Velocity and shear stress for an Oldroyd-B fluid within two cylinders
© Kang et al. 2016
Received: 29 September 2015
Accepted: 24 January 2016
Published: 10 February 2016
This paper aims to explore the possible solutions for the movement of an Oldroyd-B fluid placed under certain conditions, i.e. the fluid is present within two cylinders, which are coaxial and oscillating within. Having said that the governing model will be an Oldroyd-B fluid, we wish to achieve our goal of finding the velocity and shear stress by using some common transformations, namely the Laplace transformation and the Hankel transformation. The final results, for the sake of simplicity, will be expressed in the form of generalized G-function and they satisfy all imposed initial and boundary conditions.
Flow due to an oscillating cylinder is one of the most important and interesting problems of motion near oscillating walls. As early as 1886, Stokes  established an exact solution to the rotational oscillations of an infinite rod immersed in a Newtonian fluid. An extension of this problem to the rod undergoing both rotational and longitudinal oscillations has been realized in , while the first exact solutions for similar motions of non-Newtonian fluids are those of Rajagopal  and Rajagopal and Bhatnagar . However, all these solutions are steady-state solutions to which a transient solution has to be added in order to describe the motion of the fluid for small and large times.
The first closed-form expressions for the starting solutions corresponding to an oscillating motion seem to be those of Erdogan  for Newtonian fluids. New exact solutions for the same problem, but presented as a sum of steady-state and transient solutions, have also been established by Corina Fetecau et al. . The extension of these solutions to second grade fluids has been achieved in , while the starting solutions for the motion of the same fluids due to longitudinal and torsional oscillations of a circular cylinder have been established in . Recently, starting solutions for oscillating motions of a Maxwell fluid in cylindrical domains have been obtained in . Other interesting results regarding oscillating flows of non-Newtonian fluids have been presented in [10–15].
In this paper, we are interested in the velocity and shear stress for the movement of an Oldroyd-B fluid within two coaxial infinite oscillating cylinders oscillatory motion of a generalized Maxwell fluid between two infinite coaxial circular cylinders, both of them oscillating around their common axis with given constant angular frequencies z. The velocity field and associated tangential stress of the motion are determined by using Laplace and Hankel transforms and are presented by integral and series. It is worthy to point out that the solutions that have been obtained satisfy the governing differential equation and all imposed initial and boundary conditions as well. The solutions correspond to the ordinary Oldroyd-B fluid, performing the same motion.
1.1 Governing equations of problem
2 Theoretical description of the problem
3 Calculation of the velocity field
3.1 Calculation of shear stress
4 Particularization of the above results
The above results are of a general nature and the imposition of certain limits/conditions may bring these to particular fluids.
4.1 Ordinary Oldroyd-B fluid
4.2 Ordinary Maxwell fluid
Our above endeavors were to develop a formula for the calculation of exact solutions for the velocity field and the shear stress of the motion (flow) of an Oldroyd-B fluid present between two rotationally oscillating cylinders of infinite lengths. The use of fractional derivatives and the commonly known transformations, i.e. the Laplace and the Hankel transformations, has made the approach more accessible. The central notion depicts the phenomenon that a viscoelastic (Oldroyd-B) fluid will react under certain conditions and that can we control such flow. At first stage the inner cylinder was supposed to be at rest, i.e. fixed, whereas the movement was produced by the outer cylinder. At the second stage, we analyzed the flow of the fluid produced by the movement of the inner cylinder while considering the outer cylinder at rest or fixed. The obtained solutions satisfy the governing equations and all imposed initial and boundary conditions. The solutions, obtained by means of Laplace and Hankel transforms, are presented in integral and series forms in terms of the generalized G-function. In the end these general solutions have been particularized for ‘ordinary Oldroyd-B fluids’ and for ‘ordinary Maxwell fluids’.
This study was supported by research funds from Dong-A University.
- Stokes, GG: On the Effect of the Rotation of Cylinders and Spheres About Their Axes in Increasing the Logarithmic Decrement of the Arc of Vibration. Cambridge University Press, Cambridge (1886) Google Scholar
- Casarella, MJ, Laura, PA: Drag on oscillating rod with longitudinal and torsional motion. J. Hydronaut. 3, 180-183 (1969) View ArticleGoogle Scholar
- Rajagopal, KR: Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech. 49, 281-285 (1983) View ArticleMATHGoogle Scholar
- Rajagopal, KR, Bhatnagar, RK: Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech. 113, 233-239 (1995) View ArticleMathSciNetMATHGoogle Scholar
- Erdogan, ME: A note on an unsteady flow of a viscous fluid due to an oscillating plane wall. Int. J. Non-Linear Mech. 35, 1-6 (2000) View ArticleMathSciNetMATHGoogle Scholar
- Fetecau, C, Vieru, D, Fetecau, C: A note on the second problem of Stokes for Newtonian fluids. Int. J. Non-Linear Mech. 43, 451-457 (2008) View ArticleMATHGoogle Scholar
- Fetecau, C, Fetecau, C: Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int. J. Eng. Sci. 43, 781-789 (2005) View ArticleMathSciNetMATHGoogle Scholar
- Fetecau, C, Fetecau, C: Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Int. J. Eng. Sci. 44, 788-796 (2006) View ArticleMathSciNetMATHGoogle Scholar
- Vieru, D, Akhtar, W, Fetecau, C, Fetecau, C: Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42, 573-583 (2007) View ArticleMATHGoogle Scholar
- Hayat, T, Siddiqui, AM, Asghar, S: Some simple flows of an Oldroyd-B fluid. Int. J. Eng. Sci. 39, 135-147 (2001) View ArticleGoogle Scholar
- Hayat, T, Mumtaz, S, Ellahi, R: MHD unsteady flows due to non-coaxial rotations of a disk and a fluid at infinity. Acta Mech. Sin. 19, 235-240 (2003) View ArticleGoogle Scholar
- Hayat, T, Ellahi, R, Asghar, S, Siddiqui, AM: Flow induced by non-coaxial rotation of a porous disk executing non-torsional oscillating and second grade fluid rotating at infinity. Appl. Math. Model. 28, 591-605 (2004) View ArticleGoogle Scholar
- Hayat, T, Ellahi, R, Asghar, S: Unsteady periodic flows of a magnetohydrodynamic fluid due to non-coaxial rotations of a porous disk and fluid at infinity. Math. Comput. Model. 40, 173-179 (2004) View ArticleMathSciNetMATHGoogle Scholar
- Aksel, N, Fetecau, C, Scholle, M: Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids. Z. Angew. Math. Phys. 57, 815-831 (2006) View ArticleMathSciNetMATHGoogle Scholar
- Khan, M, Ellahi, R, Hayat, T: Exact solution for oscillating flows of generalized Oldroyd-B fluid through porous medium in a rotating frame. J. Porous Media 12, 777-788 (2009) View ArticleGoogle Scholar