1 Introduction

Considerable progress has been made in studying flows of non-Newtonian fluids over 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. Increased interest of many researchers has shown that these flows are important 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. Therefore, models have been developed for the rheological properties of non-Newtonian fluids.

Amongst them, the rate type fluid model^[1] has received great attention. These fluids exhibit the relaxation and retardation phenomena. The Maxwell and Oldroyd-B fluids are amongst the simplest types of such fluids. Finding analytical solutions is quite difficult for such fluids. In spite of several challenges, many researchers have established the analytical solutions regarding these fluids.The exact solution that has beeninvestigated by Fetecau et al. ^[2] takes into account the flow of Oldroyd-B fluid, which is due to the constantly accelerating flat plate placed between two perpendicular side walls. Fetecau et al. ^[3] scrutinized the helical flows of Oldroyd-B fluids. Tan et al. ^[4] scrutinized the exact solution for the flow of the fractional Maxwell model induced by impulsive motion of a plate, when the fluid is placed between two parallel plates. Li et al. ^[5] presented the exact solution of the problem of helical flows while taking heated generalized Oldroyd-B fluid in a porous medium. Analytical solutions were determined for the Oldroyd-B fluid caused by oscillations of an infinite flat plate, oscillating pressure gradient and pressure jumps by Aksel et al. ^[6] . Unsteady flow of a generalized Oldroyd-B fluid under the influence of uniform magnetic field was investigated by Zheng et al. ^[7] .Vieruetal.^[8] examined the flow of the generalized Oldroyd-B fluid, taking place due to the plate which is accelerating constantly. Stokes’ first problem combined with modified Darcy’s law was analyzed by Tan and Masuoka^[9] . Exact solution for the magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid passing through a porous space was investigated by Khan et al. ^[10] . However, these models cannot predict the rheological properties of some food products like cheese. In 1935, onedimensional rate type fluid model which is named as the Burgers model ^[11] was considered in the thermodynamic framework. Later, this work was extended by Krishnan and Rajagopal ^[12] for three-dimensional flow. The Burgers fluid model is used to explore the properties of earth mantle. This model is also used to describe the behavior of asphalt concrete^[13] . In addition, the Burgers model is sometimes used to describe other geological structures, such as Olivine rocks ^[14] and the propagation of seismic waves in the interior of the earth ^[15] . Here, we cite some studies ^{[16, 17, 18, 19, 20]} related to the Burgers fluid. Moreover, MHD flow involving such fluids has promising applications related to the development of energy generation, astrophysics, and geophysics fluid dynamics. Recently, the theory of MHD has received great attention (see Refs. ^{[21, 22]} and the references therein). The effects of transverse magnetic field in the porous space over the unsteady non-Newtonian fluids were analyzed by several researchers ^{[23, 24, 25, 26]} .

Motivated by the above mentioned studies, this work presents an MHD flow of the generalized Burgers fluid passing through a porous medium due to the oscillation of an infinite rigid plate. The influence of transverse magnetic field and modified Darcy’s law on the flow is considered. The resulting expressions of velocity distribution and shear stress can be described as a combination of steady-state and transient solutions. The effects of various physical parameters are discussed through graphical illustrations. 2 Problem formulation

The Cauchy stress tensor T in the generalized Burgers fluid is defined by ^{[16, 17, 18, 19, 20]}

where S denotes extra-stress tensor, p denotes the pressure, I denotes the identity tensor, μ denotes the dynamic viscosity, A₁ = L + L^T denotes the first Rivlin-Ericksen tensor with L as the gradient of velocity, λ₁ and λ₃( ≤ λ₁) represent time of relaxation and retardation, respectively, λ₂ and λ₄ represent generalized Burgersfluid material parameters, and δ/δ_t represents the upper convected time derivative defined as in which d/dt represents the material time derivative.

The governing equations for unsteady flow of incompressible fluid are expressed as follows:

where V denotes the velocity, ρ represents the fluid density, σ represents the electrical conductivity of the fluid, Rdenotes Darcy’s resistance, and B₀ represents the magnitude of applied magnetic field, whereas the induced magnetic field is ignored by choosing a small magnetic Reynolds number^[7].

Velocity and stress fields for the flow analysis are expressed as follows:

where i represents the unit vector in the direction of x. The velocity field (5) automatically satisfies the continuity equation (3).

Substitute (5) into (1), and use the initial conditions

which yields S_yy = S_yz = S_zz = S_xz = 0 and where S_xy is the tangential stress.

In view of Ref. ^[22], we have the following relation ofRfor a generalized Burgers fluid:

where k represents porous medium permeability, and φ represents the porosity.

By substituting (5) into (4), with (7) and (8) and in the absence of pressure gradient in the flow direction, the governing equation becomes

where ν(=μ/ρ) represents the kinematic viscosity of the fluid. 3 Geometry of problem

The physical model under consideration with the coordinate system is shown in Fig. 1.

4 Problem statement and solution method

Let us consider an incompressible and electrically conducting generalized Burgers fluid. The fluid occupies the porous space above the flat plate perpendicular to they-axis and permeated by an applied magnetic field B₀ normal to the flow. The plate starts to oscillate with velocit y UH(t) cos(wt) or UH(t) sin(wt) with H(·)for t > 0 as the Heaviside unit step function and U as the amplitude of plate velocity. Fluid particles move above the plate due to shear stress. The governing equation of the problem is (9) with the associated initial and boundary conditions

In the non-dimensional form, we can write the above problem as and the asterisks have been omitted for simplicity. 4.1 Calculation of velocity field

In order to obtain the exact solution related to the flow for both small and large times, Fourier sine and Laplace transforms ^[27] are used. Thus, multiplying (13) by sin(yξ)onboth sides, integrating the resulting expression with respect to yfrom 0 to infinity, and utilizing (14)-(16), we find that

while the initial conditions take the form Applying the Laplace transform on (18) and (19) subject to the (20), we obtain where Us(ξ, r) represents the Laplace transform of u_s(ξ, t), rrepresents the transform parameter, and

Rewrite (21) and (22) in a simpler form as

where (24) and (25) can also be written as where Using the inverse Laplace transform on (23), one obtains where Inverting (28) and (29) employing Fourier’s sine formulae ^[27] , we can express the starting solu-tions as Using the relations (A1) of Appendix A into (30) and (31), we get the following simplified expressions for the starting solutions:

The starting solutions, i.e., (33) and (34) are expressed as a combination of the steady-state and transient solutions. The steady-state solutions are

4.2 Calculation of shear stress

To compute the shear stress, we use the relation (7) in the non-dimensional form as follows:

The Laplace transform reduces (37) to the following form: where τ(y, r) represents the Laplace transform of τ(y, t), and the image function U(y, r)=(u(y, t)) has been obtained through (21) and (22) by applying the inverse Fourier sine transform as follows:

Substituting (39) and (40) into (38), we reach at the following expressions:

or in a simpler form, we can write as where

Inverting (43) by means of the Laplace transform and following the same way as for the velocity field, we find the following expressions for the tangential stress:

where we have used notations 5 Results and discussion

To study the significant physical effects of the obtained results, the impact of the material parameters on the fluid motion is highlighted by graphical illustration of the velocity profiles for the flow due to the sinusoidal oscillations of the infinite plate. To illustrate the difference, we depict the velocity profile for both cases, i.e., cosine oscillations and sine oscillations. The numerical results are plotted for various values of timet, M, permeability parameter K, and the rheological parametersβandγof Burgers and generalized Burgers fluids.

Figure 2 compares the velocity profiles for various values of time for cosine as well as sine oscillations of the boundary, respectively. It is noted that similar amplitudes and a phase shift persevere for all times in the case of cosine as well as sine oscillations. Here, we can observe that as the bottom plate is set into motion, the velocity near the bottom plate is developing and fluctuating around zero with the same frequency as the plate. Additionally, the fluid oscillation has maximum amplitude adjacent to the plate at y = 0 and reduces far away from the plate and approaches zero.

Figure 3 presents the profiles of the velocity for various values of the magnetic parameter M for both cosine as well as sine oscillations of the boundary, respectively. Since magnetic field is applied in the transverse direction, it is a force which resists the flow. Therefore, by increasing the value ofM, the amplitude of the oscillation tends to decrease. Also, it is scrutinized that the amplitude of oscillation is larger for the hydrodynamic case (M = 0) when compared with the hydromagnetic case (M ≠ 0). The effects of the permeability parameterKare depicted in Fig. 4. It has quite opposite effect on the velocity profile compared with that of the magnetic parameterM. It is noted that with the increasing Kthe velocity field increases.

Figure 5 presents the effect of material parameter β onthevelocityprofileforbothcosine as well as sine oscillations of the boundary, respectively. We can clearly see from the figure that the velocity decreases slightly by increasing the parameter β.Further, the effect of the rheological parameter γof the generalized Burgers fluid is shown in Fig. 6 for cosine as well as sine oscillations of the boundary, respectively. It is noted that an increase in the rheological parameterγof generalized Burgers fluid yields an effect opposite to that of the parameter β. From these figures, it is noticed that the profile of the velocity for the sine oscillation is more sensitive compared with the cosine oscillations of the boundary. Additionally, Fig. 7 delineates the influence of the magnetic parameterMon the shear stress. It is quite clear from this figure that for the larger values of M, the magnitude of shear stress decreases for both cosine as well as sine oscillations. Also, we can compare the hydromagnetic (M ≠ 0) and hydrodynamic (M = 0) cases. It might be seen from this figure that the amplitude of oscillation is larger for the later case.

6 Conclusions

In this article, the problem of an MHD flow through a porous medium involving generalized Burgers fluid has been discussed for two types of oscillations of the plate, i.e., cosine and sine oscillations. The governing equations are modeled by employing the modified Darcy’s law developed for the generalized Burgers fluid. Combined Fourier sine and Laplace transforms are employed to construct the results for the flow. These results, expressed as a combination of the steady-state and transient parts, explain the flow of the fluid after some time of the initiation. After some time, when the transient solutions vanish, the motion of the fluid is called steadystate, which does not depend on initial conditions. Considering the derived results, we have analyzed the influence of the various parameters on velocity profile of the fluid. The obtained results indicate that

(I) With the increase in the magnetic parameter M, the magnitude of velocity distribution is decreased, while the opposite effect could be seen for permeability parameter K for cosine as well as sine oscillations.

(II) For cosine as well as sine oscillations, the magnitude of velocity profile is decreased with the increasing value of the material parameter β, whereas quite opposite behavior might be seen for the rheological parameter γ.

(III) The magnitude of the shear stress is decreased for larger values of M for both cosine and sine oscillations.