Unsteady Stagnation-point Flow of a Second-grade Fluid

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1 Avestia Publishing Journal of Fluid Flow, Heat Mass Transfer (JFFHMT) Volume 3, Year 016 ISSN: DOI: /jffhmt Unsteady Stagnation-point Flow of a Second-grade Fluid Fotini Labropulu 1, Daiming Li 1 Luther College/University of Regina Regina, SK, Canada S4S 0A fotini.labropulu@uregina.ca Department of Chemical Petroleum Engineering, University of Calgary 500 University Dr. NW, Calgary, Alberta, Canada TN 1N4 Abstract - The unsteady two-dimensional stagnation point flow of second-grade fluid impinging on an infinite plate is examined solutions are obtained. It is assumed that the infinite plate at y = 0 is oscillating with velocity U cos Ωt, the fluid occupies the entire upper half plane y > 0 it impinges obliquely on the plate. The governing partial differential equations are reduced to a system of ordinary differential equations by assuming a form of the streamfunction a priori. The resulting equations are, then, solved numerically using a shooting method for various values of the Weissenberg number, We. It is observed that the effect of the Weissenberg number is to decrease the velocity near the wall as it increases. Furthermore, analytical solutions are obtained for small large values of frequency. Gree Symbols α 1, α β = Ω γ ε = U η μ ν τ 1 ν τ = Ωt ψ Ω viscoelastic parameters of the fluid frequency constant constant non-dimensional varable fluid viscosity inematic viscosity Shear stress component non-dimensional variable streamfunction frequency Keywords: Unsteady, Stagnation-point, Oscillating plate, Non-Newtonian fluid. Copyright 016 Authors - This is an Open Access article published under the Creative Commons Attribution License terms ( Unrestricted use, distribution, reproduction in any medium are permitted, provided the original wor is properly cited. Nomenclature A 1, A F(y), G(y, t) p T V u, v U We Date Received: Date Accepted: Date Published: 1 st nd Rivlin Ericsen tensors similarity variables constant fluid pressure Cauchy Stress Tensor velocity vector velocity components along x y axis constant Weissenberg number Introduction In the past, the fluid flow near a stagnation point has been investigated extensively. Hiemenz [1] derived an exact solution of the steady flow of a Newtonian fluid impinging orthogonally on an infinite flat plate. Stuart [], Tamada [3] Dorrepaal [4] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate. Beard Walters [5] used boundary-layer equations to study two-dimensional flow near a stagnation point of a viscoelastic fluid. Dorrepaal et al [6] investigated the behaviour of a viscoelastic fluid impinging on a flat rigid wall at an arbitrary angle of incidence. Labropulu et al [7] studied the oblique flow of a viscoelastic fluid impinging on a porous wall with suction or blowing. Unsteady stagnation point flow of a Newtonian fluid has also been studied extensively. Rott [8] Glauert [9] studied the stagnation point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. Srivastava [10]

2 investigated the same problem for a non-newtonian second grade fluid using the Karman-Pohlhausen method [11] to solve the resulting equations. Labropulu et al [1] used series methods to solve the unsteady stagnation point flow of a Walters' B' fluid impinging on an oscillating flat plate. Matunobu [13, 14] Kawaguti Hamano [15] examined the fundamental character of the unsteady flow near a stagnation point for a Newtonian fluid. Taemitsu Matunobu [16] studied the oblique stagnation point flow for a Newtonian fluid obtained the general features of a periodic stagnation point flow. The case when the stagnation point fluctuates along a solid boundary is especially interesting from the biomechanical point of view. This is because the wall shear stress experienced by blood vessels may be thought to be increased by pulsating blood flow near the mean position of fluctuating stagnation point [15, 17] lead to vascular diseases [18]. In this wor, the unsteady stagnation point flow of a viscoelastic second-grade fluid is examined solutions are obtained. We assume that the infinite plate at y = 0 is oscillating with velocity U cos Ωt, the fluid occupies the entire upper half plane y > 0 the fluid impinges obliquely on the plate. The governing partial differential equations are reduced to a system of ordinary differential equations by assuming a form of the streamfunction a priori. The resulting equations are, then, solved numerically using a shooting method for various values of the Weissenberg number, We. It is observed that the effect of the Weissenberg number is to decrease the velocity near the wall as it increases. Furthermore, analytical solutions are obtained for small large values of frequency.. Flow Equations The flow of a viscous incompressible non- Newtonian second-grade fluid, neglecting thermal effects body forces, is governed by div V = 0 (1) ρv = div T () when the constitutive equation for the Cauchy stress tensor T which describes second-grade fluid given by Rivlin Ericsen [19] is T = pi + μa 1 + α 1 A + α A 1 A 1 = (grad V) + (grad V) T A = A 1 + (grad V) T A 1 + A 1 (grad V)} (3) Here V is the velocity vector field, p the fluid pressure function, ρ the constant fluid density, μ the constant coefficient of viscosity α₁, α₂ the normal stress moduli. Dunn Fosdic [0] Dunn Rajagopal [1] have shown that if the second-grade fluid described by (3) is to undergo motions which are compatible with Clausius-Duhem inequality [] the assumption that the free energy density of the fluid be locally at rest, then the material constants must satisfy the following restrictions: μ 0, α 1 0, α 1 + a = 0 (4) Considering the flow to be plane, we tae V = (u(x, y, t), v(x, y, t)) p = p(x, y, t) so that the flow equations (1) to (3) tae the form u x + v y = 0 (5) u u u 1 p 1 u v u u t x y x t u u u v v u u v 4 x x x y x x x y v u u u u v y x y x y x y v v u v u x y x x x y v v v 1 p u v v v t x y y t 1 v u u u v v u v x x y x y x y x y v v u v u u 4 y x y y y x y v v v u v y y y x y (6) (7) 18

3 where is the inematic viscosity. Continuity equation (5) implies the existence of a streamfunction ( x, y, t) such that where γ is a non-dimensional constant characterizing the obliqueness of oncoming flow. It is assumed that only the real part of a complex quantity has its physical meaning. u, v y x Substitution of (8) in equations (6) (7) elimination of pressure from the resulting equations using pxy pyx yields t, x, y 1 1 t x y ,, (8) (9) Substitution of equation (11) in (9) yields ( iv ) 1 v ( iv f f f f f f f f f ) 0 (13) ν 4 g y 4 3 g t y + α 1 5 g ρ t y 4 + (f 3 g g f y3 y ) α 1 ρ (f 5 g g f(iv) y5 y ) = 0 (14) Having obtained a solution of equation (9), the velocity components are given by (8) the pressure can be found by integrating equations (6) (7). The shear stress component 1 of the Cauchy stress T is given by y x y x y x x y x y x y y x x y (10) 3. Solutions in the Fixed Frame of Reference Following Taemitsu Matunobu [16], we assume that x f ( y) g( y, t) (11) We assume that the infinite plate at y = 0 is oscillating with velocity Ucos Ωt that the fluid occupies the entire upper half plane y > 0. Furthermore, we assume the streamfunction far from the wall is given by ψ = 1 γy + xy (see Stewart []). Thus, the boundary conditions are given by U it f (0) f (0) 0, g(0, t) 0, g y (0, t) e (1a) f ( ) 1, g (, t) y (1b) y Integrating equations (13) (14) once with respect to y using the conditions at infinity, we have 1 iv f f f f f f f f f ν 3 g y 3 g t y + α 1 ρ α 1 ρ (f 4 g y 4 f 3 g 4 g t y 3 + (f g g f y y ) y 3 + g g f f y y ) = 0 Using the non-dimensional variables y, t, f ( y) g( y, t) F( ), U G(, ),, (15) (16) (17) in equations (15) (16), boundary conditions (1a) (1b), we obtain F F F F W F F F F F 1 (18a) iv e F(0) 0, F(0) 0, F( ) 1 (18b) 19

4 3 G η 3 + F G G F η η We (F 4 G η 4 3 G F η 3 F G G F η η ) β G τ η + Weβ 4 G τ η 3 = 0 G(0, τ) = 0, G η (0, τ) = εe iτ, G ηη (, τ) = γ (19a) (19b) where We = α 1 ρν is the Weissenberg number. System (18 a-b) has been solved numerically by Garg Rajagopal [3] Ariel [4, 5]. Following Bellman Kalaba [6] Garg Rajagopal [3], the quasi-linearized form of equation (18a) is F (iv) n+1 = F n+1 (F F n + 1 n We ) + F n+1 ( 1 We F n ) F n + F n+1 (F F n F n + F n+1 F [F n F n F n + F n n We ) (0) F n 1 ] + F n + We We F n where the subscript n (n + 1) represents the n th (n + 1) th approximation to the solution. Since the above equation is non-homogeneous, the solution at any approximation level can be written as = F homogeneous + F particular. Further, the homogeneous solution, F homogeneous, is a linear combination of two linearly independent solutions namely F h1 F h. The details of this technique are well described by Garg Rajagopal [3]. Using the quasi-linearization technique described by Garg Rajagopal [3], we find that F (0) = when We = 0. This value is in good agreement with the value obtained by Taemitsu Matunobu [16]. Numerical values of F (0) for different values of We are shown in Table 1. These values are in good agreement with the values obtained by Garg Rajagopal [3] Ariel [4]. Figure 1 shows the profiles of F for various values of We. We observed that as the elasticity of the fluid increases, the velocity near the wall decreases. Figure 1. Variations of F (η) for various values of We. Letting G(η, τ) = G 0 (η) + εg 1 (η)e iτ, then system (19) gives G 0 + FG 0 F G 0 We(FG 0 (iv) F G 0 + F G 0 F G 0 ) = 0 (1a) G 0 (0) = 0, G 0 (0) = 0, G 0 ( ) = γ (1b) G 1 + FG 1 F G 1 We (FG 1 (iv) F G 1 + F G 1 F G 1 ) iβ(g 1 We G 1 ) = 0 (a) G 1 (0) = 0, G 1 (0) = 1, G 1 ( ) = 0 (b) Letting G 0 (η) = γh 0 (η), then system (1 a-b) gives H 0 + FH 0 F H 0 We(FH 0 F H 0 + F H 0 (3a) F H 0 ) = 0 H 0 (0) = 0, H 0 ( ) = 1 (3b) System (3 a-b) is solved numerically using a shooting method it is found that for We = 0, H 0 (0) = Since G 0 (0) = γh 0 (0), then for We = 0, G 0 (0) = γ which is in good agreement with the value obtained by Taemitsu Matunobu [16]. Numerical values of H 0 (0) for different values of We are shown in Table 1. Figure depicts the profiles of H 0 for various values of We. 0

5 The only parameter in equation (4a) is the frequency β. Two series solutions valid for small large β respectively will be obtained. For small values of the frequency β, we assume that φ(η) = β n φ n (η) n=0 = φ 0 (η) + iβ φ 1 (η) + (iβ) φ (η) + (5) where the numerical values for φ 0 (0), φ 1 (0) φ (0) are given in Table 1 for different values of We. For large values of the frequency β, we let Figure. Variations of H 0 (η) for various values of We. Table 1. Numerical values of F (0), H 0 (0), φ 0 (0), φ 1 (0) φ (0) for different values of We. We F (0) H 0 (0) φ 0 (0) φ 1 (0) φ (0) Y = αη, α = iβ φ(η) = α n φ n (η) n=0 = φ 0 (η) + α φ 1 (η) + α φ (η) + (6) it was found that φ (0) = φ 0 (0) + αφ 1 (0) + α φ (0) + α 3 φ 3 (0) + = 1 (3 4m) 1 + m 8(1 + m) F (0) α m m α⁴ (40m3 50m + 8m 33)F (0) 18(1 + m) 1 + m α 6 + (7) where m 1 the numerical values of F (0) are given in Table 1 for different values of We. Figures 3-5 depict the variations of φ 0 (η), φ 1 (η) φ (η) for various values of We Letting φ(η) = G 1 (η), then system ( a-b) becomes φ + Fφ F φ We(Fφ F φ + F φ F φ) (4a) iβ(φ We φ ) = 0 φ(0) = 1, φ( ) = 0 (4b) Figure 3. Variations of φ 0 (η) for various values of We. 1

6 once with respect to y using the conditions at infinity, we obtain 1 iv f f f f f f f f f (30) Figure 4. Variations of φ 1 (η) for various values of We. ν 3 h y 3 h + α 1 4 h t y ρ α 1 ρ (f 4 h y 4 f 3 h + (f h h t y 3 y f ) y y 3 + h h f f y y ) (31) = (1 + iω ) U eiωt Non-dimensionalizing using y, t, f ( y) F( ), h( y, t) G(, ),, U (3) we obtain Figure 5. Variations of φ (η) for various values of We. 4. Solutions in the Moving Frame of Reference We assume that the Cartesian coordinates (x, y) are moving with the plate, the x-axis is along the plate the y-axis is normal to the plate. In this case, following Taemitsu Matunobu [16], we assume that the streamfunction is given by x f ( y) h( y, t) (8) the boundary conditions are given by f (0) f (0) 0, h(0, t) 0, h (0, t) 0 y (9a) U i t f ( ) 1, hy(, t) y e (9b) We note that the flow is oscillating with velocity U cos Ωt at infinity. Using equation (8) in (9), equating different powers of x to zero integrating iv F F F F W F F F F F 1 e (33a) F(0) 0, F(0) 0, F( ) 1 (33b) 3 G η 3 + F G G F η η We (F 4 G η 4 3 G F η 3 F G η G F η G ) β τ η + Weβ 4 G = (1 + iβ)ε eiτ (34a) τ η3 G(0, τ) = 0, G η (0, τ) = εe iτ, G η (, τ) = γη ε e iτ (34b) System (33 a-b) has been solved numerically in section 3. Letting G(η, τ) = G 0 (η) ε H(η)e iτ, system (34 a-b) gives G 0 + FG 0 F G 0 We (FG (iv) 0 F G 0 + F G 0 (35a) F G 0 ) = 0 G 0 (0) = 0, G 0 (0) = 0, G 0 ( ) = γ (35b)

7 H + FH F H We(FH (iv) F H + F H (36a) F H ) iβ(h We H ) = 1 iβ H(0) = 0, H (0) = 1, H ( ) = 1 (36b) Numerical solutions of system (35 a-b) have been obtained in section 3. It can easily be shown that the function H = F + iβg 1 iβ 1 iβ (37) is a solution of system (36 a-b) since it satisfies both the equation the boundary conditions. In equation (37), the functions F G 1 have been found in section Discussion Conclusions The unsteady second grade stagnation-point flow impinging obliquely on an oscillatory flat plate is studied. Numerical results for this flow are found for various values of the Weissenberg number We. Figure 1 shows the variations of F (η) for various values of We. The effect of the Weissenberg number, We, is to decrease the velocity F (η) near the wall as it increases. Figure depicts the variations of H 0 (η) for various values of We shows that H 0 (η) decreases near the wall as We is increasing. The variations of φ 0 (η) with various values of We are shown in Figure 3. From this figure we observed that φ 0 (η) is decreasing as We is incresing. Figure 4 shows the variations of φ 1 (η) for various values of We Figure 5 depicts the variations of φ (η) for various values of We. From Table 1, F (0) is decreasing as We is increasing. References [1] K. Hiemenz, "Die Grenzschicht an einem in den gleichformigen Flussigeitsstrom eingetauchten geraden Kreiszylinder," Dingler's Polytech., vol. 36, pp. 31, [] J. T. Stuart, "The viscous flow near a stagnation point when the external flow has uniform vorticity," J. Aerospace Sci., vol. 6, pp. 14, [3] K. J. Tamada, "Two-dimensional stagnation point flow impinging obliquely on a plane wall," J. Phys. Soc. Japan, vol. 46, pp. 310, [4] J. M. Dorrepaal, "An Exact solution of the Navier- Stoes equation which describes non-orthogonal stagnation-point flow in two dimensions," J. Fluid Mech., vol. 163, pp. 141, [5] B. W. Beard K. Walters, "Elastico-viscous boundary layer flows," in Proc. Camb. Phil. Soc., [6] J. M. Dorrepaal, O. P. Chna F. Labropulu, "The flow of a visco-elastic fluid near a point of reattachment," ZAMP, vol. 43, pp. 708, 199. [7] F. Labropulu, J. M. Dorrepaal O. P. Chna, "Viscoelastic fluid flow impinging on a wall with suction or blowing," Mech. Res. Comm., vol. 0, no., pp. 143, [8] N. Rott, "Unsteady viscous flow in the vicinity of a stagnation point," Quart. of Appl. Math, vol. 13, pp. 444, [9] M. B. Glauert, "The laminar boundary layer on oscillating plates cylinders," J. Fluid Mech., vol. 1, pp. 97, [10] A. C. Srivastava, "Unsteady flow of a second-order fluid near a stagnation point," J. Fluid Mech., vol. 4, no. Part 1, pp. 33, [11] C. H. Yih, Fluid Mechanics, New Yor: West River Press, [1] F. Labropulu, X. Xu M. Chinichian, "Unsteady Stagnation Point Flow of a non-newtonian Second- Grade Fluid," Int. J. Math. & Math. Sci., vol. 003, no. 50, pp , 003. [13] Y. Matunobu, "Structure of pulsatile Hiemenz flow temporal variation of wall shear stress near the stagnation point. I," J. Phys. Soc. Japan, vol. 4, pp. 041, [14] Y. Matunobu, "Structure of pulsatile Hiemenz flow temporal variation of wall shear stress near the stagnation point. II," J. Phys. Soc. Japan, vol. 43, pp. 36, [15] M. Kawaguti K. Hamano, "Two-dimensional model of pulsatile flow through constricted artery," in Xth Intern. Congr. Angiology, [16] N. Taemitsu Y. Matunobu, "Unsteady stagnation-point flow impinging oqliquely on an oscillating flat plate," J. Phys. Soc. Japan, vol. 47, no. 4, pp. 1347, [17] Y. Matunobu M. Araawa, "Model experiment on the post-stenotic dilatation in blood vessels," Biorhelogy, vol. 11, pp. 47, [18] B. Gessner, "Hemodynamic theories of atherogenesis," Circulation Research, vol. 33, pp. 59,

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