Permanent solutions for some oscillatory motions of fluids with power-law dependence of viscosity on the pressure and shear stress on the boundary

1 Introduction

The fact that the fluid viscosity can depend on the pressure was early recognized by Stokes [1] and many experimental studies have certified this dependence (see for instance the authoritative book of Bridgman [2] for the pertinent literature prior to 1931 and Cutler et al. [3], Johnson and Cameron [4], Johnson and Tewaarwerk [5] and Bair and Winer [6] with the therein references for later). The viscosity-dependence of pressure is important in many applications like the food processing, pharmaceutical tablet manufacturing, fuel oil pumping, fluid film lubrication, microfluidics, polymeric materials, geophysics and it is a direct result of gravity. Unfortunately, in the existing literature, there are very few studies containing exact solutions for unsteady motions of fluids with pressure-dependent viscosity in which the gravity to be taken into consideration and the first exact solutions for steady motions of such fluids seem to be those of Hron et al. [7].

Kannan and Rajagopal [8] studied the motion of such fluids between two parallel plates and found that the gravity has a significant influence on the flow characteristics as the viscosity changes with the depth. Le Roux [9] also found that the boundary layers develop in an orthogonal rheometer when slip boundary conditions are taken into consideration. Such boundary layers also develop when such fluids flow along an inclined plane and the gravity effects are taken into consideration (see Rajagopal [10]). Kannan and Rajagopal [11] shown that, due to the gravity, the viscosity of the rock glacier meaningfully varies with the depth. Interesting expressions for permanent solutions (steady-state or long time solutions) of the motion of fluids with pressure-dependent viscosity have been established by Rajagopal [12] and Prusa [13]. Prusa also showed that the starting solutions, that have been numerically obtained, converge to the steady-state solutions presented in terms of Kelvin functions.

The most general solutions for motions of the fluids with exponential or power-law dependence of viscosity on the pressure between two infinite horizontal parallel plates when the gravity effects are taken into consideration have been established by Rajagopal et al. [14] under series form in terms of the eigenfunctions of some suitable boundary value problems. The fluid motion is generated by both plates that are moving in their planes. Some uniqueness and qualitative results regarding the nature of obtained solutions are also provided. Other exact solutions for steady flows of such fluids in cylindrical or spherical domains have been established by Kalagirou et al. [15], respectively Housiadas et al. [16]. A steady extension to the couple stress fluids with exponential dependence of viscosity on the pressure has been presented by Lin et al. [17].

It is worth pointing out the fact that in all motion problems that have been above mentioned the velocity is given on the boundary although in many practical situations what is specified is the shear stress on the boundary (see Renardy [18], [19], [20]). In addition, some polymeric fluids can slip or slide on the boundary and the no-slip boundary condition is not applicable. Consequently, boundary conditions on stresses are meaningful and Renardy [20] showed how well-posed boundary value problems can be formulated. Actually, in Newtonian mechanics force is the cause and kinematics is the effect (see Rajagopal [21] for a detailed discussion on the problem) and prescribing the shear stress on the plate is tantamount to prescribe the force (shear) applied to move it. The first exact solutions for motions of fluids in which the shear stress is given on the boundary have been early enough obtained by Waters and King [22]. During the time a lot of similar solutions have been established (see for instance Jamil and Fetecau [23], Fetecau et al. [24], Zafar et al. [25] and therein references) but none of them for fluids with pressure dependent viscosities.

The purpose of this note is to provide exact expressions for the permanent components of the starting solutions corresponding to some motions with engineering applications of two classes of fluids with power-law dependence of viscosity on the pressure. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate that applies oscillatory shear stresses to the fluid. The obtained solutions are independent of the initial condition but satisfy the boundary conditions and governing equations. To validate the results that have been obtained, three limiting cases are considered and some comparative graphical representations are included. Furthermore, the similar solutions for the motion induced by the lower plate that applies a constant shear stress to the fluid are also determined. In both types of fluids, the dimensionless steady shear stress corresponding to this motion is constant on the whole flow domain although the adequate velocity fields are functions both of the spatial variable y and the pressure-viscosity coefficient α.

2 Constitutive and governing equations

The constitutive equation of incompressible Newtonian fluids with pressure-dependent viscosity is given by (see for instance Rajagopal [12] and [10])

where T is the Cauchy stress tensor, S is the extra stress tensor, A is the first Rivlin–Ericksen tensor, I is the unit tensor, p is the hydrostatic pressure and η(p) is the fluid viscosity which, for the present study, will have anyone of the forms

In the last two relations, μ is the fluid viscosity at the reference pressure p_h while the positive constant α is the dimensional pressure-viscosity coefficient. This coefficient, whose values have been obtained from measurements, is a measure of the pressure dependence of the viscosity of the liquid in elastohydrodynamic lubrication. Making α→0 in Eq. (2), η(p)→μ and Eq. (1) reduces to the constitutive equation of the ordinary incompressible Newtonian fluids. Furthermore, both relations (2) implies η(p)→∞ if p→∞, a feature that has been experimentally proved (see Bridgman [2]).

In the following, we shall determine simple exact expressions for the permanent solutions (steady-state or long time solutions) corresponding to some oscillatory motions of fluids defined by Eqs. (1) and (2)₁ or (2)₂. Let us assume that such a fluid is at rest between two infinite horizontal parallel plates at the distance h apart. At the moment t = 0⁺, the lower plate begins to apply an oscillatory shear stress S cos(ωt) or S sin(ωt) to the fluid. Here, S and ω are the amplitude, respectively the frequency of oscillations. Due to the shear, the fluid is gradually moved and following Prusa [13] and Rajagopal et al. [14] we are looking for a solution of the form

where v is the velocity vector and i is the unit vector along the x-direction of a suitable Cartesian coordinate system whose upward y-axis is perpendicular to the plates.

Substituting Eq. (3) in (1) and the result in the balance of linear momentum, we attain to the following relevant partial or ordinary differential equations

Into above relations, τ(y,t) = S_yx(y,t) is the non-trivial component of the extra-stress tensor S, ρ is the fluid density and g is the acceleration due to the gravity. In order to get Eq. (4)₃, we used the fact that the specific body force is −gj, with j the unit vector along the y-axis. Of course, the continuity equation is identically verified while Eq. (4)₃ implies

where p_h = p(h) is the dimensional pressure at the upper plate.

Now, replacing p(y) from Eq. (5) in (2), introducing the results in Eq. (4)₁ and eliminating τ(y,t) we find the governing equations for the velocity field corresponding to the motion of the two types of fluids with pressure-dependent viscosity, namely

respectively

The non-trivial shear stresses corresponding to these motions of fluids in consideration can be obtained using the relations

In addition, the initial condition

and the following boundary conditions

respectively

have to be satisfied.

Introducing the next non-dimensional variables, functions and parameter

and dropping out the star notation, the governing Eqs. (6) and (7) for the fluid velocity take the dimensionless forms

wherer Re = hV/v (V = hS / μ being a characteristic velocity) is the Reynolds number.

The corresponding initial and boundary conditions are

respectively,

The adequate non-dimensional shear stresses, which are given by

allow us to determine the dimensionless permanent frictional forces per unit area exerted by the fluid on the upper plate.

3 Permanent solutions

It is well known the fact that the starting solutions u_c(y,t) and u_s(y,t) corresponding to oscillatory motions induced by the lower plate that applies shear stresses of the form S cos(ωt), respectively S sin(ωt) to the fluid can be written as sums of the permanent and transient components, namely

Some time after the motion initiation the fluid moves according to the starting solutions u_c(y,t) or u_s(y,t). After this time, when the transients disappear or can be neglected, the fluid motion is characterized by the permanent solutions u_cp(y,t), respectively u_sp(y,t). This is the required time to reach the permanent state. It is important for those who want to eliminate the transients from their experiments. Generally, in order to determine this time for different fluid motions, it is necessary to know exact expressions at least for their permanent solutions because the corresponding starting solutions can be numerically determined.

This is the reason that, in the following, we shall establish exact expressions for the permanent components u_cp(y,t) and u_sp(y,t) corresponding to the motions of the two types of fluids with pressure-dependent viscosity. Their expressions are independent of the initial condition (15) but satisfy the boundary conditions and governing equations. In order to do determine them in a simple way, we introduce the dimensionless complex velocity field

where i is the imaginary unit. In order to determine v_p(y,t) for the oscillatory motions of the two types of fluids with power-law dependence of viscosity on the pressure, we shall use suitable changes of the spatial variable and the method of separating variables.

4 Limiting cases

In order to certify the correctness of results that have been here obtained, as well as to get some physical insight of them, we consider some particular cases whose solutions are well known in the existing literature or can be easily determined and include a few comparative graphical representations.

4.1 Case α = 0 (Ordinary incompressible Newtonian fluids)

The dimensionless permanent velocity fields u_Ncp(y,t) and u_Nsp(y,t) corresponding to incompressible Newtonian fluids performing the same motions as in previous sections have the simple forms (see Javaid et al. [29, Eqs. (42) without porous and magnetic effects])

(47)uNcp(y,t)=−1ωReReal{sh[(1−y)iωRe]ch(iωRe)eiωti},

(48)uNsp(y,t)=−1ωReIm{sh[(1−y)iωRe]ch(iωRe)eiωti},

while the adequate frictional forces exerted by the fluid on the upper plate are given by

(49)τNcp(1,t)=Real{1ch(iωRe)eiωt}, τNsp(1,t)=Im{1ch(iωRe)eiωt}.

Now, we show that the Newtonian solutions u_Ncp(y,t) and u_Nsp(y,t) can be obtained as limiting cases of u_cp(y,t) and u_sp(y,t) given by Eqs. (31), respectively (32) when α→0. Indeed, using the asymptotic approximations (A2) from Appendix, we can easy prove that

(50)ucp(y,t)≈1ωRe(α+1)[α(1−y)+1]8Real{sin{b[1−[α(1−y)+1]34]}cos{b[1−(α+1)34]}eiωt−i},

(51)usp(y,t)≈1ωRe(α+1)(α(1−y)+1)8Im{sin{b[1−[α(1−y)+1]34]}cos{b[1−(α+1)34]}eiωt−i},

for small enough values of the non-dimensional pressure-viscosity coefficient α. Using the Maclaurin series expansions for [1+α(1−y)]3/4 and (1+α)3/4 and the identities (A3) from Appendix and taking the limit of Eqs. (50) and (51) when α→0 we recover Eqs. (47), respectively (48). Furthermore, following the same way as before, it is easy to show that the frictional forces τ_Ncp(1,t) and τ_Nsp(1,t) given by Eqs. (49) can be also obtained as limiting cases of τ_cp(1,t) and τ_sp(1,t) given by Eqs. (33), respectively (34) when α→0.

As regards the dimensionless permanent solutions u_cp(y,t) and u_sp(y,t) given by Eqs. (43) and (44) corresponding to the same oscillatory motions of the second type of fluids with power-law dependence of viscosity on the pressure, we provide Figures 1 and 2. From these figures, as expected, it clearly results the convergence of these solutions to the Newtonian solutions u_Ncp(y,t), respectively u_Nsp(y,t) when the dimensionless pressure-viscosity coefficient α→0. In addition, Figures 3 and 4 show that the diagrams of τ_cp(1,t) and τ_sp(1,t) given by Eqs. (45) and (46) tend to superpose over those of τ_Ncp(1,t), respectively τ_Nsp(1,t) given by Eq. (49)_{1 and 2} when the same parameter α→0. As expected, in both cases the oscillations’ amplitude is the same and the phase difference is obviously observed.

Figure 1:

Profiles of the permanent solutions u_cp(y, t) for three values of the pressure-viscosity coefficient α and u_Ncp(y) given by Eqs. (43) and (47), for ω = π/6 and t = 10.

Figure 2:

Profiles of the permanent solutions u_sp(y,t)for three values of the pressure-viscosity coefficient α and u_Nsp(y)given by Eqs. (44) and (48), for ω = π/6 and t = 10.

Figure 3:

Time evolution of frictional forces τ_cp(1, t)for three values of the pressure-viscosity coefficient α and τ_Ncp(1, t) given by Eqs (45) and (49)₁, for ω = π/6 and Re = 100.

Figure 4:

Time evolution of frictional forces τ_sp(1, t)for three values of the pressure-viscosity coefficient α and τ_Nsp(1, t)given by Eqs. (46) and (49)2, for ω = π/6 and Re = 100.

4.2 Case ω = 0 (flow due to a constant shear stress S on the boundary)

Dimensionless permanent velocity fields corresponding to this motion of the two different types of fluids with η(p)=μ[α(p−ph)+1]1/2 or η(p)=μ[α(p−ph)+1]2, namely

(52)uSp1(y)=2α[1−α(1−y)+1], respectively uSp2(y)=1α[1α(1−y)+1−1],

are immediately obtained solving the ordinary differential equations (see Eqs. (13) and (14))

(53)ddy[α(1−y)+1duSp(y)dy]=0, respectively ddy{[α(1−y)+1]2duSp(y)dy}=0,

with the boundary conditions

(54)duSp(y)dy|y=0=1, uSp(1)=0.

Figure 5 show that, for decreasing values of the oscillations’ frequency ω, the diagrams of the dimensionless velocity field u_cp(y,t) given by Eq. (43) tend to superpose over the profile of the velocity u_Sp2(y) given by Eq. (52)₂ when ω→0. More exactly, the dimensionless velocity field u_Sp2(y) corresponding to the motion induced by the lower plate that applies a constant shear stress S to the fluid can be obtained as a limiting case of u_cp(y,t) given by Eq. (43) when ω→0 and this convergence can be also proved by direct computations. In addition, the fluid velocity in absolute value smoothly decreases from maximum values on the lower plate to the zero value on the stationary plate.

Figure 5:

Profiles of the permanent solutions u_cp(y, t) for three values of the frequency ω and u_Sp2(y) given by Eqs. (43), respectively (52)₂, for α = 0.9 and Re = 100.

Finally, it is worth pointing out the fact the dimensionless shear stresses corresponding to this motion of the two types of fluids with power-law dependence of viscosity on the pressure are constants on the whole flow domain. They are equal with the non-dimensional shear stress applied by the lower plate to the fluid, namely τ_Sp1=τ_Sp2 = 1, although the adequate velocity fields given by Eqs. (52)_{1 and 2} are functions both of the spatial variable y and the dimensionless pressure-viscosity coefficient α.

4.3 Case α = ω = 0 (flow of an incompressible Newtonian fluid induced by a constant shear stress S on the boundary)

Making ω→0 in Eq. (47) or α→0 in anyone of the equalities (52)_{1 and 2}, and bearing in mind the final remark from the previous subsection, we recover the dimensionless permanent velocity and shear stress fields (see Javaid et al. [29])

(55)uNSp(y)=y−1, τNSp=1,

corresponding to the motion of incompressible Newtonian fluids induced by the lower plate that applies a constant shear stress S to the fluid. The sign difference between the present results and those of Javaid et al. [29] appears due to the boundary conditions on the lower plate which are taken with opposite signs. The convergence of the velocity fields given by Eqs. (52)_{1 and 2} to u_NSp(y) is also shown in Figure 6. Indeed, from these figures, it is clearly results that for α→0 the diagrams of the two solutions tend to the same straight line whose equation is given by Eq. (55)₁. Moreover, as expected, the fluid velocity in absolute value is a decreasing function with regard to the pressure-viscosity coefficient α. This is normal because if α increases the viscosity increases and the fluid flows slower.

Figure 6:

Profiles of the permanent solutions u_Sp1(y) and u_Sp2(y) given by Eq. (52)₁, respectively (52)₂ for three values of the pressure-viscosity coefficient α.

5 Conclusions

In this paper, we established exact expressions for the dimensionless permanent solutions corresponding to some unsteady motions of two classes of incompressible Newtonian fluids with power-law dependence of viscosity on the pressure between two infinite horizontal parallel plates. The fluid motion is induced by the lower plate that applies an oscillatory shear stress to the fluid and the gravity effects are taken into consideration. These solutions, which are new in the literature, are presented in simple forms under the polynomial form or in terms of some standard Bessel functions of the first and second kind. They can be useful for those who want to eliminate the transients from their experiments or can be used as tests to verify different numerical methods that are developed to study more complex motion problems. In addition, these solutions have been already used here to determine dimensionless permanent frictional forces per unit area exerted by the fluid on the upper plate. Another relevant physical entity such as the vorticity can be also determined.

In order to demonstrate the correctness of results that have been obtained, as well as to get some physical insight of them, three special cases are considered and some comparative graphical representations are presented in Figures 1–6. From Figures 1 and 2, it clearly results that the diagrams of velocity fields u_cp(y,t) and u_sp(y,t) given by Eqs. (43) and (44) tend to superpose over those of u_Ncp(y,t), respectively u_Nsp(y,t) given by Eqs. (47) and (48) when the dimensionless pressure-viscosity coefficient α→0. More precisely, as it was to be expected, the dimensionless permanent velocity fields corresponding to oscillatory motions of fluids with power-law dependence of viscosity on the pressure converge to those of ordinary Newtonian fluids performing the same motions if α→0. The convergence of the first two solutions u_cp(y,t) and u_sp(y,t) given by Eqs. (31) and (32) to u_Ncp(y,t), respectively u_Nsp(y,t) has been analytically proved.

The convergence of the upper wall shear stresses τ_cp(1,t) and τ_sp(1,t) given by Eqs. (45) and (46) to the corresponding Newtonian solutions τ_Ncp(1,t) and τ_Nsp(1,t) given by Eq. (49)₁, respectively (49)₂ has been graphically proved by Figures 3 and 4 for α→0. In these figures, the oscillatory characteristics of the motion are better underlined. The oscillations’ amplitudes, of the same order of magnitude in both motions, diminish for decreasing values of the pressure-viscosity coefficient α and the phase difference is easy observed. Figure 5 is also devoted to the convergence of the velocity field u_cp(y,t) given by Eq. (43) to u_Sp(y) from Eq. (52)₂ when ω→0. Figure 6 show that the diagrams of both velocity fields given by Eqs. (52)_{1 and 2} tend to a straight line if α→0. This straight line, as expected, represents just the profile of the velocity field u_NSp(y) given by Eq. (55)₁. From this figure, it also results that the fluid velocity in absolute value grows for decreasing values of α. Consequently, the ordinary fluids flow faster as compared to fluids with pressure-dependent viscosity. This is possible since the fluid viscosity diminishes if the coefficient αdecreases.

The time variations of the midplane velocity fields u_cp(y,t) and u_sp(y,t) given by Eqs. (43), respectively (44) are presented in Figures 7 and 8 for two values of Reynolds number Re. From these figures, as well as from Figures 3 and 4, the oscillatory feature of the motion is better captured. The amplitude of the oscillations, which is of the same order of magnitude in both motions, is an increasing function with respect to the dimensionless pressure-viscosity coefficient α and decreases for increasing values of the Reynolds number Re. The phase difference between the profiles of the velocities corresponding to the two motions due to cosine or sine oscillations of the shear stress on the boundary is obviously observed.

Figure 7:

Midplane velocity as a function of time corresponding to u_cp(y, t) given by Eq. (43) for three values of the pressure-viscosity coefficient α, for ω = π/6 and y = 0.5.

Figure 8:

Midplane velocity as a function of time corresponding to u_sp(y, t) given by Eq. (44) for three values of the pressure-viscosity coefficient α, for ω = π/6 and y = 0.5.

Figures 9 and 10 have been presented to provide approximate values for the required time to reach the permanent state for oscillatory motions of the two types of fluids with pressure-dependent viscosities induced by cosine oscillations of the shear stress on the boundary. This is the time after which the fluid moves according to the permanent solutions. More exactly, it is the time after which the diagrams of starting solutions are almost identical to those of the corresponding permanent solutions. As it results from these figures, this time decreases for increasing values of the Reynolds number Re. It also diminishes for increasing values of the pressure-viscosity coefficient α (see Vieru et al. [30]). In addition, as it was to be expected, the convergence of the starting solutions u_c(y,t) (numerical solutions) to the corresponding permanent components u_cp(y,t) can be easy observed from these figures.

Figure 9:

Comparison of the permanent solution u_cp(y, t) given by Eq. (31) with the corresponding starting solution (numerical solution), for α = 0.9,  ω = π/6.

Figure 10:

Comparison of the permanent solution u_cp(y, t) given by Eq. (43) with the corresponding starting solution (numerical solution), for α = 0.9, ω = π/6.

Spatial profiles of the dimensionless starting solutions corresponding to both oscillatory motions of fluids with power-law dependence of viscosity on the pressure are presented in Figures 11 and 12 for fixed values of the parameters α and ω and the Reynolds number Re. From these graphical representations, it is easy to see that the initial condition (15), as well as the boundary condition on the upper plate, is clearly satisfied. Furthermore, the oscillatory characteristics of the fluid motion, as well as the phase difference between motions due to cosine or sine oscillations of the shear stress on the boundary, are more easily observed. For these graphs, “the Maple pde-solver” has been used.

Figure 11:

Spatial profiles of the velocity fields u_c(y, t) and u_s(y, t) (numerical solutions) whose permanent components u_cp(y, t), respectively u_sp(y, t) are given by Eqs. (31) and (32), for Re = 100, α = 0.9, ω = π/6.

Figure 12:

Spatial profiles of the velocity fields u_c(y, t) and u_s(y, t) (numerical solutions) whose permanent components u_cp(y, t), respectively u_sp(y, t) are given by Eqs. (43) and (44), for Re = 100, α = 0.9, ω = π/6.

In Figures 13 and 14, the three-dimensional distribution of the velocity fields u_c(y,t) and u_s(y,t) has been visualized with the help of the two-dimensional contour graphs (see for instance Fullard and Wake [31]) for Re = 100, α = 0.9 and ω = π/6. They are based on the numerical solutions of the partial differential Eqs. (13), respectively (14) with the initial condition (15) and the corresponding boundary conditions. The trajectories of contour graphs are marked with the black color. Light and dark black colors are representing the maximum, respectively minimum values of the solutions while the middle value is coded with medium black. Their values between maximum and minimum are viewed with the gradient of the color between light and dark black. The oscillatory behavior of the solutions is observed by viewing the alternation of the two distinct sets of almost closed trajectories along the time t. Along the y-axis, the color of all curves tends to become middle black for increasing values of y. It means that, as expected, the amplitude of oscillations decreases for increasing values of the spatial variable y.

Figure 13:

Contour profiles of the velocity fields u_c(y, t) and u_s(y, t) (numerical solutions) whose permanent components u_cp(y, t), respectively u_sp(y, t) are given by Eqs. (31) and (32), for Re = 100, α = 0.9, ω = π/6.

Figure 14:

Contour profiles of the velocity fields u_c(y, t) and u_s(y, t) (numerical solutions) whose permanent components u_cp(y, t), respectively u_sp(y, t) are given by Eqs. (43) and (44), for Re = 100, α = 0.9, ω = π/6.

The principal outcomes that can be brought to light from the previous presentation are:

1. Permanent solutions corresponding to some motions of Newtonian fluids with power-law dependence of viscosity on the pressure have been established in simple forms.

2. The amplitude of oscillations of velocity and frictional force diminishes for decreasing values of the pressure-viscosity coefficient α or increasing values of Reynolds number Re.

3. Required time to reach the permanent state has been approximated for motions due to cosine oscillations of the shear on the boundary. It diminishes for increasing values of Re.

4. Similar solutions for the motion induced by a constant shear on the boundary have been also established. Ordinary fluids flow faster than fluids with pressure-dependent viscosities.

The shear stresses corresponding to this motion of both types of fluids are constant over the entire flow field although the adequate velocities depend on the spatial variable y and α.