Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

Transcription

1 THEORETICAL & APPLIED MECHANICS LETTERS, Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders M. Imran, a M. Kamran, M. Athar Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Received 03 November 00; accepted 6 January 0; published online 0 March 0 Abstract The velocity field the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace finite Hankel transforms. Initially the fluid is at rest, at time t =0 +, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations all imposed initial boundary conditions. The corresponding solutions for ordinary second grade Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material fractional parameters on the fluid motion a comparison between models, are underlined by graphical illustrations. c 0 The Chinese Society of Theoretical Applied Mechanics. doi:0.063/.00] Keywords second grade fluid, fractional derivative, longitudinal flow, velocity field, shear stress, Laplace finite Hankel transforms Navier-Stokes equations are nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions in which the nonlinear inertial terms do not disappear automatically. Exact solutions are very important not only because they are solutions of some fundamental flows but also because they serve as accuracy checks for experimental, numerical, asymptotic methods. The inadequacy of the classical Navier-Stokes theory to describe rheologically complex fluids such as polymer solutions, blood, paints, certain oils greases, has led to the development of several theories of non-newtonian fluids. Amongst the many models which have been used to describe the non- Newtonian behavior exhibited by certain fluids, the fluids of different types have received special attention. The fluids of second grade, which form a subclass of the fluids of different types, have been studied successfully in various types of flow situations. Here, we mention some of the studies. Sometimes since the equations governing the flow of second-grade fluids are one order higher than the Navier-Stokes equations, one would require boundary conditions in addition to the nonslip condition to have a well-posed problem. In recent years, fractional calculus has achieved much success in the description of complex dynamics. Fractional derivative models are used quite often to describe viscoelastic behavior of polymers in the glass transition the glassy state. The starting point is usually classical differential equation which is modified by replacing the classical, time derivatives of an integer order by the so-called Riemann-Liouville operator. This generalization allows one to define precisely non integer order integrals or derivatives.,3 Fractional calculus has also gained much fame in the description a Corresponding author. drmimranchaudhry@gmail.com. of viscoelasticity. 4,5 Tan Xu 6 discussed the flow of a generalized second grade fluid due to the impulsive motion of a flat plate. Relevant studies involving generalized second grade fluid in bounded domains are presented. 7 4 In this work, we consider the viscoelastic fluid to be modeled as fractional second grade fluid FSGF study the flow starting from rest due to the sliding of the inner cylinder along its axis with a constant acceleration. The velocity adequate shear stress, obtained by means of the finite Hankel Laplace transforms, are presented under series form in terms of the generalized G functions. The similar solutions for the ordinary second grade or Newtonian fluids, performing the same motion, are respectively obtained as special cases when β, or β α 0. The flows to be considered here have the form of 9,5 v = vr, t =vr, te z, where e z is the unit vector in the z-direction of the cylindrical coordinates system r, θ z. For such flows, the constraint of incompressibility is automatically satisfied. Furthermore, if initially the fluid is at rest, then vr, 0 = 0. The governing equations, corresponding to such motions for second grade fluid are 9,5 τr, t = vr, t t vr, t μ + α, 3 t r = ν + α t r + vr, t, 4 r r

2 000- M. Imran, M. Kamran M. Athar Theor. Appl. Mech. Lett., where μ is the dynamic viscosity, ν = μ/ρ is the kinematic viscosity, ρ being the constant density of the fluid, α is a material constant α = α /ρ. The governing equations corresponding to an incompressible fractional second grade fluid, performing the same motion, are vr, t t =ν + αd β t r + r vr, t, 5 r τr, t =μ + α D β vr, t t, 6 r where the fractional differential operator D β t is defined by Ref. 6 as D β d t ft = Γ β dt t 0 fτ t τ β dτ; 0 β<, 7 Γ is the Gamma function. Of course, the new material constant α, although for simplicity we keep the same notation, tends to the original α as β. In the following the system of fractional partial differential Eqs. 5 6, with appropriate initial boundary conditions, will be solved by means of Laplace finite Hankel transforms. In order to avoid lengthy calculations of residues contour integrals, the discrete inverse Laplace transform method will be used. Let us consider an incompressible FSGF at rest in the annular region between two straight circular cylinders of radii R R >R. At time t =0 +,theinner cylinder suddenly begins to slide along the common axis with constant accelerations A. Owing to the shear, the fluid is gradually moved, its velocity being of the form of Eq.. The governing equations are Eqs. 5 6, while the appropriate initial boundary conditions are vr, 0 = 0, r R,R ], 8 vr,t=at, vr,t=0 for t 0, 9 where A is a real constant. Applying the Laplace transform to Eqs. 5 9, we get qvr, q = ν + αq β r + vr, q, 0 r r vr,q= A q, vr,q=0, where r R,R, vr, q is the Laplace transform of function vr, t. In the following, let us denote 7 v H r n,q= R R rvr, qbr, r n dr,,, 3,... by the finite Hankel transform of the function vr, q Br, r n =J 0 rr n Y 0 R r n J 0 R r n Y 0 rr n, 3 where r n are the positive roots of the transcendental equation BR,r = 0, J p Y p are Bessel functions of the first second kind of order p. The inverse Hankel transform of w H r n,qisgiven by 7 vr, q = π r nj 0 R r n Br, r n J 0 R r n J 0 R r n v Hr n,q. Multiplying now both sides of Eq. 0 byrbr, r n, then integrating it with respect to r from R to R, taking into account Eq. along with the following relations d dr Br, r n= r n J rr n Y 0 R r n J 0 R r n Y rr n ], J 0 zy z J zy 0 z = πz, the result which we can easily prove R R r r + r r J 0R r n vr,q πj 0 R r n we find that vr, qbr, r n dr = r nv H r n,q, 4 v H r n,q= ν + αq β AJ 0 R r n πj 0 R r n q q + νrn + αrnq β. 5 It can also be written in the equivalent form of v H r n,q= AJ 0R r n πr nj 0 R r n AJ 0 R r n πr nj 0 R r n q + qq + νr n + αr nq β. 6 Applying the inverse Hankel transform to Eq. 6, using identity ln R /r ln R /R = π J 0 R r n J 0 R r n Br, r n J0 R r n J0 R, r n we get vr, q= lnr /r A lnr /R q + J0 R r n J 0 R r n Br, r n J 0 R r n J0 R r n qq + νrn + αrnq β. 7

3 000-3 Exact analytical solutions for axial flow of a fluid Theor. Appl. Mech. Lett., Using identity qq + νrn + αrnq β = νr n k q βkβ q β + αr n k+, Equation 7 can be further simplified to vr, q= lnr /r A lnr /R q + J 0 R r n J 0 R r n Br, r n J 0 R r n J0 R r n νrn k q βkβ q β + αrn k+. 8 In order to determine velocity field vr, t, applying the discrete inverse Laplace transform to Eq. 8 using A from Appendix, 8 we find that vr, t = lnr /r lnr /R At+ J 0 R r n J 0 R r n Br, r n J 0 R r n J0 R r n νr k n Gβ,βkβ,k+ αr n,t ]. 9 Applying the Laplace transform to Eq. 6, we get τr, q =μ + α q β vr, q. 0 r Using Eq. 8, it becomes τr, q = μ + α q β A rq lnr /R where r n J 0 R r n J 0 R r n B r, r n J0 R r n J0 R r n νr n k q βkβ q β + αr n k+ ], B r, r n =J rr n Y 0 R r n J 0 R r n Y rr n. Now, applying the inverse Laplace transform to Eq., we get the shear stress of the form of τr, t = A r ln μt+ α t β R /R Γ β r n J 0 R r n J 0 R r n B r, r n J0 R r n J0 R r n νr k n μgβ,βkβ,k+ αr n,t + α G β,βk,k+ αr n,t ]. 3 By introducing Eq. 9 intoeq.6 using the results of D β t G a, b, c d, t =G a,b+β,c d, t, 4 D β t t γ = Γγ +tγβ Γγ β +, 5 we can also find shear stress as τr, t = A r ln μt+ α t β R /R Γ β r n J 0 R r n J 0 R r n B r, r n J0 R r n J0 R r n νr k n μgβ,βkβ,k+ αr n,t + α G β,βk,k+ αr n,t ]. 6 Making β ineqs.9 3 we find the velocity field the shear stress v SG r, t = lnr /r lnr /R At+ J 0 R r n J 0 R r n Br, r n J 0 R r n J0 R r n νr k n G0,k,k+ αr n,t, 7 τ SG r, t = μa r ln R /R t + α μ r n J 0 R r n J 0 R r n B r, r n J0 R r n J0 R r n νr k n μg0,k,k+ αr n,t + α G 0,k,k+ αr n,t ], 8

4 000-4 M. Imran, M. Kamran M. Athar Theor. Appl. Mech. Lett., corresponding to an ordinary second grade fluid, performing the same motion. These solutions can also be simplified to see also Eqs. A A 3 from Appendix v SG r, t = lnr /r lnr /R At+ J 0 R r n J 0 R r n Br, r n ν r nj0 R r n J0 R r n ] ] exp νr nt +αrn, 9 μa τ L r, t = r ln R /R πρa t + α μ J 0 R r n J 0 R r n B r, r n r n J 0 R r n J 0 R r n ], 34 whicharethesameforbothtypesoffluids. τ SG r, t = μa r ln t + α R /R μ J 0 R r n J 0 R r n B r, r n πρa r n J0 R r n J0 R r n ] ] +αrn exp νr nt +αrn. 30 Now making α 0inEqs.9 30, we find the velocity field the shear stress v N r, t = lnr /r lnr /R At+ ν J 0 R r n J 0 R r n Br, r n r n J 0 R r n J 0 R r n ] e νr n t, 3 τ N r, t = μa r ln t R /R πρa J 0 R r n J 0 R r n B r, r n r n J 0 R r n J 0 R r n ] e νr n t, 3 corresponding to the Newtonian fluid, performing the same motion. 3 For large values of t, these solutions as well as those corresponding to second grade fluids approach to large time solutions v L r, t = lnr /R lnr /R At + J 0 R r n J 0 R r n Br, r n ν rn J0 R r n J0 R r n ], 33 Fig.. The time required to reach large time solutions for Newtonian, second grade fractional second grade fluids, for R =0.3, R =0.5, A =,α=0.003, ν=0.00 β =0.5.

5 000-5 Exact analytical solutions for axial flow of a fluid Theor. Appl. Mech. Lett., In conclusion, after some time, the behavior of the second grade fluid can be well enough approximated by that of a Newtonian fluid. Our interest here is to show that such a property also holds for fractional second grade fluid whose exact solutions, as well as those of the second grade fluid see Eqs. 7 8,arewritten in terms of generalized G a, b c., t functions. More exactly, we must show that large time solutions corresponding to fractional second grade fluid is also given by Eqs Indeed from Fig. we also have the time required to reach the large-time solutions for these fluids. This time 0s is the lowest for Newtonian fluid the highest 30s for fractional second grade fluid. In this paper, we find the velocity field the adequate tangential shear stress, corresponding to the flow of a fractional second grade fluid between two infinite circular cylinders in which the inner cylinder slides along the axis, are determined. At time t =0 +,theinner cylinder begins to move along the common axis with constant acceleration A. The solutions, determined by employing the Laplace finite Hankel transforms, presented in terms of Bessel functions J 0, Y 0, J Y generalized G functions, satisfy the corresponding governing equations as well as all imposed initial boundary conditions. In the special cases, when β orβ α 0, the corresponding solutions for a second grade fluid or for the Newtonian fluid, performing the same motion, are respectively obtained from the general solutions. Fig. 3. Profiles of the velocity field vr, t given by Eq. 9 for R =0.3, R =0.5, t=5s, A =,α=0.00, β=0.8 different values of ν Fig. 4. Profiles of the velocity field vr, t given by Eq. 9 for R =0.3, R =0.5, t=5s, A =,ν=0.005, β=0. different values of α motion. Its effect on the fluid motion is qualitatively thesameasthatofα. More exactly, the velocity vr, t is a decreasing function with regards to β. Finally, for comparison, the profiles of vr, t corresponding to the motion of the Newtonian, ordinary second grade fractional second grade are depicted together in Fig. 6 for the same t the same common Fig.. Profiles of the velocity field vr, t given by Eq. 9 for R =0.3, R =0.5, A=,α=0.0003, ν=0.003,β = 0. different values of t Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity vr, t are depicted against r for different values of time t, for different material fractional parameters. Figure clearly shows that the velocity is an increasing function of t. The influence of the kinematic viscosity ν on the fluid motion is shown in Fig. 3. The velocity vr, t is a decreasing function of ν. The influences of material parameter α on the velocity is shown in Fig. 4. As expected, the velocity is a decreasing function with respect to α. Figure 5 shows the influence of the fractional parameter β on the fluid Fig. 5. Profiles of the velocity field vr, t given by Eq. 9 for R =0.3, R =0.5, t=6s, A =,ν=0.003, α=0.0 different values of β

6 000-6 M. Imran, M. Kamran M. Athar Theor. Appl. Mech. Lett., Fig. 6. Profiles of the velocity field vr, t corresponding to the Newtonian, Second grade fractional second grade fluids, for R =0.3, R =0.5, t=3s,a=,α=0., ν= β =0. material fractional parameters. The fractional second grade fluid, as resulting from these figures, is the slowest the Newtonian fluid is the swiftest on the whole flow domain. In practice, it is necessary to know the approximate time after which the fluid is moving according to the large time solutions. This time, as resulting from Fig., is the smallest for the Newtonian fluid the highest for fractional second grade fluid. The units of the material constants are SI units in all figures, the roots r n have been approximated by nπ/r R. Appendix = L { q b } q a d c = G a,b,c d, t d j Γc + j t c+jab ΓcΓj + Γc + ja b], A j=0 where Reac b > 0, d/q a <, νrn k G 0, k,k+ αrn,t k=o = +αrn exp νr nt +αrn, A νrn k G 0, k,k+ αrn,t = νr n k=o exp νr nt +αrn ]. A 3. C. Truesdell W. Noll, The Non-linear Field Theories of Mechanics. In Hbuch der Physik. III/3 Springer Berlin, A. S. Gupta, T. Y. Na, Int. J. Non-Lin Mech. 8, K. R. Rajagopal, J. Non-Newtonian Fluid Mech. 5, A. M. Benharbit, A. M. Siddiqui, Acta Mech. 94, A. M. Siddiqui, T. Hayat, S. Asghar, Int. J. Non-Lin Mech. 34, T. Hayat, S. Asghar, A. M. Siddiqui, Acta Mech. 3, C. Fetecau, Corina Fetacau, J. Zierep, Int. J. Non-Lin Mech. 37, T. Hayat, K. Masood, A. M. Sidiqqui, S. Asghar, Int. J. Non-Lin Mech. 39, C. Fetecau, Corina Fetecau, Int. J. Eng. Sci. 44, T. Hayat, M. Khan, M. Ayub, Math. Comp. Modelling, 43, C. Fetecau, T.Hayat, M. Khan, Corina Fetecau, Acta Mech. 98, K. S. Miller, B. Ross, An Introduction to the Fractional Calculus Fractional Differential Equations Wiley, S. G. Samko, A. A. Kilbas, I. MarichevO, Fractional Integrals Derivatives: Theory Applications Gordon Breach New York, L. I. Palade, P. Attane, R. R. Huilgol, B. Mena, Int. J. Engng. Sci. 37, Y. A. Rossikhin, M. V. Shitikova, Int. J. Engng. Sci. 39, W. C. Tan, M. Y. Xu, Mech. Res. Comm. 9, W. C. Tan, X. Feng, W. Lan, Chinese Sci. Bull. 47, in Chinese 8. W. C. Tan, P. Wenxiao, X. Mingyu, Int. J. Non-Linear Mech. 38, M. Nazar, Corina Fetecau, A. U. Awan, Nonlinear Analysis: Real World Applications,, M. Athar, M. Kamran, C. Fetecau, Nonlinear Analysis: Modelling Control 5, A. Mahmood, C. Fetecau, N. A. Khan, M. Jamil, Acta Mech. Sin. 6, M. Athar, M. Kamran, M. Imran, Nonlinear Analysis: Modeling Control 5, C. Fetecau, A. U. Awan, M. Athar, Nonlinear Analysis: Modelling Control 5, M.Athar, M. Kamran, M. Imran, Meccanica DOI 0.007/s R. Belli, K. R. Rajagopal, Int. J. Non-Linear Mech. 30, I. Podlubny, Fractional Differential Equations Academic Press San Diego, I. N. Sneddon, Functional Analysis in: Encyclopedia of Physics, vol. II Springer Berlin, C. F. Lorenzo, T. T. Hartley, Generalized functions for the fractional calculus NASA/TP /REV.