Effects of slip on oscillating fractionalized Maxwell fluid

1 Introduction

It is well-established fact that mechanics of nonlinear fluids provides interesting challenges to the researchers in the field. This is due to various rheological parameters appearing in the constitutive equations of non-Newtonian fluids. These parameters add complexities to the governing equations in terms of nonlinearity and higher order.

Unlike the viscous fluid, the non-Newtonian fluids cannot be described by a single constitutive equation. Hence, different models for such fluids have been suggested in the forms of differential, rate and integral. Literature survey shows that the differential fluids have been studied much in comparison with the fluids of rate and integral types. There is a simplest subclass of rate type fluids known as the Maxwell fluid, which we intend to study here. The constitutive equations, especially the Maxwell model with fractional derivative have been proved to be valuable tool to handle the complex viscoelastic behavior of fluids. Now a days fractional calculus approach are finding increasing use in many areas of science, engineering and technology from the nano- to the macro-scale [1–3]. Some recent investigations dealing with the flows of ordinary and fractionalized Maxwell fluid are discussed by the various authors in References [4–10].

It is known that no-slip condition in polymeric liquids with high molecular weight is not appropriate. This condition also fails in many problems like thin film problems, rarefied fluid problems and flow on multiple interfaces. Fluid motion inside the human body also involves many complex processes in which heat is generated such as hemodialysis, oxygenation of blood, heat conduction in tissues, metabolic processes, etc. A general view of the literature shows that the slip effects on the flows of non-Newtonian fluids has been given not much attention. Especially, polymer melts exhibit a macroscopic wall slip. The fluids exhibiting a boundary slip are important in technological applications, for example, the polishing of artificial heart valves. Experimental observations show that [11–13] non-Newtonian fluids, such as polymer melts, often exhibit macroscopic wall slip, which, in general, is described by a nonlinear and non monotone relation between the wall slip velocity and the traction. A more realistic class of slip flows are those in which the magnitude of the shear stress reaches some critical value, here called the slip yield stress, before slip occurs. In fact, some experiments show that the onset slip and slip velocity may also depend on the normal stress at the boundary [11, 14, 15]. Much of the research involving slip presumes that the slip velocity depends on the shear stress. The slip condition is an important factor in sharskin, spurt, and hysteresis effects, but the existing theory for non-Newtonian fluids with wall slippage is scant. We mention here some recent attempt regarding exact analytical solutions of non-Newtonian fluids with slip effects [16–20].

The purpose of current work is to investigate the oscillating flow over an infinite plate of fractionalized Maxwell fluid with slip effects, which is important due to their practical applications. More precisely, our aim is to find the velocity field and the shear stress corresponding to the motion of a fractionalized Maxwell fluid due to an oscillating plate, where no-slip assumption is no longer valid. The general solutions are obtained using the discrete Laplace transforms. They are presented in series form in term of the H-functions, and presented as sum of the slip contribution and the corresponding no-slip contributions. The similar solutions for ordinary Maxwell and Newtonian fluids, can easily be obtained as special cases of general solutions. Furthermore, the solutions for fractionalized and ordinary Maxwell fluid for no-slip condition are also obtained as a special cases and they are equivalent to the classical Stokes’ first problem of fractionalized and ordinary Maxwell fluid, if oscillating parameter ω = 0. Finally, the influence of the material, slip and fractional parameters on the motion of fractionalized and ordinary Maxwell fluids is underlined by graphical illustrations. The difference among fractionalized Maxwell, ordinary Maxwell and Newtonian fluid models is also highlighted.

2 Partial differential equations governing the flow

The equations governing the flow of an incompressible fluid include the continuity equation and the momentum equation. In the absence of body forces, they are

where ρ is the fluid density, V is the velocity field, t is the time and ▽ represents the gradient operator. The Cauchy stress T in an incompressible Maxwell fluid is given by [4–20]

where −pI denotes the indeterminate spherical stress due to the constraint of incompressibility, S is the extra-stress tensor, L is the velocity gradient, A = L + L^T is the first Rivlin Ericsen tensor, μ is the dynamic viscosity of the fluid, λ is relaxation time, the superscript T indicates the transpose operation and the superposed dot indicates the material time derivative. The model characterized by the constitutive equations (2) contains as special case the Newtonian fluid model for λ→0. For the problem under consideration we assume a velocity field V and an extrastress tensor S of the form

where i is the unit vector along the x-coordinate direction. For these flows the constraint of incompressibility is automatically satisfied. If the fluid is at rest up to the moment t = 0, then

and Eqs. (1)–(3) yield the meaningful equation

where τ(y, t) = S_xy(y, t) is the non-zero shear stress and v = μ/ρ is the kinematic viscosity of the fluid.

The governing equations corresponding to an incompressible Maxwell fluid with fractional derivatives, performing the same motion in the absence of a pressure gradient in the flow direction are [6]

where α is the fractional parameter, and the fractional differential operator so called Caputo fractional operator D_t^αdefined by [21, 22]

and Γ(•) is the Gamma function. In the following the system of fractional partial differential equations (6), with appropriate initial and boundary conditions, will be solved by means of discrete Laplace transform. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method will be used [5–13].

3 Problem formulation

Consider an incompressible Maxwell fluid with fractional derivatives occupying the space lying over an infinitely extended plate which is situated in the (x, z) plane and perpendicular to the y-axis. Initially, the fluid is at rest and at the moment t = 0⁺ the plate start to oscillate in its plane. Here we assume the existence of slip boundary between the velocity of the fluid at the plate u(0, t) and the speed of the plate, the relative velocity between u(0, t) and the plate is assumed to be proportional to the shear rate at the plate. Due to the shear, the fluid above the plate is gradually moved. Its velocity is of the form (3)₁ while the governing equations are given by Eqs. (6). The appropriate initial and boundary conditions are

where H(t) is the Heaviside function and θ is the slip strength or slip coefficient. If θ = 0 then the general assumed no-slip boundary condition is obtained. If θ is finite, fluid slip occurs at the plate but its effect depends upon the length scale of the flow. Furthermore, the natural conditions

have to be also satisfied.

4 Case-I: Solution for Cosine oscillations

5 Case-II: Solutions for sine oscillations

Proceeding in a similar manner as before, we find the corresponding velocity field under the form

and the associate shear stress

for sine oscillations.

6 The special cases

7 Numerical Results and Conclusions

The aim of this communication is to provide exact analytical solutions for fractionalized Maxwell fluid over an oscillating infinite plate where the no-slip assumption between the plate and the fluid is no longer valid, by employing the discrete Laplace transforms. The motion of the fluid is due to the plate that at time t = 0⁺ is start to oscillate with velocity U cos(ωt) or Usin(ωt) in its own plane. Exact analytical solutions are obtained for cosine and sine oscillations of the velocity field u_C(y, t) and u_S(y, t) and the shear stress T_C(y, t) and T_S(y, t) in series form in terms of the H-functions. These solutions, presented as a sum of the slip contribution and the corresponding no-slip contributions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell fluids with and without slip effects are also obtained from general solutions for α→1. For λ →0, the Newtonian solutions are obtained as a special cases from the general solutions. In the special case when θ → 0, the general solution reduces to the results for Stokes’ second problem. Furthermore, making ω = 0, in Eqs. (17) and (24), we recovered [24,25] the classical solutions for Stokes’ first problem for fractionalized and ordinary Maxwell fluid.

In order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field u_C(y, t) and the shear stress T_C(y, t) have been drawn against y and t for different values of t and the material constants ν, λ and a. From all figures it is clear that increasing the slip parameter at the plate the velocity decreases at the plate. Figs. 1 are prepared to show the effect of time on velocity and shear stress profiles with and without slip effects. As expected, it is clear that when slip parameter is nonzero, the velocity and shear stress are smaller in comparison without slip effects. It also noted that the amplitude of velocity and shear stress profiles are sometime decreases or sometime increases, this due to the fact that plate is oscillate about its own plane. Figs. 2 and 3 depicted to show the influence of relaxation time λ and kinematic viscosity ν on the oscillating fluid with and without slip parameter. As expected, the effect of these parameter on the fluid motion is opposite. The velocity u_C(y, t) and the shear stress T_C(y, t), for instance are decreases with increasing values of relaxation time. The impact of frequency of oscillating plate ω on fluid motion is shown in Figs. 4. It is observed that frequency of oscillation and amplitude of velocity and shear stress profiles have inverse relation for both values of slip parameter θ = 0.5 or 0.0. We may also say that the time period of the oscillating plate is directly proportional to the amplitude of velocity and shear stress profiles. The influence of slip parameter decrease the amplitude of velocity field and shear stress profiles as it is clear from Figs. 5. Figs. 6 are prepared to put into action, the fractional parameter a on fluid motion. It is observed that the amplitude of the fluid oscillation increase with regard to fractional parameter a, for θ = 0.5 or θ = 0.0. From Figs. 7, it can easily be seen that the amplitude of the fluid oscillation decays away from the plate and moves to zero. Finally, for comparison, the velocity field u_C(y, t) and the shear stress T_C(y, t) corresponding to the three models (fractionalized Maxwell, ordinary Maxwell, Newtonian) are together depicted in Fig. 8 for smaller and larger slip parameter and the same values of t and of the material constants. It is clearly seen from Figs. 8 that the amplitude of ordinary Maxwell fluid is largest and smallest for Newtonian fluid, either slip effect is small or large. It is also clear from these figures that, the velocity of ordinary Maxwell fluid approaches to zero much faster in comparison with fractionalized Maxwell and Newtonian fluids away from the plate, wether slip effect small or large. The units of the material constants in all figures are SI units.

Fig. 1.

Profiles of the velocity field u_C(y, t) and the shear stress τ_C(y, t) for fractionalized Maxwell fluid given by Eqs. (17) and (24), for U = 1, y = 1, ν = 0.379, μ = 33, λ = 1.5, a = 0.2, ω = 1 and different values of t.

Fig. 2.

Profiles of the velocity field u_C(y, t) and the shear stress τ_c(y, t) for fractionalized Maxwell fluid given by Eqs. (17) and (24), for U = 1, y = 1, v= 0.379, μ = 33, α = 0.2, ω = 1, t = 4s and different values of λ.