Exact solutions in generalized Oldroyd-B fluid

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1 Appl. Math. Mech. -Engl. Ed., 33(4, (01 DOI /s c Shanghai University and Springer-Verlag Berlin Heidelberg 01 Applied Mathematics and Mechanics (English Edition Exact solutions in generalized Oldroyd-B fluid T. HAYAT 1, S. ZAIB 1, S. ASGHAR, A.A.HENDI 3 (1. Department of Mathematics, Quaid-i-Azam University 4530, Islamabad 44000, Pakistan;. Department of Mathematics, Comsats Institute of Information Technology, Islamabad 44000, Pakistan; 3. Department of Physics, Faculty of Science, King Saud University, P. O. Box 1846, Riyadh 1131, Saudi Arabia Abstract This investigation deals with the influence of slip condition on the magnetohydrodynamic (MHD and rotating flow of a generalized Oldroyd-B (G. Oldroyd-B fluid occupying a porous space. Fractional calculus approach is used in the mathematical modeling. Three illustrative examples induced by plate oscillations and periodic pressure gradient are considered, and the exact solutions in each case are derived. Comparison is provided between the results of slip and no-slip conditions. The influence of slip is highlighted on the velocity profile by displaying graphs. Key words slip condition, exact solution, fractional calculus Chinese Library Classification O Mathematics Subject Classification 76W99 1 Introduction Investigation of flow dynamics of non-newtonian fluids has shown an increasing amount of attention during the last several decades. This is in view of the relevance of non-newtonian flows to a number of industrial applications. For instance, processed food, pharmaceutical, foams, and chemical and biochemical fluid media generally show non-newtonian characteristics. Such fluids have a non-linear shear stress-strain rate behavior. The constitutive relationships of non-newtonian fluids give rise to equations which in general are more non-linear and higher order than the Navier-Stokes equations. Hence, for a unique solution, the researchers in the field require an extra boundary condition(s. An excellent survey regarding this issue has been made in Refs. [1 3]. A large amount of research has been devoted in the past to the flows of differential and rate type fluids in the non-rotating frame. Some recent investigations dealing with such flows have been mentioned in Refs. [4 15]. There also have been very recent studies [16 17] to discuss the steady and unsteady rotating flows of non-newtonian fluids. The review article [18] may be very useful in this direction. In Ref. [19], the authors have seen the influence of slip condition on steady rotating flows of a third grade fluid. Received May 16, 011 / Revised Jan. 4, 01 Corresponding author T. HAYAT, Professor, Ph. D., pensy t@yahoo.com

2 41 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI To the best of our knowledge, the slip effects on the flows of a differential type fluids in a rotating frame have received very little attention. However, the slip effects on flows of rate type fluids are even not discussed so far. The object of the present paper is to examine the slip effects on the rotating flows of an Oldroyd-B fluid in a porous space. The fluid is permeated by a transverse magnetic field. The governing equations include the modified Darcy s law for an Oldroyd-B fluid. In fact, this analysis aims to assess and quantify the extent of slip parameter on the three oscillatory flows. Fractional calculus approach [0 30] is adopted in the description of mathematical analysis. Results are presented for the velocity profiles. It is shown that the results of viscous, second grade, and Maxwell fluids can be obtained as the limiting cases of the present closed form solutions. Basic equations The fundamental equations which can govern an incompressible flow in a rotating frame are ( dv ρ dt +Ω V +Ω (Ω r = ρ +divs + J B + R, (1 div V =0, ( where V =(u, v, w is the velocity field, ρ is the fluid density, p is the pressure, Ω is the angular velocity in a rotating system, r = x + y, and an extra stress S in a generalized Oldroyd-B (G. Oldroyd-B fluid has the form [4] (1+λ α Dα Dt α S = μ (1+θ β Dβ Dt β A 1. (3 Here, μ is the dynamic viscosity, λ and θ are the relaxation and retardation time, α and β are fractional calculus parameters such that 0 α β 1andα β. The first Rivlin-Erickson tensor A l is given by A 1 = V +( V T, (4 where is the gradient operator, T indicates the matrix transpose and D α S Dt α = D α S Dt α +(V S ( V S S( V T, (5 where the fractional derivative of order α with respect to t is defined by the following expression: D α f(t Dt α = 1 d Γ(1 α dt t 0 (t ε α f(tdε, 0 <α<1, (6 in which Γ( is the Gamma function. For α = β = 1, (3 reduces to an Oldroyd-B fluid. When λ =0andμθ = α 1 (material parameter of fluid, we get a generalized second grade (G. second grade fluid, and for θ = 0, we have a generalized Maxwell (G. Maxwell fluid. For λ = θ =0 and α = β = 1, we deduce the Navier-Stokes fluid. The Maxwell equations can be written as div B =0, curl B = μ m J, curl E = B t, (7 in which J is the current density, B is the total magnetic field so that B = B 0 + b, B 0 and b are the applied and induced magnetic fields, respectively, μ m is the magnetic permeability, and E is the electric field.

3 Exact solutions in generalized Oldroyd-B fluid 413 In the present analysis, it is assumed that the magnetic Reynolds number is small and hence an induced magnetic field is negligible. The magnetic field B 0 is applied in z-direction and E = 0. In view of these assumptions, the Lorentz force becomes Expression for Darcy s resistance R in a G. Oldroyd-B fluid is [4] J B = σb 0V. (8 (1+λ α Dα Dt α R = μφ (1+θ β Dβ K Dt β V, (9 where φ and K are the porosity and permeability of the medium, respectively. 3 Governing equation Here, we consider the MHD rotating flow of an incompressible G. Oldroyd-B fluid over a non-conducting rigid plate at z = 0. The fluid and the plate are rotating with a uniform angular velocity parallel to the z-axis (taken perpendicular to the plate. For an infinite plate, all the physical quantities except the pressure depend upon z and t. The extra stress tensor S and velocity profile V for the present analysis are S(z, t = S xx S xy S xz S yx S yy S yz, (10 S zx S zy S zz V (z, t =(u(z, t, v(z, t, 0. (11 By (11, the continuity equation is identically satisfied, and (1 and (3 yield (1+λ α α t α S xz = μ (1+θ β β u t β z, (1 (1+λ α α t α S yz = μ (1+θ β β v t β z, (13 ( (1+λ α α u t α t Ωv = 1 (1+λ α α p ( ρ t α x + ν 1+θ β β u t β z σb 0 (1+λ α α ρ t α u νφ k (1+θ β β t β u, (14 ( (1+λ α α v t α t +Ωu = 1 (1+λ α α p ( ρ t α y + ν 1+θ β β v t β z σb 0 (1+λ α α ρ t α v νφ k (1+θ β β t β v, (15 p z =0,

4 414 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI where ν is the kinematic viscosity, the modified pressure p = p (ρ/ω r,andtheabove equation indicates that p p(z. Write F = u +iv, (16 equations (14 and (15 can be combined into the following differential equation: ( (1+λ α α F t α t +iωf = 1 ( (1+λ α α p +i p ρ t α + ν (1+θ β β F x y t β z σb 0 (1+λ α α ρ t α F νφ k In the next section, three flow examples will be considered. 4 Flow induced by general periodic oscillation (1+θ β β t β F. (17 In this section, flow is engendered by general periodic oscillation of the plate with a slip condition. The slip condition is defined in terms of the shear stress, which in mathematical form can be expressed as u(0,t γ μ S xz = 0 Here, γ is a slip parameter and The Fourier series coefficients in (18 are a k e ikω0t, v(0, t γ μ S yz =0. (18 u, v 0 as z. (19 a k = 1 T 0 T 0 f(te ikω0t dt. (0 In terms of F, the boundary conditions reduce to (1+λ α α t α F (z, t γ (1+θ β β F t β z = 0 a k ( 1+λ α α t α e ikω0t, z =0, (1 F (, t=0. Considering (17 in the absence of modified pressure gradient and then using z = z 0 ν, F = F, t = t 0 0 ν, ω 0 = ω 0ν, 0 λ 1 = λ 0 ν, θ = θ 0 ν, Ω = Ων, 0 ( (3 M = σb 0 ν ρ, 0 1 K = φν k 0, γ = 0γ ν,

5 Exact solutions in generalized Oldroyd-B fluid 415 we have and ( (1+λ α α F t α t +iωf = (1+θ β β F t β z M ( 1+λ α α t α F z =0, 1 K (1+λ α α t α F (z, t γ (1+θ β β F t β z = F(,t=0, (1+θ β β t β F (4 a k ( 1+λ α α t α e ikω0t, (5 where the asteriks have been suppressed. By Fourier transform method, one can write ψ(z, ω = u(z, te iωt dt, (6 u(z, t = 1 ψ(z, ωe iωt dω, (7 π D β Dt β [u(z, t] e iωt dt =(iω β ψ(z, ω, (8 (iω β = ω β e iβπ sign ω = ω β ( cos βπ The solution to the problem consisting of (4 and (5 is where F (z,t = βπ +isignωsin. (9 a k (1 + λ α (ikω 0 α e m kz+i(kω 0t n k z (1 + λ α (ikω 0 α +γ(m k +in k (1+θ β (ikω 0 β, (30 L L r + L i m k = + L r r + L i, n k = L r, (31 L r = 1 ( K + M ( 1+λ α kω 0 α cos απ ( 1+θ β kω 0 β cos βπ + λ α θ β kω 0 α+β (sign(kω 0 sin απ βπ sin (kω 0 +Ωsign(kω 0 ( λ α kω 0 α sin απ ( 1+θ β kω 0 β cos βπ θ β kω 0 β sin βπ ( 1+λ α kω 0 α cos απ /(( 1+θ β kω 0 β cos βπ ( + θ β kω 0 β sign(kω 0 sin βπ, (3

6 416 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI ( (( L i = (kω 0 +Ω 1+λ α kω 0 α cos απ ( 1+θ β kω 0 β cos βπ + λ α θ β kω 0 α+β (sign(kω 0 sin απ ( θ β kω 0 β sin βπ ( 1+λ α kω 0 α cos απ λ α kω 0 α sin απ ( 1+θ β kω 0 β cos βπ /(( 1+θ β kω 0 β cos βπ ( + βπ sin M sign(kω 0 θ β kω 0 β sign(kω 0 sin βπ. (33 It is interesting to note that (30 is a result that corresponds to the general periodic oscillation of a plate. As a special case of this oscillation, the flow fields are obtained by an appropriate choice of the Fourier coefficients a k which give rise to different plate oscillations. For example, the flow fields F j (j =1,,, 5 due to five oscillations { 1, t <T1, exp(iω 0 t, cos(ω 0 t, sin(ω 0 t, δ(t kt 0, 0, T 1 < t <T 0 /, are, respectively, deduced by the following expressions: F 1 (z,t = (1 + λ α (iω 0 α e m1z+i(ω0t n1z (1 + λ α (iω 0 α +γ(m 1 +in 1 (1+θ β (iω 0 β, (34 F (z,t= 1 ((1 + λα (iω 0 α e m1z+i(ω0t n1z /((1 + λ α (iω 0 α +γ(m 1 +in 1 (1 + θ β (iω 0 β +(1+λ α ( iω 0 α e m 1z i(ω0t+n 1z /((1 + λ α ( iω 0 α + γ(m 1 +in 1 (1 + θ β ( iω 0 β, (35 F 3 (z,t = 1 i ((1 + λα (iω 0 α e m1z+i(ω0t n1z /((1 + λ α (iω 0 α +γ(m 1 +in 1 (1 + θ β (iω 0 β (1 + λ α ( iω 0 α e m 1z i(ω0t+n 1z /((1 + λ α ( iω 0 α F 4 (z,t = F 5 (z,t = 1 T 0 + γ(m 1 +in 1 (1 + θ β ( iω 0 β, (36 sin(kω 0 T 1 ((1 + λ α (ikω 0 α e m kz+i(kω 0t n k z kπ /((1 + λ α (ikω 0 α +γ(m k +in k (1 + θ β (ikω 0 β, k 0, (37 (1 + λ α (ikω 0 α e m kz+i(kω 0t n k z (1 + λ α (ikω 0 α +γ(m k +in k (1 + θ β (ikω 0 β. (38

7 5 Periodic flow between two plates Exact solutions in generalized Oldroyd-B fluid 417 This section deals with the flow between the two rigid plates with distant d apart. The resulting non-dimensional mathematical problem is defined by (4, (5, and (1+λ α α t α F (z,t+γ (1+θ β β F t β z =0, z =1, (39 where z = z d, F = F 0, t = t ( d ν, ω0 = ω 0 ( ν d, λ = λ ( d ν, θ = θ ( d ν, Ω = Ω ( ν d, M = σb 0 ( μ d, 1 K = φ ( k d, γ = γ d. (40 The solution is written in the following form: F (z,t= a k ((1 + λ α (ikω 0 α ((1 + λ α (ikω 0 α sinh β k (1 z +γ(1 + θ β (ikω 0 β β k cosh β k (1 z /(γ(1 + θ β (ikω 0 β (1+λ α (ikω 0 α β k cosh β k +sinhβ k (γ (1 + θ β (ikω 0 β β k+(1 + λ α (ikω 0 α e ikω0t, (41 β k = i(kω 0 +Ω(1+λ α (ikω 0 α + 1 K (1 + θβ (ikω 0 β +M (1 + λ α (ikω 0 α 1+θ β (ikω 0 β. (4 The flow fields induced by the five specific oscillations (in the previous section are F 1 (z,t =(1+λ α (iω 0 α ((1 + λ α (iω 0 α sinh β 1 (1 z +γ(1 + θ β (iω 0 β β 1 cosh β 1 (1 ze iω0t /(γ(1 + θ β (iω 0 β (1 + λ α (iω 0 α β 1 cosh β 1 +sinhβ 1 (γ (1 + θ β (iω 0 β β 1 +(1 + λ α (iω 0 α, (43

8 418 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI F (z,t = 1 ((1 + λα (iω 0 α ((1 + λ α (iω 0 α sinh β 1 (1 z +γ(1 + θ β (iω 0 β β 1 cosh β 1 (1 ze iω0t /(γ(1 + θ β (iω 0 β (1 + λ α (iω 0 α β 1 cosh β 1 +sinhβ 1 (γ (1 + θ β (iω 0 β β1+(1 + λ α (iω 0 α +(1+λ α ( iω 0 α ((1 + λ α ( iω 0 α sinh β 1 (1 z +γ(1 + θ β ( iω 0 β β 1 cosh β 1 (1 ze iω0t /(γ(1 + θ β ( iω 0 β (1 + λ α ( iω 0 α β 1 cosh β 1 +sinhβ 1 (γ (1 + θ β ( iω 0 β β 1 +(1 + λα ( iω 0 α, (44 F 3 (z,t = 1 i ((1 + λα (iω 0 α ((1 + λ α (iω 0 α sinh β 1 (1 z +γ(1 + θ β (iω 0 β β 1 cosh β 1 (1 ze iω0t /(γ(1 + θ β (iω 0 β (1 + λ α (iω 0 α β 1 cosh β 1 +sinhβ 1 (γ (1 + θ β (iω 0 β β1+(1 + λ α (iω 0 α (1 + λ α ( iω 0 α ((1 + λ α ( iω 0 α sinh β 1 (1 z +γ(1 + θ β ( iω 0 β β 1 cosh β 1 (1 ze iω0t F 4 (z,t = /(γ(1 + θ β ( iω 0 β (1 + λ α ( iω 0 α β 1 cosh β 1 +sinhβ 1 (γ (1 + θ β ( iω 0 β β 1 +(1 + λα ( iω 0 α, (45 sin(kω 0 T 1 ((1 + λ α (ikω 0 α ((1 + λ α (ikω 0 α sinh β k (1 z kπ +γ(1 + θ β (ikω 0 β β k cosh β k (1 ze ikω0t /(γ(1 + θ β (ikω 0 β (1 + λ α (ikω 0 α β k cosh β k F 5 (z,t = 1 T 0 +sinhβ k (γ (1 + θ β (ikω 0 β βk +(1 + λ α (ikω 0 α, k 0, (46 ((1 + λ α (ikω 0 α ((1 + λ α (ikω 0 α sinh β k (1 z +γ(1 + θ β (ikω 0 β β k cosh β k (1 ze ikω0t /(γ(1 + θ β (ikω 0 β (1 + λ α (ikω 0 α β k cosh β k +sinhβ k (γ (1 + θ β (ikω 0 β β k+(1 + λ α (ikω 0 α. (47

9 Exact solutions in generalized Oldroyd-B fluid Poiseuille flow Here, the flow is considered between two infinite stationary plates separated by h. The flow is induced due to imposition of the following periodic pressure gradient: p +i p x y = ρq 0e iω0t. (48 The problems governing the flow in this section are ( (1+λ α α F t α t +iωf = (1+λ α α t α Q 0 e iω0t + (1+θ β β F t β z where M ( 1+λ α α t α F 1 (1+θ β β K t β F, (49 (1+λ α α t α F (z, t+γ (1+θ β β F t β =0 z at z =1, (50 (1+λ α α t α F (z, t γ (1+θ β β F t β =0 z at z = 1, (51 z = z (υ/(h/t, F = F (h/t, t = ω 0 = ω 0 ((h/t /υ, λ = 1 K = φ (k(h/t /υ, Ω = t (ν/(h/t, λ (ν/(h/t, θ = γ γ = (υ/(h/t, Q 0 = Q 0 ((h/t 3 /υ, Ω ((h/t /υ, M = θ (ν/(h/t, σb 0 (ρ (h/t /μ, and the solution of the problem here is ( Q 1+λ α (iω 0 α 0 1+θ F (z,t = β (iω 0 β (β 1 ((1 + λ α (iω 0 α (cosh(β 1 z cosh(β 1 with γ(1 + θ β (iω 0 β (β 1 sinh(β 1 e iω0t 7 Graphical results and discussion (5 /((1 + λ α (iω 0 α cosh(β 1 +γ(1 + θ β (iω 0 β (β 1 sinh(β 1 (53 β 1 = β k k=1. (54 The aim of this section is to present the salient features of the three considered flows. In particular, the emphasis here is given to examine the difference in the results of no-slip and slip conditions. For this purpose, Figs. 1 3 have been displayed. The values for the flow

10 40 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI Fig. 1 Variation of γ on velocity parts for general periodic oscillations when M = t =0.5, ω 0 =0.1, and Ω = 0.3

11 Exact solutions in generalized Oldroyd-B fluid 41 Fig. Variation of γ on velocity parts for periodic flow between two plates when M = t =0.5, ω 0 =0.1, and Ω = 0.3

12 4 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI Fig. 3 Variation of γ on velocity parts for Poiseuille flow when M = t =0.5, ω 0 =0.1, Ω = 0.3, and Q 0 = 1

13 Exact solutions in generalized Oldroyd-B fluid 43 cases of five fluid models are tabulated. These tables also provide the comparison between the velocity profiles in five fluid models namely Newtonian, G. second grade, G. Maxwell, Oldroyd- B, and G. Oldroyd-B. Furthermore, the left panel in all figures shows the variation of u, whereas the right panel gives the variation of v. Note that in the case of general periodic oscillation, the flow due to cos ω 0 t is sketched only. The variations of u and v for the general periodic oscillation are shown in Fig. 1. It is noticed that u decreases in all the five types of fluids when the slip parameter γ increases. However, the magnitude of v increases with an increase in γ. Figure describes the influence of γ for the flow due to cos ω 0 t between two parallel plates. Here, the velocity u first decreases and then increases. Moreover, the magnitude of v decreases by an increase in γ except in the G. second grade fluid model. The influence of γ on the Poiseuille flow is depicted in Fig. 3. This figure shows that u increases in all types of fluids. The velocity v decreases in all fluid models except in the G. Oldroyd-B fluid model. Tables 1 3 are made to provide the comparison of velocity components in three considered problems for the five types of fluids in the presence of a slip condition. In Tables 1 and, the values have been computed for the oscillation cos ω 0 t. Table 1 indicates that the magnitudes of u and v are largest for a G. Maxwell fluid. Such magnitudes are smallest in the G. second grade fluid. For Table, it is found that the magnitude of u and v are largest and smallest in the G. Maxwell and the G. second grade fluids, respectively. Table 3 provides the comparison of velocity components in Poiseuille flow. We find here that the magnitudes of u and v in a G. Maxwell fluid are the largest. However, u is minimum in the G. second grade fluid and the magnitude of v is minimum in the G. Oldroyd-B fluid. Tables 4 6 are developed for the three oscillatory flows with no-slip condition. Here, Table 4showsthatu is maximum in an Oldroyd-B fluid and minimum for the G. second grade fluid. The magnitude of v is maximum in the G. second grade fluid and minimum in Newtonian fluid. Table 5 depicts that u is maximum in an Oldroyd-B and minimum in the G. second grade fluid. However, the magnitude of v is maximum in the G. Maxwell fluid and minimum in the G. second grade fluid. In case of Poiseuille flow (see Table 6, u is maximum in a G. Maxwell fluid and minimum in the G. second grade fluid. Furthermore, the magnitude of v is maximum in the G. Maxwell fluid and minimum in the Oldroyd-B fluid. Table 1 General periodic oscillations with slip condition (ω 0 =0.1, Ω = 0.3, M = t =0.5, z =0.5, and γ =0.5 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β = Table Periodic flow between two plates with slip condition (ω 0 =0.1, Ω = 0.3, M = t =0.5, z =0.5, and γ =0.5 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β =

14 44 T. HAYAT, S. ZAIB, S. ASGHAR, and A. A. HENDI Table 3 Poiseuille flow with slip condition (Q 0 = 1, ω 0 =0.1, Ω = 0.3, M = t =0.5, z =0.5, and γ =0.5 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β = Table 4 General periodic oscillations with no-slip condition (ω 0 = 0.1, Ω = 0.3, M = t = 0.5, z =0.5, and γ =0 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β = Table 5 Periodic flow between two plates with no-slip condition (ω 0 =0.1, Ω = 0.3, M = t =0.5, z =0.5, and γ =0 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β = Table 6 Poiseuille flow with no-slip condition (Q 0 = 1, ω 0 =0.1, Ω = 0.3, M = t =0.5, z =0.5, and γ =0 Type of fluid Rheological parameters u v Newtonian λ =0,θ = G. second grade λ =0,θ =1(α = β = G. Maxwell λ =,θ =0(α = β = Oldroyd-B λ =,θ =1(α = β = G. Oldroyd-B λ =,θ =1(α = β = Concluding remarks Three oscillatory flows have been analyzed for the closed form solutions. The slip condition in terms of shear stress in a G. Oldroyd-B fluid is introduced. This condition even is not introduced yet in the classical rate type fluids (Maxwell, Oldroyd-B, and Burgers. The established results explicitly show the contribution that arises because of slip parameter. The velocity components in the five fluid models are compared. The significant observations of the present analysis are described as follows: (i The x-component of velocity in the presence of slip condition is less than the no-slip condition. (ii The magnitude of y-component of velocity in the case of slip situation is greater when compared with that of no-slip case. (iii In Poiseuille flow, the magnitudes of u and v in slip case are greater in comparison to no-slip condition.

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