Volume 3  Year 2016  Pages 1724
DOI: 10.11159/jffhmt.2016.003
Unsteady Stagnationpoint Flow of a Secondgrade Fluid
Fotini Labropulu^{1}, Daiming Li^{2}
^{1}Luther College/University of Regina
Regina, SK, Canada S4S 0A2
fotini.labropulu@uregina.ca
^{2}Department of Chemical and Petroleum Engineering, University of Calgary
2500 University Dr. NW, Calgary, Alberta, Canada T2N 1N4
Abstract  The unsteady twodimensional stagnation point flow of secondgrade fluid impinging on an infinite plate is examined and solutions are obtained. It is assumed that the infinite plate at is oscillating with velocity , the fluid occupies the entire upper half plane and 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, . 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 and large values of frequency.
Keywords: Unsteady, Stagnationpoint, Oscillating plate, NonNewtonian fluid
© Copyright 2016 Authors  This is an Open Access article published under the Creative Commons Attribution License terms Creative Commons Attribution License terms. Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.
Date Received: 20150904
Date Accepted: 20160126
Date Published: 20160317
Nomenclature

1^{st} and 2^{nd} Rivlin Ericksen tensors 

similarity variables 

constant 

fluid pressure 

Cauchy Stress Tensor 

velocity vector 

velocity components along and axis 

constant 

Weissenberg number 
Greek Symbols 


viscoelastic parameters of the fluid 

frequency 

constant 

constant 

nondimensional varable 

fluid viscosity 

kinematic viscosity 

Shear stress component 

nondimensional variable 

streamfunction 

frequency 
1. 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 [2], Tamada [3] and Dorrepaal [4] independently investigated the solutions of a stagnation point flow when the fluid impinges obliquely on the plate. Beard and Walters [5] used boundarylayer equations to study twodimensional 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] and Glauert [9] studied the stagnation point flow of a Newtonian fluid when the plate performs harmonic oscillations in its own plane. Srivastava [10] investigated the same problem for a nonNewtonian second grade fluid using the KarmanPohlhausen method [11] to solve the resulting equations. Labropulu et al [12] used series methods to solve the unsteady stagnation point flow of a Walters' B' fluid impinging on an oscillating flat plate. Matunobu [13, 14] and Kawaguti and Hamano [15] examined the fundamental character of the unsteady flow near a stagnation point for a Newtonian fluid. Takemitsu and Matunobu [16] studied the oblique stagnation point flow for a Newtonian fluid and 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] and lead to vascular diseases [18].
In this work, the unsteady stagnation point flow of a viscoelastic secondgrade fluid is examined and solutions are obtained. We assume that the infinite plate at is oscillating with velocity , the fluid occupies the entire upper half plane and 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, . 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 and large values of frequency.
2. Flow Equations
The flow of a viscous incompressible nonNewtonian secondgrade fluid, neglecting thermal effects and body forces, is governed by
when the constitutive equation for the Cauchy stress tensor which describes secondgrade fluid given by Rivlin and Ericksen [19] is
Here is the velocity vector field, the fluid pressure function, the constant fluid density, the constant coefficient of viscosity and the normal stress moduli. Dunn and Fosdick [20] and Dunn and Rajagopal [21] have shown that if the secondgrade fluid described by (3) is to undergo motions which are compatible with ClausiusDuhem inequality [22] and the assumption that the free energy density of the fluid be locally at rest, then the material constants must satisfy the following restrictions:
Considering the flow to be plane, we take and so that the flow equations (1) to (3) take the form
where is the kinematic viscosity.
Continuity equation (5) implies the existence of a streamfunction such that
Substitution of (8) in equations (6) and (7) and elimination of pressure from the resulting equations using yields
Having obtained a solution of equation (9), the velocity components are given by (8) and the pressure can be found by integrating equations (6) and (7).
The shear stress component of the Cauchy stress is given by
3. Solutions in the Fixed Frame of Reference
Following Takemitsu and Matunobu [16], we assume that
We assume that the infinite plate at is oscillating with velocity and that the fluid occupies the entire upper half plane Furthermore, we assume the streamfunction far from the wall is given by (see Stewart [2]). Thus, the boundary conditions are given by
where is a nondimensional constant characterizing the obliqueness of oncoming flow. It is assumed that only the real part of a complex quantity has its physical meaning.
Substitution of equation (11) in (9) yields
and
Integrating equations (13) and (14) once with respect to using the conditions at infinity, we have
and
Using the nondimensional variables
in equations (15) and (16), and boundary conditions (12a) and (12b), we obtain
and
where is the Weissenberg number.
System (18 ab) has been solved numerically by Garg and Rajagopal [23] and Ariel [24, 25]. Following Bellman and Kalaba [26] and Garg and Rajagopal [23], the quasilinearized form of equation (18a) is
where the subscript and represents the and approximation to the solution. Since the above equation is nonhomogeneous, the solution at any approximation level can be written as . Further, the homogeneous solution, , is a linear combination of two linearly independent solutions – namely and . The details of this technique are well described by Garg and Rajagopal [23].
Using the quasilinearization technique described by Garg and Rajagopal [23], we find that when . This value is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of for different values of are shown in Table 1. These values are in good agreement with the values obtained by Garg and Rajagopal [23] and Ariel [24]. Figure 1 shows the profiles of for various values of . We observed that as the elasticity of the fluid increases, the velocity near the wall decreases.
Letting , then system (19) gives
and
Letting , then system (21 ab) gives
System (23 ab) is solved numerically using a shooting method and it is found that for , Since , then for , which is in good agreement with the value obtained by Takemitsu and Matunobu [16]. Numerical values of for different values of are shown in Table 1. Figure 2 depicts the profiles of for various values of
Table 1. Numerical values of and for different values of .
0.0  23259  0.60777  0.81107  0.49348  0.09471 
0.1  1.13425  0.54392  0.76774  0.50612  0.06023 
0.2  1.05818  0.49546  0.73291  0.51309  0.02785 
0.3  0.99689  0.45677  0.70364  0.51685  0.00204 
0.4  0.94588  0.42465  0.67826  0.51881  0.02953 
0.5  0.90248  0.39774  0.65619  0.51922  0.05474 
1.0  0.75276  0.30691  0.57522  0.51155  0.15428 
2.0  0.59677  0.21662  0.48170  0.48461  0.27270 
5.0  0.41288  0.12046  0.35721  0.41638  0.39192 
8.0  0.33533  0.08503  0.29916  0.37114  0.40062 
10  0.30283  0.07127  0.27371  0.34885  0.38807 
20  0.21857  0.03978  0.20475  0.28684  0.37758 
50  0.14008  0.01735  0.13476  0.19505  0.16073 
100  0.09951  0.00897  0.09591  0.14291  0.33255 
200  0.07053  0.00453  0.06490  0.08305  1.48595 
500  0.04469  0.00180  0.02550  0.04243  3.12615 
Letting , then system (22 ab) becomes
The only parameter in equation (24a) is the frequency . Two series solutions valid for small and large respectively will be obtained. For small values of the frequency, we assume that
where the numerical values for and are given in Table 1 for different values of .
For large values of the frequency , we let
and
and it was found that
where and the numerical values of are given in Table 1 for different values of . Figures 35 depict the variations of , and for various values of .
4. Solutions in the Moving Frame of Reference
We assume that the Cartesian coordinates are moving with the plate, the axis is along the plate and the axis is normal to the plate. In this case, following Takemitsu and Matunobu [16], we assume that the streamfunction is given by
and the boundary conditions are given by
We note that the flow is oscillating with velocity at infinity. Using equation (28) in (9), equating different powers of to zero and integrating once with respect to using the conditions at infinity, we obtain
and
Nondimensionalizing using
we obtain
and
System (33 ab) has been solved numerically in section 3. Letting , system (34 ab) gives
and
Numerical solutions of system (35 ab) have been obtained in section 3. It can easily be shown that the function
is a solution of system (36 ab) since it satisfies both the equation and the boundary conditions. In equation (37), the functions and have been found in section 3.
4. Discussion and Conclusions
The unsteady second grade stagnationpoint flow impinging obliquely on an oscillatory flat plate is studied. Numerical results for this flow are found for various values of the Weissenberg number . Figure 1 shows the variations of for various values of . The effect of the Weissenberg number, , is to decrease the velocity near the wall as it increases. Figure 2 depicts the variations of for various values of and shows that decreases near the wall as is increasing. The variations of with various values of are shown in Figure 3. From this figure we observed that is decreasing as is incresing. Figure 4 shows the variations of for various values of and Figure 5 depicts the variations of for various values of . From Table 1, is decreasing as is increasing.
References
[1] K. Hiemenz, "Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder," Dingler's Polytech., vol. 326, pp. 321, 1911. View Article
[2] J. T. Stuart, "The viscous flow near a stagnation point when the external flow has uniform vorticity," J. Aerospace Sci., vol. 26, pp. 124, 1959. View Article
[3] K. J. Tamada, "Twodimensional stagnation point flow impinging obliquely on a plane wall," J. Phys. Soc. Japan, vol. 46, pp. 310, 1979. View Article
[4] J. M. Dorrepaal, "An Exact solution of the NavierStokes equation which describes nonorthogonal stagnationpoint flow in two dimensions," J. Fluid Mech., vol. 163, pp. 141, 1986. View Article
[5] B. W. Beard and K. Walters, "Elasticoviscous boundary layer flows," in Proceedings of Camb. Phil. Soc., 1964.
[6] J. M. Dorrepaal, O. P. Chandna and F. Labropulu, "The flow of a viscoelastic fluid near a point of reattachment," ZAMP, vol. 43, pp. 708, 1992. View Article
[7] F. Labropulu, J. M. Dorrepaal and O. P. Chandna, "Viscoelastic fluid flow impinging on a wall with suction or blowing," Mech. Res. Comm., vol. 20, no. 2, pp. 143, 1993. View Article
[8] N. Rott, "Unsteady viscous flow in the vicinity of a stagnation point," Quart. of Appl. Math, vol. 13, pp. 444, 1956. View Article
[9] M. B. Glauert, "The laminar boundary layer on oscillating plates and cylinders," J. Fluid Mech., vol. 1, pp. 97, 1956. View Article
[10] A. C. Srivastava, "Unsteady flow of a secondorder fluid near a stagnation point," J. Fluid Mech., vol. 24, no. Part 1, pp. 33, 1966. View Article
[11] C. H. Yih, Fluid Mechanics, New York: West River Press, 1977.
[12] F. Labropulu, X. Xu and M. Chinichian, "Unsteady Stagnation Point Flow of a nonNewtonian SecondGrade Fluid," Int. J. Math. & Math. Sci., vol. 2003, no. 50, pp. 37973807, 2003. View Article
[13] Y. Matunobu, "Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. I," J. Phys. Soc. Japan, vol. 42, pp. 2041, 1977. View Article
[14] Y. Matunobu, "Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. II," J. Phys. Soc. Japan, vol. 43, pp. 326, 1977. View Article
[15] M. Kawaguti and K. Hamano, "Twodimensional model of pulsatile flow through constricted artery," in Xth Intern. Congr. Angiology, 1976.
[16] N. Takemitsu and Y. Matunobu, "Unsteady stagnationpoint flow impinging oqliquely on an oscillating flat plate," J. Phys. Soc. Japan, vol. 47, no. 4, pp. 1347, 1979. View Article
[17] Y. Matunobu and M. Arakawa, "Model experiment on the poststenotic dilatation in blood vessels," Biorhelogy, vol. 11, pp. 427, 1974. View Article
[18] B. Gessner, "Hemodynamic theories of atherogenesis," Circulation Research, vol. 33, pp. 259, 1973. View Article
[19] R. S. Rivlin and J. L. Ericksen, "Stressdeformation relations for isotropic materials," J. Rat. Mech. Anal., vol. 4, pp. 323, 1955. View Article
[20] J. E. Dunn and R. L. Fosdick, "Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade," Arch. Rational Mech. Anal., vol. 56, pp. 191, 1974. View Article
[21] J. E. Dunn and K. R. Rajagopal, "Fluids of differential type:critical review and thermodynamic analysis," Int. J. Eng. Sci., vol. 33, pp. 689, 1995. View Article
[22] C. Truesdell, "The Mechanical foundations of elasticity and fluid dynamics," Journal of Rational Mechanics and Analysis, vol. 1, pp. 125, 1952. View Article
[23] V. K. Garg and K. R. Rajagopal, "Stagnation point flow of a nonNewtonain fluid," Mech. Res. Comm., vol. 17, no. 6, pp. 415, 1990.
[24] P. D. Ariel, "A hybrid method for computing the flow of viscoelastic fluids," Int. J. Num. Meth. in Fluids, vol. 14, pp. 323, 1992. View Article
[25] P. D. Ariel, "On extra boundary condition in the stagnation point flow of a second grade fluid," Int. J. Engng. Sci, vol. 40, pp. 145, 2002. View Article
[26] R. E. Bellman and R. E. Kalaba, in Quasilinearization and NonLinear Boundary Value Problems, New York, Elsevier, 1965. View Article