The present work is concerned with exact solutions of Stokes second problem for magnetohydrodynamics (MHD) flow of a Burgers' fluid. The fluid over a flat plate is assumed to be electrically conducting in the presence of a uniform magnetic field applied in outward transverse direction to the flow. The equations governing the flow are modeled and then solved using the Laplace transform technique. The expressions of velocity field and tangential stress are developed when the relaxation time satisfies the condition γ = λ2/4 or γ>λ2/4. The obtained closed form solutions are presented in the form of simple or multiple integrals in terms of Bessel functions and terms with only Bessel functions. The numerical integration is performed and the graphical results are displayed for the involved flow parameters. It is found that the velocity decreases whereas the shear stress increases when the Hartmann number is increased. The solutions corresponding to the Stokes' first problem for hydrodynamic Burgers' fluids are obtained as limiting cases of the present solutions. Similar solutions for Stokes' second problem of hydrodynamic Burgers' fluids and those for Newtonian and Oldroyd-B fluids can also be obtained as limiting cases of these solutions.
Citation: Khan I, Ali F, Shafie S (2013) Stokes' Second Problem for Magnetohydrodynamics Flow in a Burgers' Fluid: The Cases γ = λ2/4 and γ>λ2/4. PLoS ONE 8(5): e61531. https://doi.org/10.1371/journal.pone.0061531
Editor: Assad Anshuman Oberai, Rensselaer Polytechnic Institute, United States of America
Received: December 14, 2012; Accepted: March 11, 2013; Published: May 8, 2013
Copyright: © 2013 Khan et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors have no funding or support to report.
Competing interests: The authors have declared that no competing interests exist.
Magnetohydrodynamics is the study of flow of electrically conducting fluids in electric and magnetic fields. This phenomenon is essentially one of the mutual interaction between the fluid velocity and electromagnetic field i.e. the motion of the fluid affects the magnetic field and the magnetic field affects the fluid motion. Basically, magnetohydrodynamics is a research area that involves the study of motion of electrically conducting fluids such as plasma and salt water. MHD flows are found to have influential applications in many natural and man made flows. They are frequently used in industry to heat, pump, stir and levitate liquid metals. Another application for MHD is the magnetohydrodynamic generator in which electrically conducting fluid is used to generate electric power. The flows of an electrically conducting fluid in the presence of a magnetic field have important applications in various areas of technology such as, accelerators centrifugal separation of solid from fluid, purification of crude oils, astrophysical flows, petroleum industry, polymer technology, solar power technology, nuclear engineering applications and other industrial areas , .
The literature on the study of MHD viscous fluid is abundant (see for example - and the references therein). However, such studies for non-Newtonian fluids are limited. To the best of author's knowledge, MHD flow of non-Newtonian fluids was first studied by Sarpkaya . Subsequently, several other investigations considering the MHD flow of non-Newtonian fluids were carried out and currently this field has become an active area of research. Ersoy  examined the MHD flow between eccentric rotating disks for an Oldroyd-B fluid. Hayat and Hutter  obtained exact solutions for flows of an electrically conducting Oldroyd-B fluid over an infinite oscillatory plate in the presence of a transverse magnetic field. Khan et al  developed exact solutions of Stokes second problem for MHD Oldroyd-B fluid. Liu et al  and Zheng et al  and  analyzed the MHD flow of generalized Oldroyd-B fluid for different fluid motions using frictional derivatives. On the other hand, studies on MHD flow of Burgers' fluid are very limited. Therefore, any MHD analysis of this model will be genuine contribution towards the enhancement of the theory of non-Newtonian fluid mechanics. Hayat et al.  studied the MHD flow of Burger's fluid whereas with heat transfer analysis was investigated by Siddiqui et al. , . Very recently, Khan et al  studied MHD flow of Burger's fluid and obtained exact solutions of Stokes' first problem by using the Laplace and Fourier sine transforms. The MHD flows of these fluid models and some other well known non-Newtonian fluids models such as second grade fluid –, third grade fluid , Maxwell fluid , , generalized Burgers' fluid , , Micropolar fluid , , Walters-B liquid fluid , Jeffery fluid  and Nanofluid  are used to describe stress relaxation, shear thinning or shear thickening, normal stress effects, earth's mantle, asphalt and asphalt mixes, food products and soil, dilute polymeric solutions, hydrocarbons, paints and several other industrial and geomechanical fluids.
Khan et al  extended the work of Fetecau et al  to the MHD flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate. The analytical solutions are carried out by using the Fourier sine and Laplace transforms. Vieru et al  determined exact solutions corresponding to the flow of a Burgers' fluid over a suddenly moved flat plate when the relaxation times satisfy the condition or They used the Laplace transform technique to find the expressions for velocity and shear stress fields which were reduced to the similar solutions for Newtonian and Oldroyd-B fluids as limiting cases. Recently, Khan et al  extended the work of Vieru et al  to the flow of a Burgers' fluid over an oscillatory moved flat plate. They used a similar method of solution and obtained the exact solutions.
From the literature survey, it is found that there are very few problems of Newtonian fluids for which the exact solutions are available. However, these solutions become even more rare if the constitutive equations of non-Newtonian fluids are considered. The importance of exact solutions is not only that they can explain the physics of some fundamental flows but also that such solutions can be used as checks against complicated numerical codes that have been developed for much more complex flows. Moreover, one of the most common mistakes that has been overlooked for the last coupled of decades has been identified by Christov . Christov pointed out that in the case of Stokes first and second problems, the plate's velocity is given by , where denotes the Heaviside step function, and is some smooth function. This inclusion of Heaviside step function was ignored previously. There are several comments and errata published in the literature for the modification of such erroneous results. It is important to mention here that such type of mistakes reported by Christov  are avoided in the present communication.
The main purpose of the present investigation is to extend the work of Vieru et al.  and Khan et al.  for the MHD flow of an electrically conducting Burgers' fluid past an oscillating plate when the magnetic field is acting perpendicular to the flow direction. It is also interesting to study the flow of non- Newtonian fluids with externally imposed magnetic fields which control the boundary layer and increase the performance of many systems. For example, when we use the electrically conducting fluid in MHD power generators, their performance increase in comparison to conventional electric generators where solid conductors are used to generate electric power. The present work can also be helpful to study underground oil, where there is a natural magnetic field and the motion of blood through arteries , .
The rest of the paper is arranged as follows. The governing equations of the problem are given in section 2. The mathematical formulation of the problem is given in Section 3.The solution of the problem is given in section where the Laplace transform technique is used and the expressions for velocity and shear stress fields are obtained when the relaxation time satisfies the condition or Limiting solutions are given in section 5. Graphical results are displayed in section and discussed for the embedded flow parameters. This paper ends with some conclusions given in section
The unsteady incompressible flow of an electrically conducting fluid is governed by the following equations(1)(2)(3)where is the velocity vector, is the density of the fluid, is the pressure,is the the extra stress tensor, is the current density, is the total magnetic field where denotes the applied magnetic field and is the induced magnetic field, is the magnetic permeability, is the electric field and is the electrical conductivity of the fluid.
The extra stress tensor for non-Newtonian Burgers' fluid constitutes the following equation , (4)in whichis the dynamic viscosity,is the first Rivlin Ericksen tensor, is the velocity gradient,is the transpose of the velocity gradient,andare the relaxation and retardation times respectively and is the material constant of Burgers' fluid multiplies the upper second order convected time derivative of defined as(5)where is the material time derivative.
In order to calculate Lorentz force, it is assumed that the polarization effects are zero (, the magnetic field is applied in outward perpendicular direction to the flow and the induced magnetic field is negligible compare to the applied magnetic field under the assumption of small magnetic Reynolds number, is the strength of applied magnetic field. Thus in view of these assumptions and using Eq. (3), the Lorentz force becomes (7)
Thus using Eq. (6), the continuity Eq. is identically satisfied and the momentum Eq. (2) in the absence of a pressure gradient in the flow direction and Eq. (4) after using Eqs. and (7) and having in mind the initial conditions give the following governing equations(8)(9)where is the non-trivial shear stress.
Mathematical Formulation of the Problem
We consider the unsteady incompressible flow of an electrically conducting Burgers' fluid occupying the upper half space of plane over a rigid flat plate. The axis is taken parallel to the flow direction whereas axis is taken normal to the plate. The magnetic field is applied in outward transverse direction to the flow. Initially, we assume that both fluid and plate are at rest. After time the plate begins to oscillate in its own plane and the fluid is gradually moved as shown in Fig. 1.
For such type of motions the governing equations are (8) and (9) with the following initial and boundary conditions(10)(11)where is the characteristic velocity, is the imposed frequency of the velocity of the plate and is the Heaviside step function.
Solution of the Problem
In order to solve the initial and boundary-value problem we consider two different cases and and use the Laplace transform.
Case-I: Solution of the problem for
In order to determine exact solutions for our problem, we substitute into Eq. apply the Laplace transform to Eqs. and and use the initial conditions We find that(19)(20)where is the transform parameter. In view of the boundary conditions, the Laplace transforms and of and have to satisfy the conditions
(22)The solutions of Eqs. and satisfying the boundary conditions are(23)(24)(25)(26)where the subscripts and denote the solutions corresponding to the cosine and sine oscillations of the boundary, respectively.
Case-II: Solution of the problem for
Here the second grade equation has complex roots.
In this section, for the accuracy of results, we consider a limiting case of our solutions. More exactly, we substitute into equations and and recover the solutions(87)(88)obtained by Vieru et al. [40, Eqs. and ]. Similarly, we can also obtained the solutions of Khan et al.  from the present solution as special cases by taking the magnetic parameter Furthermore, the solutions corresponding to Newtonian and Oldroyd-B fluid also appear as the limiting cases of the present solutions.
Results and Discussion
The objective of the present paper is to study the unsteady MHD flow of a Burgers' fluid over an oscillating plate when the relaxation time satisfies the conditions The closed form solutions involve integrals of Bessel functions, terms with only Bessel functions and other integrals are obtained using the Laplace transform technique. These solutions of velocity and shear stress are plotted using the symbolic computational software Mathematica by performing the ordinary numerical integrations. The profiles of velocity fields and shear stresses for both sine and cosine oscillations of the plate are presented in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 for different values of the embedded flow parameters. These parameters include the magnetic parameter, also called Hartmann number fluid parameters and oscillating frequency and dimensionless time
Figs. 2, 3, 4, 5, 6 are drawn so as to show the velocity profiles when the relaxation time satisfies the condition equivalently The influence of the Hartmann number and then of the magnetic field on the fluid motion is shown in Figs. and for and The magnetic field has a significant influence on the velocity field. It is clearly seen from these figures that the velocity of the fluid and the boundary layer thickness decrease if increases for both types of oscillations of the boundary. This is not a surprise as the transverse magnetic field produces a resistance force (Lorentz force) that is similar to the drag force that tends to oppose the flow and to reduce the velocity of the fluid. It is further concluded from the comparison of Figs. (a) and (a) that when the velocity profiles decay early compare to The influence of the parameter on the velocity profile is shown in Fig. The velocity of the fluid is an increasing function of for both types of oscillations of the boundary. However, as expected, for large values of the velocity of the fluid tends to zero.
Figs. 5 & 6 show the periodic nature of the flow. In Fig. 5, the velocity profiles for different values of are shown. It is observed that the velocity is developing and fluctuating around zero. For both types of oscillations of the boundary, the velocity has its maximum value at the boundary with gradual decay in its amplitude of oscillation and tends to zero away from the plate. Fig. depicts the variation of velocity with oscillating frequency This figure displays the periodic response of the flow to the cosine and sine oscillations of the plate. For it is clear that the velocity corresponding to the cosine oscillations of the boundary has its maximum value whereas for the sine oscillations it is zero. This fact also results from the imposed boundary conditions However, for large values of the fluctuation reduces and the velocity approaches zero.
Figs. 7, 8, 9 are displayed for the velocity profile when or equivalently for both the cosine and sine oscillations of the plate. From first two Figs. & , we noticed that the effects of and on the velocity profiles are qualitatively similar to those observed in Figs. for However, these results are different quantitatively. It is further observed from these figures that the velocity profiles decay early for compare to Physically, it is due to the fact that for large values of rheological parameterthe fluid motion retards and the velocity profiles approaches to zero before than for which the velocity changes are more moderately. Fig. shows the variation of velocity for different values of It is found that the velocity and boundary layer thickness decrease when increases. However, it is observed that the decrease in the boundary layer thickness for the cosine oscillations of the plate is more visible than the sine oscillations of the plate.
Figs. 10, 11, 12, 13, 14, 15, 16, 17 are prepared to discuss the variations of the shear stress for both cosine and sine oscillations of the plate. The first three figures (10, 11, 12) are plotted forand the last five (13, 14, 15, 16, 17) are displayed for As expected, the behaviors of the velocity and shear stress with respect to (Figs. & and & ), (Figs. & ), (Figs. & ) and (Figs. & ) are qualitatively the same. Their behavior with respect to (Figs. & 3 & 11 and 7 & 13) are opposite near the plate and the same elsewhere. The velocity of the fluid decreases with respect to in the whole flow domain while the shear stress increases near the plate and decreases everywhere else.
In this paper, we have studied the MHD flow of Burgers' fluid when the relaxation time satisfies the conditions and The governing equations are modelled and the closed form solutions are obtained using the Laplace transform technique. The analytical results are displayed graphically and the effects of various emerging flow parameters on the velocity and shear stress are shown. It is found that the magnetic parameter and the rheological fluid parameters have strong influence on the velocity and shear stress fields. It is observed that for large values of rheological parameterthe fluid motion retards and the velocity profiles approaches to zero early than for which the velocity changes are more moderately. Furthermore, these solutions also show the periodic nature of the flow. The existing solutions in the literature are recovered as a special case of the obtained solutions. Hence we are confident at the accuracy of our presented results. For future studies, we have planned to extend this work to the case when the relaxation time satisfies the condition The present problem can also be extended to the MHD flow of Burgers' fluid over a plate embedded in a porous medium. There are several other directions where the present work can be continued.
Conceived and designed the experiments: IK FA SS. Performed the experiments: IK FA SS. Analyzed the data: IK FA SS. Contributed reagents/materials/analysis tools: IK FA SS. Wrote the paper: IK FA SS.
- 1. Andersson HI (1992) MHD flow of a viscoelastic fluid past a stretching surface. Acta Mechanica 95: 227–230.
- 2. Hammouch Z (2008) Multiple solutions of steady MHD flow of dilatant fluids. Eur J Pure App Math 1: 11–20.
- 3. Siddiqui AM, Irum S, Ansari AR (2008) Unsteady squeezing flow of a viscous MHD fluid between parallel plates, a solution using the homotopy perturbation method. Math Model Anal 13: 565–576.
- 4. Sreeharireddy P, Nagarajan AS, Sivaiah M (2009) MHD flow of a dusty viscous con-ducting liquid between two parallel plates. J Sci Res 1: 220–225.
- 5. Kumar A, Varshney CL, Lal S (2010) Perturbation technique to unsteadyMHD periodic flow of viscous.uid through a planer channel. J Eng Tech Res 2: 73–81.
- 6. Seth GS, Nandkeolyar R, Ansari MS (2010) Unsteady MHD convective flow within a parallel plate rotating channel with thermal source/sink in a porous medium under slip boundary conditions. Int J Eng Sci Tech Res 2: 1–16.
- 7. Ali F, Norzieha M, Sharidan S, Khan I (2011) On accelerated MHD flow in a porous medium with slip condition. Eur J Sci Res 57: 293–304.
- 8. Khan I, Fakhar K, Sharidan S (2011) Magnetohydrodynamic free convection flow past an oscillating plate embedded in a porous medium. J Phys Soc Japan 80: 104401.
- 9. Khan I, Ali F, Sharidan S, Norzieha M (2011) Effects of Hall current and mass transfer on the unsteady MHD flow in a porous channel. J Phys Soc Japan 80: 064401.
- 10. Sweeta E, Vajravelua K, Gordera RAV, Pop I (2011) Analytical solution for the unsteady MHD flow of a viscous fluid between moving parallel plates. Commu Nonlinear Sci Num Simul 16: 266–273.
- 11. Sarpkaya T (1961) Flow of non-Newtonian fluids in a magnetic field. AICHEJ 7: 324–328.
- 12. Ersoy HV (1999) MHD flow of an Oldroyd-B fluid due to non-coaxial rotations of a porous disk and the fluid at infinity. Int J Eng Sci 37: 1973–1984.
- 13. Hayat T, Hutter K (2002) MHD flows of an Oldroyd-B fluid. Math Comput Modell 36: 987–995.
- 14. Khan M, Arshad M, Anjum A (2012) On exact solutions of Stokes second problem for MHD Oldroyd-B fluid. Nuclear Eng Design 243: 20–32.
- 15. Liu Y, Zheng L, Zhang X (2011) Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative. Comput Math Appl 61: 443–450.
- 16. Zheng L, Liu Y, Zhang X (2012) Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Anal Real World Appl. 13: 513–523.
- 17. Zheng L, Liu Y, Zhang X (2011) Exact solutions for MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate, Math Comput Modell. 54: 780–788.
- 18. Hayat T, Hussain M, Khan M (2007) Effect of Hall current on the flows of a Burgers' fluid through a porous medium. Transp Porous Med 68: 249–263.
- 19. Siddiqui AM, Rana MA, Ahmed N (2008) E¤ects of Hall current and heat transfer on MHD flow of a Burgers' fluid due to a pull of eccentric rotating disks, Commu Nonlinear Sci Numer Simul. 13: 1554–1570.
- 20. Siddiqui AM, Rana MA, Ahmed N (2008) Hall effect on Hartmann flow and heat transfer of a Burgers' fluid. Phys Lett A 372: 562–568.
- 21. Khan M (2013) Stokes first problem for an MHD Burgers' fluid. Commun Theor Phys: 99–104.
- 22. Rao AR, Rao PR (1985) MHD flow of a second grade fluid in an orthogonal rheometer. Int J Eng Sci 23: 1387–1395.
- 23. Hayat T, Ahmed N, Sajid M, Asghar S (2007) On the MHD flow of a second grade fluid in a porous channel. Comput Math Appl 54: 407–414.
- 24. Hayat T, Fetecau C, Sajid M (2008) Analytical solution for MHD transient rotating flow of a second grade fluid in a porous space. Non-Linear Anal.: Real World Appl: 1619–1627.
- 25. Hayat T, Khan I, Ellahi R, Fetecau C (2008) Some MHD flows of a second grade fluid through the porous medium. J Porous Med 11: 389–400.
- 26. Khan I, Ali F, Norzieha M, Sharidan S (2010) Exact solutions for accelerated flows of a rotating second grade fluid in a porous medium. World App Sci Journal (Special Issue of Applied Math) 9: 55–68.
- 27. Hameed M, Nadeem S (2007) UnsteadyMHD flow of a non-Newtonian fluid on a porous plate. J Math Anal Appl 325: 724–733.
- 28. Fakhar K, Kara AH, Khan I, Sajid M (2011) On the computation of analytical solutions of an unsteady magnetohydrodynamics flow of a third grade fluid with Hall effects. Comput Math Appl 61: 980–987.
- 29. Hayat T, Fetecau C, Sajid M (2008) On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame. Phys Lett A 372: 1639–1644.
- 30. Hayat T, Qasim M (2010) Influence of thermal radiation and Joule heating on Magne-tohydrodynamic flow of a Maxwell fluid in the presence of thermophoresis. Int J Heat Mass Transfer 53: 4780–4788.
- 31. Zheng L, Liu Y, Zhang X (2011) MHD flow and heat transfer of a generalized Burgers' fluid due to an exponential accelerating plate with the effect of radiation. 62: 3123–3131.
- 32. Khan I, Fakhar K, Sharidan S (2012) Magnetohydrodynamic rotating flow of a general-ized Burgers' fluid in a porous medium with Hall current. Transp Porous Med 9: 49–58.
- 33. Kasiviswanathan SR, Gandhi MV (1992) A class of exact solutions for the magnetohy-drodynamic flow of a micropolar fluid. Int J Eng Sci 30: 409–417.
- 34. Hayat T, Qasim M (2010) Effects of thermal radiation on unsteady magnetohydrody-namic flow of a micropolar fluid with heat and mass transfer. Zeitschrift Fur Natur-forschung A 65: 950–960.
- 35. Ghasemi E, Bayat M, Bayat M (2011) Viscoelastic MHD flow of Walters liquid B fluid and heat transfer over a non-isothermal stretching sheet. Int. J. Phys. Sci. 6: 5022–5039.
- 36. Hayat T, Qasim M, Abbas Z, Hendi AA (2010) Magnetohydrodynamic flow and mass transfer of a Jeffery fluid over a nonlinear stretching surface. Zeitschrift Fur Natur-forschung A 65: 1111–1120.
- 37. Uddin MJ, Khan WA, Ismail AI (2012) (2012) MHD free convective boundary layer flow of a Nanofluid past a.at vertical plate with Newtonian heating boundary condition. PLoS One. 7: e49499 .
- 38. Khan M, Malik R, Anjum A (2011) Analytical solutions for MHD flows of an Oldroyd-B fluid between two side wall perpendicular to the plate. Chem. Eng. Comm. 198 (2011) 1415–1434.
- 39. Fetecau C, Hayat T, Khan M, Fetecau C (2011) Erratum to: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate. Acta Mech 216: 359–361.
- 40. Khan M, Anjum A, Fetecau C (2011) On exact solutions of Stokes second problem for a Burgers' fluid, II. The Cases γ = λ2/4 and γ> λ2/4. Z AngewMath Phys 62: 749–759.
- 41. Vieru D, Hayat T, Fetecau C, Fetecau C (2008) On the first problem of Stokes. For Burgers' fluid, II: The Cases γ = λ2/4 and γ > λ2/4: Appl Math Comput 197: 76–86.
- 42. Christov IC (2010) Stokes' first problem for some non-Newtonian fluids: Results and mistakes. Mech Res Commun 37: 717–723.
- 43. Singh J, Rathee R (2011) Analysis of non-Newtonian blood flow through stenosed vessel in porous medium under the effect of magnetic field. Int J Phys Sci 6: 2497–2506.
- 44. Suri PK, Suri PR (1981) Effect of static magnetic field on blood flow in a branch. Ind J Pure Appl Math 12: 907–918.