The paper present a different semi-analytical solutions for the Stokes flow in a rectangular cavity by using the biorthogonality condition of the eigenfunctions. Firstly, the biorthogonality condition for rectangular cavity is applied on Stokes Flow problem. Secondly, the biorthogonality condition and the new solution is verified against results available in the literature. Further, it has been found that this solution mechanism is effi cient and it can be successfully used to obtain flow structures and bifurcations

Bilgil, Halis

Aksaray University Institutional Repository

Applied Mathematics E-Notes, 19(2019), 456-464 c© ISSN 1607-2510Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/An Application Of Khuri’s BiorthogonalityCondition On Stokes Flow In A Rectangular Cavity∗Halis Bilgil†Received 3 July 2018AbstractThe paper present a different semi-analytical solutions for the Stokes flow ina rectangular cavity by using the biorthogonality condition of the eigenfunctions.Firstly, the biorthogonality condition for rectangular cavity is applied on StokesFlow problem. Secondly, the biorthogonality condition and the new solution isverified against results available in the literature. Further, it has been found thatthis solution mechanism is effi cient and it can be successfully used to obtain flowstructures and bifurcations.1 IntroductionFlow inside a rectangular cavity has been studied extensively far more than two decadesand is one of the most popular fluid problems in the fluid mechanics. In the available lit-erature, it is possible to find numerous studies on lid-driven cavity flow [1, 2, 7, 8, 11, 15].Joseph and Sturges [12] were the first to formulate and solve the Stokes flow problemanalytically by expressing the streamfunction, ψ, as an infinite series of eigenfunctionsand determining the associated series coeffi cients using Smith’s biorthogonality con-dition [19]. Subsequently, Shankar [17] applied a least-squares method rather thana biorthogonality condition to determine the coeffi cients in the infinite series for thesame problem. Gaskell et al. [4] compared the accuracy of the above least-squaresand biorthogonal approaches by examining the convergence of the streamfunction andthe horizontal velocity component to the upper boundary conditions. Their results forthe ψ = 0 condition are better via the biorthogonal method than via the least-squaresmethod, but for the velocity conditions the situation is reversed. They also comparedvalues of the streamfunction in the cavity interior and found that the streamfunctionvalues are extremely close and almost unaffected by the choice of either the Shankar[17] or the Joseph and Sturges [12] coeffi cient determination methods.A new eigenfunction expansion method for solving Stokes flow problems in cavitiesthat arise in fluid dynamics has been developed by Khuri [13]. The method leads tothe development of a set of eigenfunctions, adjoint eigenfunctions, biorthogonality con-ditions, and an algorithm for the computation of the coeffi cients of the eigenfunction∗Mathematics Subject Classifications: 00A05, 35Q35, 76D99, 82B23, 65N99.†Department of Mathematics, Aksaray University, Aksaray 68100, Turkey456H. Bilgil 457expansion. Subsequently, Khuri [14] introduced a biorthogonality condition for rectan-gular regions. This biorhogonality conditions aims to solve the Stokes flow problemmore easily and apparent than above analytical solution methods.This article has two aims: first, to find a new analytical solution of Stokes flowproblem in a rectangular cavity by using Khuri’s biorthogonality conditions; secondly,to verify the accuracy and effi ciency of Khuri’s theorem to solve Stokes flow problemsin comparison to the previous solutions.2 Biorthogonality ConditionsThe fourth-order boundary value problem and the boundary conditions are given by,(P0(r)y′′(r))′′+ (P1(r;α)y′(r))′+ P2(r;α)y(r) = 0, r ∈ [r1, r2], (1)y(r1) = y(r2) = y′(r1) = y′(r2) = 0. (2)The biorthogonality condition for the boundary value problem given in following Khuri’sTheorem [1].THEOREM 1. Consider the boundary value problem given in Eq.(1) and (2) whereP0(r), P ′′1 (r;α), P2(r;α) are continuous and P0(r) 6= 0 on r1 ≤ r ≤ r2. Pi in Eq.(1)is a polynomial of degree at most i in the parameter α, in particular, let P1(r;α) =p11(r)α+ p12(r), and we requireP 21 (r;α)− 4P0(r)P2(r;α) = p31(r)α+ p32(r),p211(r) + p231(r) 6≡ 0.Then with P ∗n defined byP ∗n =∫ r2r1[φ(n)2 (r), φ(n)1 (r)]B(r)[φ(n)1 (r)φ(n)2 (r)]dr,we have the following biorthogonality condition:∫ r2r1[φ(m)2 (r), φ(m)1 (r)]B(r)[φ(n)1 (r)φ(n)2 (r)]dr = P ∗nδmn,where δmn is the Kronecker’s delta,B(r) =(− p11(r)2P0(r) 012p′′11(r) +14p31(r)2P0(r)− p11(r)2P0(r))withφ(n)1 (r) = yn(r),φ(n)2 (r) = P0(r)y′′n(r) +12P1(r;αn)yn(r).Here yi is an eigenfunction of Eq.(1) corresponding to the eigenvalue αi.458 An Application of Khuri’s Biorthogonality Condition3 The Boundary Value Problem For Rectangular Cav-ity and Biorthogonality Condition3.1 The Formulation of Flow ProblemA two-dimensional rectangular cavity ν = {x, y : −1 ≤ x ≤ 1,−A ≤ y ≤ A} is filledwith incompressible fluid (Fig. 1). The boundaries x = ±1 are fixed walls, y = A andy = −A are two moving lids which translate with speeds U1 and U2 in the horizontaldirection respectively, thus setting the fluid into motion.Under Stokes’s approximation for the creeping flow the streamfunction ψ(x, y) sat-isfies the two-dimensional biharmonic equation∇4ψ(x, y) =(∂2∂x2+∂2∂y2)2ψ(x, y) = 0. (3)The streamfunction is set equal to zero on the boundaries of the cavity, and the no-slipconditions for the upper and lower lids and side walls areψ(±1, y) = 0, ∂ψ∂x(±1, y) = 0, (4)ψ(x,±A) = 0, (5)∂ψ∂y(x,A) = U1,∂ψ∂y(x,−A) = U2. (6)The boundary-value problem has a separation of variables solution for ψ of the form,ψ(x, y) =∞∑−∞[Aneλn(y−A) +Bne−λn(y+A)] φ(n)1 (x, λn)λ2n(7)whereAn andBn are complex coeffi cients, the λn are complex eigenvalues and φ(n)1 (x, λn)are even Papkovich-Fadle eigenfunctions [2-4]. The eigenfunctions are given byφ(n)1 (x, λn) = λn sin(λn) cos(λnx)− λnx sin(λnx) cos(λn). (8)The parameters λn are complex eigenvalues determined by the side wall conditionsEq.(4), which yield the eigenvalue equation, sin 2λn = −2λn. The λn eigenvalues maybe determined by the simple Newton iteration procedure described by Robbins andSmith [5]. Table of the eigenvalues can be seen in [10]. The flow is governed by twophysical control parameters: the cavity aspect ratio A and the ratio (S = U2U1 ) of thelower to the upper lid speed.3.2 The Biorthogonality Condition For Rectangular Cavitiesand SolutionThe following biorthogonality condition for rectangular cavities is given by Khuri [7],∫ 1−1[φ(m)2 (x), φ(m)1 (x)]B(x)[φ(n)1 (x)φ(n)2 (x)]dx = P ∗nδmn (9)H. Bilgil 459Fig. 1: The boundary value problem for the lid-driven cavity.whereB(x) =(−1 00 −1)andφ(n)2 (x, λn) =(φ(n)1 (x))′′+ λ2nφ(n)1 (x) = −2λ2n cos(λn) cos(λnx). (10)Eq.(7) must satisfy the boundary conditions Eq.(5) and (6); that is,U1 =∞∑−∞[An −Bne−2λnA] φ(n)1 (x)λn, (11)0 =∞∑−∞[An +Bne−2λnA] φ(n)1 (x)λ2n, (12)U2 =∞∑−∞[Ane−2λnA −Bn] φ(n)1 (x)λn, (13)0 =∞∑−∞[Ane−2λnA +Bn] φ(n)1 (x)λ2n, (14)where U1 and U2 represent the velocities of the top and bottom lids. The other bound-ary conditions are used to find the λn eigenvalues. The solution procedure up to hereis almost the same with previous studies [6, 7, 8].The critical step in determining An and Bn is to apply the biorthogonality conditionEq.(9) to equations (11)—(14). To prefare for this application of the biorthogonalitycondition, equations (11)—(12) are combined to give∞∑−∞1λ2n(An +Bne−2λnA)(φ(n)1 (x)φ(n)2 (x))−∞∑−∞1λ2n(An +Bne−2λnA)( 0φ(n)2 (x))460 An Application of Khuri’s Biorthogonality Condition+∞∑−∞1λn(An −Bne−2λnA)( 0φ(n)1 (x))=(0U1). (15)Similarly, by coupling equations (13)—(14), rearranging terms, we get∞∑−∞1λ2n(Ane−2λnA +Bn)(φ(n)1 (x)φ(n)2 (x))−∞∑−∞1λ2n(Ane−2λnA +Bn)( 0φ(n)2 (x))+∞∑−∞1λn(Ane−2λnA −Bn)( 0φ(n)1 (x))=(0U2). (16)The operator ∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[··]dx = P ∗nδmnis then applied to Eq.(15) and (16) to yield∞∑−∞1λ2n(An +Bne−2λnA)Pm −∞∑−∞1λ2n(An +Bne−2λnA)Vm+∞∑−∞1λ2n(An −Bne−2λnA)= Gm,∞∑−∞1λ2n(Ane−2λnA +Bn)Pm −∞∑−∞1λ2n(Ane−2λnA +Bn)Vm+∞∑−∞1λ2n(Ane−2λnA −Bn)= gm,whereGm =∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[0U1]dx = −4U1,gm =∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[0U2]dx = −4U2,Pm =∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[φ(n)1 (x)φ(n)2 (x)]dx,Vmn =∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[0φ(n)2 (x)]dx,andWmn =∫ 1−1[φ(m)2 (x), φ(m)1 (x)]( −1 00 −1)[0φ(n)1 (x)]dx.Equations (15) and (16) form an infinite set to be solved for the coeffi cients An, Bn forn = ±1, ±2, .... They are determined as Eq.(15) and (16) are truncated after N terms.In this case Eq.(15) and (16) yield 4N equations for the 4N unknowns.H. Bilgil 4614 ResultsFigure 2 shows examples of streamline plots for S = −1 at which the lids move inopposite directions with equal speed ratio. It is seen that, the flow structure is sym-metrical about x and y-axis for all values of A. A single eddy occupies the cavity forA = 0.36 and A = 0.7, see Fig. 2a and 2b respectively. The single eddy has a centerstagnation point in the cavity center. As the aspect ratio is increased from 0.7 there arefour main stages in the development of the second and third eddies. In the first stage,a bifurcation appears at a critical value of A and two additional stagnation points aregenerated in the cavity, (see Fig. 2c where A = 1).(a) (b) (c)(d) (e) (f)Fig. 2: Streamline for a double lid-driven cavity for S = −1. (a) A = 0.36 (b) A = 0.7(c) A = 1 (d) A = 2.64 (e) A = 2.8 (f) A = 3.462 An Application of Khuri’s Biorthogonality ConditionAs A is increased further, two degenerate critical points appear on the two stationaryside walls. In the third stage, the heteroclinic connections coalesce with each other atthe interior saddle point to produce four heteroclinic connections between the saddlepoint and the four separation points on the side walls, as shown in Fig. 2d. As theaspect ratio is increased further, it is seen that there are two complete eddies withinthe cavity and between them a third is about to be created. Indeed, as A is increasedmore the heteroclinic connections from the four separation points separate from thesaddle point on the cavity center. There are now two heteroclinic connections crossingthe cavity, each connected to separation points on the side walls, and between thesetwo streamlines lies the saddle point on (x, y) = (0, 0) with its separatrix enclosingtwo sub-eddies. As A increases, the sub-eddy center lying left of the saddle on y = 0approach the saddle point on (x, y) = (0, 0) and coalesce, disappearing at a criticalvalue of A (Fig. 2e). This is a cusp (saddle-node) bifurcation. At this critical aspectratio the formation of a third eddy, between the other two, is complete so that threeeddies now occupy the cavity (Fig. 2f). This is a mechanism for eddy generation inwhich one eddy becomes three.Fig. 3: Typical flow structures. (a) S = 0, A = 1.4 (b) S = 0, A = 1.7(c) S = 1, A = 2.Figure 3 shows the some flow structures for S = 0 (where U2 = 0) and S = 1 atwhich the lids move in opposite directions with equal speed ratio respectively. Theseeddy genesis mechanisms are not new. They have been found before by several authors[6, 8, 9, 17, 10]. For this reason, the more detailed representation of the flow structuresis not necessary.H. Bilgil 463Table 1: Satisfaction of horizontal wall boundary conditions for U1 = 2, U2 = 1,A = 5.Table 1 shows values of the streamfunction from the present study compared withthose of Shankar and Joseph-Sturges on the cavity boundary. Here, averages are shownover 1000 equally spaced points in x ∈ [0, 1). All the tabulated results relate to a typicalcase in which U1 = 2, U2 = 1 and A = 5. Remarkably good aggreement is seen betweenthe present computations and those of Shankar and Joseph-Sturges [17, 12, 6].References[1] H. Bilgil and F. Gürcan, Effect of the Reynolds number on flow bifurcations andeddy genesis in a lid-driven sectorial cavity, Jpn. J. Ind. Appl. Math., 33(2016),343—360.[2] A. Deliceoğlu and S. H. Aydın, Topological flow structures in an L-shaped cavitywith horizontal motion of the upper lid, J. Comput. Appl. Math., 259(2014), partB, 937—943.[3] P. H. Gaskell, M. D. Savage, J. L. Summers and H. M. Thompson, Modeling andanalysis of meniscus roll coating, J. Fluid Mech., 298(1995), 113—137.[4] P. H. Gaskell, M. D. Savage., J. L. Summers and H. M. Thompson, Creeping flowanalyses of free surface cavity flows, J. Theor. Comput. Fluid Dynamics, 8(1996),415—433.[5] P. H. Gaskell, Gürcan, F., M. D. Savage. and H. M. Thompson, Stokes flow ina double-lid-driven cavity with free surface side-walls, Proc. Instn. Mech. Engrs,Part C: J. Mechanical Engineering Science, 212(1998), 387—403.[6] P. H. Gaskell, M. D. Savage, J. L. Summers and H. M. Thompso, Stokes flow inclosed, rectangular domains, Appl. Math. Model., 22(1998), 727—743,[7] F. Gürcan, Streamline topologies in Stokes flow within lid-driven cavities, Theor.Comput. Fluid Dyn., 17(2003), 19—30.464 An Application of Khuri’s Biorthogonality Condition[8] F. Gürcan, P. H. Gaskell, M. D. Savage and M. Wilson, Eddy genesis and trans-formation of Stokes flow in a double-lid-driven cavity, Proc. of The Instn. Mech.Eng. Part-C: J. Mec. Eng. Sci, 217(2003), 353—364.[9] F. Gürcan, M. Wilson and M. D. Savage, Eddy genesis and transformation ofStokes flow in a double-lid-driven cavity. Part 2: deep cavities, Proc. of The Instn.Mech. Eng. Part-C: J. Mec. Eng. Sci, 220(2006), 1765—1774.[10] F. Gürcan and H. Bilgil, Bifurcations and eddy genesis of Stokes flow within asectorial cavity, European Journal of Mechanics-B/Fluids, 39(2013), 42—51[11] C. J. Heaton, On the appearance of Moffatt eddies in viscous cavity flow as theaspect ratio varies, Physics of Fluid, 20(2008), 103102.[12] D. D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonicand stokes flow edge problems: Part II, Siam., J. Appl. Math., 34(1978), 7—27.[13] S. A. Khuri, Biorthogonal series solution of stokes flow problems in sectorial re-gions, Siam., J. Appl. Math., 56(1996), 19—39.[14] S. A. Khuri, Biorthogonality condition for stokes flow in rectangular regions, Int.J. Comput. Math., 70(1999), 411—415.[15] V. V. Meleshko, Steady Stokes flow in a rectangular cavity, Proc. R. Soc. LondonSer. A, 452, (1996), 1999—2022.[16] M. Scholle, A. Haas, N. Aksel, M. C. T. Wilson, H. M. Thompson and P. H.Gaskell, Competing geometric and inertial effects on local flow structure in thickgravity-driven fluid films, Physics of Fluids, 20(2008), 123101.[17] P. N. Shankar, Three-dimensional eddy structure in a cylindrical container, J.Fluid Mechanics, 342(1997), 97—118.[18] P. N. Shankar and M. D. Despande, Fluid mechanics in the driven cavity, Ann.Rev. Fluid Mech., 136(2000), 93—136..[19] R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser.A, 5(1952), 227—237.

An application of khuri’s biorthogonality condition on stokes flow in a rectangular cavity

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