Abstract
This chapter presents some new exact solutions corresponding to unsteady magnetohydrodynamic (MHD) flow of Jeffrey fluid in a long porous rectangular duct oscillating parallel to its length. The exact solutions are established by means of the double finite Fourier sine transform (DFFST) and discrete Laplace transform (LT). The series solution of velocity field, associated shear stress and volume flow rate in terms of Fox H-functions, satisfying all imposed initial and boundary conditions, have been obtained. Also, the obtained results are analyzed graphically through various pertinent parameter.
Keywords
- porous medium
- Jeffrey fluid
- oscillating rectangular duct
- Fox H-function
- MSC (2010): 76A05
- 76A10
1. Introduction
Considerable progress has been made in studying flows of non-Newtonian fluids throughout the last few decades. Due to their viscoelastic nature non-Newtonian fluids, such as oils, paints, ketchup, liquid polymers and asphalt exhibit some remarkable phenomena. Amplifying interest of many researchers has shown that these flows are imperative in industry, manufacturing of food and paper, polymer processing and technology. Dissimilar to the Newtonian fluid, the flows of non-Newtonian fluids cannot be explained by a single constitutive model. In general the rheological properties of fluids are specified by their so-called constitutive equations. Exact recent solutions for constitutive equations of viscoelastic fluids are given by Rajagopal and Bhatnagar [1], Tan and Masuoka [2, 3], Khadrawi et al. [4] and Chen et al. [5] etc. Among non-Newtonian fluids the Jeffrey model is considered to be one of the simplest type of model which best explain the rheological effects of viscoelastic fluids. The Jeffrey model is a relatively simple linear model using the time derivatives instead of convected derivatives. Nadeem et al. [6] obtained analytic solutions for stagnation flow of Jeffrey fluid over a shrinking sheet. Khan [7] investigated partial slip effects on the oscillatory flows of fractional Jeffrey fluid in a porous medium. Hayat et al. [8] examined oscillatory rotating flows of a fractional Jeffrey fluid filling a porous medium. Khan et al. [9] discussed unsteady flows of Jeffrey fluid between two side walls over a plane wall.
Much attention has been given to the flows of rectangular duct because of its wide range applications in industries. Gardner and Gardner [10] discussed magnetohydrodynamic (MHD) duct flow of two-dimensional bi-cubic B-spline finite element. Fetecau and Fetecau [11] investigated the flows of Oldroyd-B fluid in a channel of rectangular cross-section. Nazar et al. [12] examined oscillating flow passing through rectangular duct for Maxwell fluid using integral transforms. Unsteady magnetohydrodynamic flow of Maxwell fluid passing through porous rectangular duct was studied by Sultan et al. [13]. Tsangaris and Vlachakis [14] discussed analytic solution of oscillating flow in a duct of Navier-Stokes equations.
In the last few decades the study of fluid motions through porous medium have received much attention due to its importance not only to the field of academic but also to the industry. Such motions have many applications in many industrial and biological processes such as food industry, irrigation problems, oil exploitation, motion of blood in the cardiovascular system, chemistry and bio-engineering, soap and cellulose solutions and in biophysical sciences where the human lungs are considered as a porous layer. Unsteady MHD flows of viscoelastic fluids passing through porous space are of considerable interest. In the last few years a lot of work has been done on MHD flow, see [15, 16, 17, 18, 19] and reference therein.
According to the authors information up to yet no study has been done on the MHD flow of Jeffrey fluid passing through a long porous rectangular duct oscillating parallel to its length. Hence, our main objective in this note is to make a contribution in this regard. The obtained solutions, expressed under series form in terms of Fox H-functions, are established by means of double finite Fourier sine transform (DFFST) and Laplace transform (LT). Finally, the obtained results are analyzed graphically through various pertinent parameter.
2. Governing equations
The equation of continuity and momentum of MHD flow passing through porous space is given by [7]
where velocity is represented by
For an incompressible and unsteady Jeffrey fluid the Cauchy stress tensor is defined as [9]
where
where
where
In the following problem we consider a velocity field and extra stress of the form
where
therefore from Eqs. (2), (5) and (6), it results that
where
where
3. Statement of the problem
We take an incompressible flow of Jeffrey fluid in a porous rectangular duct under an imposed transverse magnetic field whose sides are at
or
The solutions of problems (8)–(10) and (8), (9), (11) are denoted by
which is the solution of the problem
The solution of the problem (13)–(15) will be obtained by means of the DFFST and LT.
The DFFST of function
4. Calculation of the velocity field
Multiplying both sides of Eq. (13) by
where
The Fourier transform
We apply LT to Eq. (17) and using initial conditions (18) to get
We will apply the discrete inverse LT technique [20] to obtain analytic solution for the velocity fields and to avoid difficult calculations of residues and contour integrals, but first we express Eq. (19) in series form as
We apply the discrete inverse LT to Eq. (20), to obtain
Taking the inverse Fourier sine transform we get the analytic solution of the velocity field
To obtain a more compact form of velocity field we write Eq. (22) in terms of Fox H-function,
or
where
From Eq. (24), we obtain the velocity field due to cosine oscillations of the duct
and the velocity field due to sine oscillations of the duct
We use the following property of the Fox H-function [21] in the above equation
5. Calculation of the shear stress
We denote the tangential tensions for the cosine oscillations of the duct by
If we introduce
in Eq. (7), we get
We apply LT to Eqs. (29) and (30), to obtain
Taking the inverse Fourier transform of Eq. (19) to get
or
where
We express Eq. (34) in series form in order to obtain a more suitable form of
Using the inverse LT of the last equation, we obtain
Lastly, we write the stress field in a more compact form by using Fox H-function
From Eq. (37), we obtain the tangential tension due to cosine oscillations of the duct
and the tangential tension corresponding to sine oscillations of the duct
In the similar fashion we can find
6. Volume flux
The volume flux due to cosine oscillations is given by
putting
Similarly, we obtain the volume flux of the rectangular duct due to the sine oscillations
7. Numerical results and discussion
We have presented flow problem of MHD Jeffrey fluid passing through a porous rectangular duct. Exact analytical solutions are established for such flow problem using DFFST and LT technique. The obtained solutions are expressed in series form using Fox H-functions. Several graphs are presented here for the analysis of some important physical aspects of the obtained solutions. The numerical results show the profiles of velocity and the adequate shear stress for the flow. We analyze these results by variating different parameters of interest.
The effects of relaxation time
References
- 1.
Rajagopal KR, Srinivasa A. Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mechanica. 1995; 113 :233-239 - 2.
Tan WC, Masuoka T. Stoke’s first problem for second grade fluid in a porous half space. International Journal of Non-Linear Mechanics. 2005; 40 :515-522 - 3.
Tan WC, Masuoka T. Stoke’s first problem for an Oldroyd-B fluid in a porous half space. Physics of Fluids. 2005; 17 :023101 - 4.
Khadrawi AF, Al-Nimr MA, Othman A. Basic viscoelastic fluid problems using the Jeffreys model. Chemical Engineering Science. 2005; 60 :7131-7136 - 5.
Chen CI, Chen CK, Yang YT. Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate. International Journal of Heat and Mass Transfer. 2004; 40 :203-209 - 6.
Nadeem S, Hussain A, Khan M. Stagnation flow of a Jeffrey fluid over a shrinking sheet. Zeitschrift für Naturforschung. 2010; 65a :540-548 - 7.
Khan M. Partial slip effects on the oscillatory flows of a fractional Jeffrey fluid in a porous medium. Journal of Porous Media. 2007; 10 :473-487 - 8.
Hayat T, Khan M, Fakhar K, Amin N. Oscillatory rotating flows of a fractional Jeffrey fluid filling a porous medium. Journal of Porous Media. 2010; 13 (1):29-38 - 9.
Khan M, Iftikhar F, Anjum A. Some unsteady flows of a Jeffrey fluid between two side walls over a plane wall. Zeitschrift für Naturforschung. 2011; 66 (a):745-752 - 10.
Gardner LRT, Gardner GA. A two-dimensional bi-cubic B-spline finite element used in a study of MHD duct flow. Computer Methods in Applied Mechanics and Engineering. 1995; 124 :365-375 - 11.
Fetecau C, Fetecau C. Unsteady flows of Oldroyd-B fluids in a channel of rectangular cross-section. International Journal of Non-Linear Mechanics. 2005; 40 :1214-1219 - 12.
Nazar M, Shahid F, Akram S, Sultan Q. Flow on oscillating rectangular duct for Maxwell fluid. Applied Mathematics and Mechanics (English Edition). 2012; 33 :717-730 - 13.
Sultan Q, Nazar M, Akhtar W, Ali U. Unsteady flow of a Maxwell fluid in a porous rectangular duct. Scientific International Journal. 2013; 25 (2):181-194 - 14.
Tsangaris S, Vlachakis NW. Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle. Zeitschrift für Angewandte Mathematik und Physik. 2003; 54 :1094-1100 - 15.
Vajravelu K. Hydromagnetic flow and heat transfer over a continuous moving porous flat surface. Acta Mechanica. 1986; 64 :179-185 - 16.
Amakiri ARC, Ogulu A. The effect of viscous dissipative heat and uniform magnetic field on the free convective flow through a porous medium with heat generation/absorption. European Journal of Scientific Research. 2006; 15 (4):436-445 - 17.
Singh KD. Exact solution of an oscillatory MHD flow in a channel filled with porous medium. International Journal of Applied Mechanics and Engineering. 2011; 16 :277-283 - 18.
Samiulhaq, Fetecau, Khan I, Ali F, Shafie S. Radiation and porosity effects on the magnetohydrodynamic flow past an oscillating vertical plate with uniform heat flux. Zeitschrift für Naturforschung. 2012; 67 (a):572-580 - 19.
Khan I, Fakhar K, Shafie S. Magnetohydrodynamic free convection flow past an oscillating plate embedded in a porous medium. Journal of the Physical Society of Japan. 2011; 80 :104-110 - 20.
Sneddon IN. Fourier Transforms. New York: McGraw-Hill; 1951 - 21.
Mathai AM, Saxena RK, Haubold HJ. The H-functions: Theory and Applications. New York: Springer; 2010