( y) d dy O( ) () t e sin( ) O( ). () y d dy Evaluating all integrals as well as in [] and [] we attain to the following approximate expressions L L O 4t t O. () 4t t ( ) O () 8 t t Equations (4) (6) () and () also imply d d 9(5 ) E E O. (4) dt dt 68 t t The results corresponding to L and de / dt are in accordance with those obtained in [] for the motion due to a suddenly moved plate. More exactly into both motions L and de / dt increase for second grade fluids in comparison with ewtonian fluids. On the other hand the boundary layer thickness δ increases for the present motion and decreases for the first problem of Stokes. 5. Analogy to the Teipel Series Expansion As early as 98 Teipel [7] studied the first problem of Stokes for second grade fluids and provided a series solution. This series has been determined until an arbitrary constant although all initial and boundary conditions have been fulfilled. In order to determine that constant Teipel used a new initial condition namely (). Later in order to determine this constant Erdogan and Imrak [] used the condition that the volume flux is zero at time t. Following the Teipel s idea [7] and having in mind Equation () we are loog for a velocity field of the form uyt ( ) y (5) At f( ) f( ) O ; t t t for the flow induced by a constantly accelerating plate in a second grade fluid. Introducing (5) into the governing Equation () and identifying the coefficients of the same powers we find that f ( ) f ( ) 4 f( ) (6) and f ( ) f ( ) f ( ). The appropriate boundary conditions being f() f( ) and f() f( ) (7) it results that in contrast with the results of Teipel the functions f () and f () are completely determined and have the forms f( ) 4i Erfc( ) f( ) e (8) Consequently an approximate solution for the velocity field is Copyright SciRes. EG

4 C. FETECAU ET AL. 469 u( y t) t t 4t t (9) Introducing (9) into () and keeping the terms of the y Ay t y 4Ati Erfc exp O. first order into /( t) we get for the shear stress the approximate expression ( yt ) y A tierfc t A t y y exp O. t t 4t t 6. Conclusions () In this note the power due to the shear stress at the wall dissipation and the boundary layer thickness corresponding to the unsteady flow of a second grade fluid induced by a constantly accelerating plate have been determined in exact and approximate forms. Their exact expressions corresponding to ewtonian fluids performing the same motion are recovered as limiting cases of general results. As a consequence the changing of the etic energy with time is obtained from the energetic balance. Finally series solutions are established both for the velocity field u( y t ) and the shear stress ( yt ). In conclusion from Equations ()-(4) it clearly results that L and de / dt increase for second grade fluids in comparison with ewtonian fluids. Furthermore the results corresponding to L and de / dt are in accordance with those obtained in [] for the motion due to a suddenly moved plate. As regards the boundary layer thickness the present result is opposite to that from [] where decreases for second grade fluids in comparison with the ewtonian fluids. Finally in order to reveal some relevant physical aspects of the obtained results the diagrams of the velocity uyt ( ) and the shear stress ( yt ) against y are presented for different values of the time t. Figures and clearly show that the approximate solutions (9) and () give a very good approximation for the exact solutions (7) and (8). The variations of the power due to the shear stress at the wall L(t) and the boundary layer thickness () t are also depicted in Figures and 4 for different values of the material constant. These graphs show that L(t) and () t are increasing functions with respect to t and decreasing ones with regard to. Furthermore as expected for they are going to L and corresponding to ewtonian fluids. Figure. Comparison between the exact solution (7) curves u(y) u(y) u(y) and the asymptotic solution (9) curves u(y) u4(y) u5(y) for A =. =. = 97 =. and different values of t. Figure. Comparison between the exact solution (8) curves τ (y) τ (y) τ 4(y) and the asymptotic solution (9) curves τ (y) τ 4(y) τ 5(y) for A =. =. = 97 =. and different values of t. Figure. Power of the shear at the wall L(t) corresponding to ewtonian fluid curve L(t) and second grade fluid curves L(t) L(t) L(t) for A =. =. = 97 and different values of. Copyright SciRes. EG

5 47 C. FETECAU ET AL. Figure 4. The boundary layer thickness (t) corresponding to ewtonian fluid curve (t) and second grade fluid curves (t) (t) (t) for =. and different values of. 7. Acknowledgements The authors Corina Fetecau and Constantin Fetecau acknowledge support from the Ministry of Education and Research CCSIS through P II-Ideas Grant o. 6/8-9-7 Cod CCSIS ID_59. The authors also would like to express their gratitude to the referees both for their careful assessment and the fruitful comments and suggestions regarding the first form of this note. 8. References [] K. Bühler and J. Zierep Energetische Betrachtungen Zum Rayleigh-Stokes Problem Proceedings in Applied Mathematics and Mechanics Vol. 5 o. 5 pp [] J. Zierep and C. Fetecau Energetic Balance for the Rayleigh-Stokes Problem of a Second Grade Fluid International Journal of Engineering Science Vol. 45 o. 7 pp [] J. Zierep and C. Fetecau Energetic Balance for the Rayleigh-Stokes Problem of a Maxwell Fluid International Journal of Engineering Science Vol. 45 o pp [4] R. Bandelli K. R. Rajagopal and G. P. Galdi On some Unsteady Motions of Fluids of Second Grade Archives of Mechanics Vol. 47 o pp [5] C. Fetecau and J. Zierep On a Class of Exact Solutions of the Equations of Motion of a Second Grade Fluid Acta Mechanica Vol. 5 o. - pp [6] M. E. Erdogan and C. E. Imrak On Unsteady Motions of a Second-Order Fluid over a Plane Wall International Journal of on-linear Mechanics Vol. 8 o. 7 pp [7] I. Teipel The Impulsive Motion of a Flat Plate in a Visco-Elastic Fluid Acta Mechanica Vol. 9 o pp [8] C. Fetecau S. C. Prasad and K. R. Rajagopal A ote on the Flow Induced by a Constantly Accelerated Plate in an Oldroyd-B Fluid Applied Mathematical Modelling Vol. o. 4 7 pp [9] C. Fetecau D. Vieru A. Mahmood and C. Fetecau On the Energetic Balance for the Flow of a Maxwell Fluid Due to a Constantly Accelerating Plate Acta Mechanica Vol. o. - 9 pp [] M. E. Erdogan and C. E. Imrak On the Comparison of Two Different Solutions in the Form of Series of the Governing Equation of an Unsteady Flow of a Second Grade Fluid International Journal of on-linear Mechanics Vol. 4 o. 4 5 pp Copyright SciRes. EG

SUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS. where most of body fluids like blood and mucus are non-newtonian ones.

SUMMARY A STUDY OF VISCO-ELASTIC NON-NEWTONIAN FLUID FLOWS Non-Newtonian fluids abound in many aspects of life. They appear in nature, where most of body fluids like blood and mucus are non-newtonian ones.