MHD Accelerated Flow of Maxwell Fluid in a Porous Medium and Rotating Frame

Abstract

The magnetohydrodynamic (MHD) and rotating flow of Maxwell fluid induced by an accelerated plate is investigated. The Maxwell fluid saturates the porous medium. Both constant and variable accelerated cases are considered. Exact solution in each case is derived by using Fourier sine transform. Many interesting available results in the relevant literature are obtained as the special cases of the present analysis. The graphical results are presented and discussed.

1. Introduction

Several fluids including butter, cosmetics and toiletries, paints, lubricants, certain oils, blood, mud, jams, jellies, shampoo, soaps, soups, and marmalades have rheological characteristics and are referred to as the non-Newtonian fluids. The rheological properties of all these fluids cannot be explained by using a single constitutive relationship between stress and shear rate which is quite different than the viscous fluids [1, 2]. Such understanding of the non-Newtonian fluids forced researchers to propose more models of non-Newtonian fluids.

In general, the classification of the non-Newtonian fluid models is given under three categories which are called the differential, the rate, and the integral types [3]. Out of these, the differential and rate types have been studied in more detail. In the present analysis we discuss the Maxwell fluid which is the subclass of rate-type fluids which take the relaxation phenomenon into consideration. It was employed to study various problems due to its relatively simple structure. Moreover, one can reasonably hope to obtain exact solutions from Maxwell fluid. This motivates us to choose the Maxwell model in this study. The exact solutions are important as these provide standard reference for checking the accuracy of many approximate solutions which can be numerical or empirical in nature. They can also be used as tests for verifying numerical schemes that are being developed for studying more complex flow problems [4–9].

On the other hand, these equations in the non-Newtonian fluids offer exciting challenges to mathematical physicists for their exact solutions. The equations become more problematic, when a non-Newtonian fluid is discussed in the presence of MHD and porous medium. Despite this fact, various researchers are still making their interesting contributions in the field (e.g., see some recent studies [1–15]). Few investigations which provide the examination of non-Newtonian fluids in a rotating frame are also presented [1–19]. Such studies have special relevance in meteorology, geophysics, and astrophysics.

To the best of our knowledge, no investigation has been reported so far which discusses the accelerated flows of non-Newtonian fluids in a rotating frame. This is the objective of the present study. Here, we examine the rotating and MHD flow induced by an accelerated plate. Two explicit examples of acceleration subject to a rigid plate are taken into account. Constitutive equations of a Maxwell fluid are used and modified Darcy’s law has been utilized. The exact solution to the resulting problem is developed by Fourier sine transform. With respect to physical applications, the graphs are plotted in order to illustrate the variations of embedded flow parameters. The mathematical results of many existing situations are shown as the special cases of the present study.

2. Formulation of the Problem

We choose a Cartesian coordinate system by considering an infinite plate at 𝑧=0. An incompressible fluid occupying the porous space is electrically conducting by exerting an applied magnetic field 𝐵_∘ parallel to the 𝑧-axis. The electric field is not taken into consideration, the magnetic Reynolds number is small, and the induced magnetic field is not accounted for. The Lorentz force 𝐉×𝐵_∘ under these conditions is equal to −𝜎𝐵²_∘𝐕. Here, 𝐉 is the current density, 𝐕 is the velocity field, and 𝜎 is electrical conductivity of fluid. Both plate and fluid possess solid body rotation with a uniform angular velocity Ω about the 𝑧-axis.

The governing equations are 𝜌𝜕𝐕+()(𝜕𝑡𝐕⋅∇𝐕+2Ω×𝐕+Ω×Ω×𝑟)=−∇𝑝+div𝐒−𝜎𝐵²_∘𝐕+𝐑,div𝐕=0, (1)(2) in which 𝜌 is the fluid density, 𝑟 is a radial vector with 𝑟²=𝑥²+𝑦², 𝑝 is the pressure, and 𝐑 is Darcy’s resistance.

The extra stress tensor 𝐒 for a Maxwell fluid satisfies 𝐓=−𝑝𝐈+𝐒,𝐒+𝜆𝑑𝐒𝑑𝑡−𝐋𝐒−𝐒𝐋^𝑇=𝜇𝐀, (3) where 𝐓 is the Cauchy stress tensor, 𝐈 is the identity tensor, 𝐋 is the velocity gradient, 𝐀=𝐋+𝐋^𝑇 is the first Rivlin-Eriksen tensor, 𝜆 is the relaxation time, 𝜇 is dynamic viscosity of fluid and 𝑑/𝑑𝑡 indicates the material derivative.

According to Tan and Masuoka [4], Darcy’s resistance in an Oldroyd-B fluid satisfies the following expression: 𝜕1+𝜆𝜕𝑡𝐑=−𝜇𝜙𝑘1+𝜆_𝑟𝜕𝜕𝑡𝐕, (4) where 𝜆_𝑟 is the retardation time, 𝜙 is the porosity, and 𝑘 is the permeability of the porous medium. For Maxwell fluid 𝜆_𝑟=0, and hence, 𝜕1+𝜆𝜕𝑡𝐑=−𝜇𝜙𝑘𝐕. (5)

We seek a velocity field of the form: 𝐕=(𝑢(𝑧,𝑡),𝜐(𝑧,𝑡),𝑤(𝑧,𝑡)), (6) which together with (2) yields 𝑤=0. Then, using (1) and (3) into (5) we arrive at 𝜌𝜕𝑢𝜕𝑝𝜕𝑡−2Ω𝜐=−+𝜕𝑥𝜕𝑆_𝑥𝑧𝜕𝑧−𝜎𝐵²_∘𝑢+𝑅_𝑥,𝜌𝜕𝜐𝜕𝑝𝜕𝑡+2Ω𝑢=−+𝜕𝑦𝜕𝑆_𝑦𝑧𝜕𝑧−𝜎𝐵²_∘𝜐+𝑅_𝑦, (7) where 𝜕1+𝜆𝑆𝜕𝑡^𝑥𝑧=𝜇𝜕𝑢,𝜕𝜕𝑧1+𝜆𝑆𝜕𝑡^𝑦𝑧=𝜇𝜕𝜐.𝜕𝑧 (8)𝑅_𝑥 and 𝑅_𝑦 are 𝑥- and 𝑦-components of Darcy’s resistance 𝑅; and 𝑧-component of (1) indicates that 𝑝≠𝑝(𝑧), and modified pressure 𝑝 is 𝑝=𝑝−(𝜌/2)Ω²𝑟².

Invoking (4) and (8) in (7) and then neglecting the pressure gradient, we arrive at 𝜌𝜕1+𝜆𝜕𝑡𝜕𝑢𝜕𝜕𝑡−2Ω𝜐=𝜇²𝑢𝜕𝑧²−𝜎𝐵²_∘𝜕1+𝜆𝜕𝑡𝑢−𝜇𝜙𝑘𝜌𝜕𝑢,1+𝜆𝜕𝑡𝜕𝜐𝜕𝜕𝑡+2Ω𝑢=𝜇²𝜐𝜕𝑧²−𝜎𝐵²_∘𝜕1+𝜆𝜕𝑡𝜐−𝜇𝜙𝑘𝜐. (9) Mathematically when the pressure gradient is ignored, the equation of motion becomes simplified and manageably solvable, and physically the fluid is still in motion as required by the boundary conditions. We have 𝐓=−𝑝𝐈+𝐒, and when the pressure gradient is negligible, the stress tensor 𝐓 is equivalent to the extra stress tensor 𝐒, portraying the fluid motion.

The initial and boundary conditions for a constant accelerated plate are 𝑢()()𝑢=𝜐=0at𝑡=0,𝑧>0,0,𝑡=𝐴𝑡,𝜐0,𝑡=0for𝑡>0,𝑢,𝜕𝑢𝜕𝑧,𝜐,𝜕𝜐𝜕𝑧⟶0,as𝑧⟶∞,𝑡>0, (10)(11)(12) where 𝐴 has a dimension of 𝐿/𝑇².

3. Solution for Constant Accelerated Flow

Setting 𝐹=𝑢+𝑖𝜐 and introducing the following dimensionless quantities: 𝐴𝜉=𝑍𝜈^21/3𝐴,𝜏=𝑡²𝜈^1/3,𝐹𝐺=(𝜈𝐴)^1/3𝐴,𝛽=𝜆²𝜈^1/3,𝑀=𝜎𝐵²_∘𝜌𝜈𝐴^21/3Ω𝜈,𝜔=^1/3𝐴^2/3,1𝐵=𝜙𝑘𝜈²𝐴^2/3,𝑐=2𝑖𝜔+𝑀. (13) The above problem statement reduces to 𝛽𝜕²𝐺(𝜉,𝜏)𝜕𝜏²+()1+𝛽𝑐𝜕𝐺𝜉,𝜏+1𝜕𝜏𝑐+𝐵𝐺()=𝜕𝜉,𝜏₂𝐺(𝜉,𝜏)𝜕𝜉²𝐺()𝐺()=(),𝜉,𝜏>0,0,𝜏=𝜏,𝜏>0,𝜉,0𝜕𝐺𝜉,𝜏𝐺(),()𝜕𝜏=0,𝜉>0,𝜉,𝜏𝜕𝐺𝜉,𝜏𝜕𝜉⟶0as𝜉⟶∞,𝜏>0. (14) After using Fourier sine transform, (14) becomes 𝛽𝜕²𝐺_𝑠𝜂,𝜏𝜕𝜏²+1+𝛽𝑐𝜕𝐺_𝑠𝜂,𝜏+1𝜕𝜏𝑐+𝐵+𝜂²𝐺_𝑠=𝜂,𝜏2𝜋𝐺𝜂𝜏,𝜂,𝜏>0,_𝑠𝜂,0=𝜕𝐺_𝑠𝜂,𝜏𝜕𝜏=0,𝜂>0, (15)(16) in which 𝐺_𝑠(𝜂,𝜏) indicates the Fourier sine transform of 𝐺(𝜉,𝜏).

The solution of (15) satisfying condition (16) can be expressed as 𝐺_𝑠𝜂,𝜏=2𝜋𝜂𝑐+1/𝐵+𝜂²×𝑟₃+𝑐𝑟₂𝑒^𝑟1𝜏−𝑟₄+𝑐𝑟₁𝑒^𝑟2𝜏𝑐+1/𝐵+𝜂²𝑟₂−𝑟₁𝛽+𝜏−1+𝛽𝑐𝑐+1/𝐵+𝜂² (17) or 𝐺_𝑠𝜂,𝜏=2𝜋𝜂𝑐+1/𝐵+𝜂²×𝑟²₁𝑒^𝑟2𝜏−𝑟²₂𝑒^𝑟1𝜏𝑐+1/𝐵+𝜂²𝑟₂−𝑟₁𝛽+𝜏−1+𝛽𝑐𝑐+1/𝐵+𝜂² (18)

with 𝑟₁,𝑟₂=−1+𝛽𝑐±1+𝛽𝑐²−4𝛽𝑐+(1/𝐵)+𝜂²,𝑟2𝛽₃,𝑟₄=1−𝛽𝑐±1+𝛽𝑐²−4𝛽𝑐+(1/𝐵)+𝜂².2𝛽 (19) Inversion of Fourier sine transform in (17) gives 𝐺(𝜉,𝜏)=𝜏𝑒^{−(1/𝐵+𝑐)𝜉}−2𝜋^∞₀𝑟1+𝛽𝑐−²¹𝑒^𝑟2𝜏−𝑟²₂𝑒^𝑟1𝜏𝑟₂−𝑟₁𝛽×𝜂sin𝜉𝜂𝑐+1/𝐵+𝜂²²𝑑𝜂. (20) The above expression for hydrodynamic fluid 𝑀=0 in a nonporous space 1/𝐵=0 is 𝐺(𝜉,𝜏)=𝜏𝑒^{√−(1+𝑖)𝜔𝜉}−2𝜋^∞₀𝑟1+2𝑖𝜔𝛽−²⁵𝑒^𝑟6𝜏−𝑟²₆𝑒^𝑟5𝜏𝑟₆−𝑟₅𝛽×𝜂sin𝜉𝜂2𝑖𝜔+𝜂²²𝑟𝑑𝜂,₅,𝑟₆=−1+2𝑖𝜔𝛽±1+2𝑖𝜔𝛽²−4𝛽2𝑖𝜔+𝜂².2𝛽 (21)(22) Letting 𝜔=0 in (21) and (22), we arrive at 𝐺()2𝜉,𝜏=𝜏−𝜋^∞₀𝑟1−^2𝑎𝑒^𝑟𝑏𝜏−𝑟²_𝑏𝑒^𝑟𝑎𝜏𝑟_𝑏−𝑟_𝑎𝛽sin𝜉𝜂𝜂³𝑟𝑑𝜂_𝑎,𝑟_𝑏=−1±1−4𝛽𝜂².2𝛽 (23) The velocity field 𝐺(𝜉,𝜏) given by (21) is equivalent to that obtained by Fetecau et al. [3, Equation 25] and [5, Equation 31].

Result (20) for a magnetohydrodynamic viscous fluid 𝜆=0 in a porous space is 𝐺(𝜉,𝜏)=𝜏𝑒^{−(1/𝐵+𝑐)𝜉}−2𝜋^∞₀1−𝑒^{(𝑐+1/𝐵+𝜂2)𝜏}𝜂sin𝜉𝜂𝑐+1/𝐵+𝜂²²𝑑𝜂. (24) In the above expression, if we put 𝑐=0 and 1/𝐵=0, we obtain the equivalent solution which was obtained by Fetecau et al. [5, Equation 22], 𝐺()2𝜉,𝜏=𝜏−𝜋^∞₀1−𝑒^𝜂2𝜏sin𝜉𝜂𝜂³𝑑𝜂. (25)

To see the variations of embedded flow parameters in the solution expressions, Figures 1 to 5 have been displayed in order to illustrate such variations for the constant accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.

(a)

(b)

Figure 1 

Velocity profiles for different values of 𝛽.

4. Solution for Variable Accelerated Flow

In this section, the problem statement consists of (9) to (10), (12) and 𝑢(0,𝑡)=ℜ𝑡²,𝜐(0,𝑡)=0for𝑡>0, (26) where ℜ has a dimension of 𝐿/𝑇³.

The flow problem after defining in terms of the dimensionless quantities, ℜ𝜁=𝑧𝜈^31/5ℜ,𝛿=𝑡²𝜈^1/5,𝐹𝑆=𝜈²ℜ^1/5ℜ,𝑃=𝜆²𝜈^1/5,𝐻=𝜎𝐵²_∘𝜌𝜈ℜ^21/5Ω𝜈,𝜓=^1/5ℜ^2/5,1𝐿=𝜙𝑘𝜈³ℜ^2/5,𝐷=2𝑖𝜓+𝐻, (27) becomes 𝑃𝜕²𝑆(𝜁,𝛿)𝜕𝛿²+()()1+𝑃𝐷𝜕𝑆𝜁,𝛿+1𝜕𝛿𝐷+𝐿𝑆()=𝜕𝜁,𝛿₂𝑆(𝜁,𝛿)𝜕𝜁²𝑆();𝜁,𝛿>0,0,𝛿=𝛿²𝑆(),(),𝛿>0,𝜁,𝛿𝜕𝑆𝜁,𝛿𝑆()=()𝜕𝜁⟶0as𝜁⟶∞;𝛿>0,𝜁,0𝜕𝑆𝜁,𝛿𝜕𝛿=0,𝜁>0. (28) Following similar methodology of solution as in the previous section, we obtain 𝑆()=𝛿𝜁,𝛿₂−𝜁[𝛿(1+𝑃𝐷)+𝑃]+𝜁()1/𝐿+𝐷1+𝑃𝐷²1+𝜁1/𝐿+𝐷41/𝐿+𝐷³𝑒^{−(1/𝐿+𝐷)𝜁}+4𝜋^∞₀𝑟₇𝑟₁₀𝑒^𝑟8𝛿−𝑟₈𝑟₉𝑒^𝑟7𝛿+𝑟₇𝑒^𝑟8𝛿−𝑟₈𝑒^𝑟7𝛿𝐷+𝑃𝐷²−1𝐿−𝜂²×𝑟₈−𝑟₇⁻¹𝑃×𝜂sin𝜁𝜂𝐷+1/𝐿+𝜂²³+𝑑𝜂4𝐷𝜋^∞₀𝑒^𝑟8𝛿−𝑒^𝑟7𝛿𝑟₈−𝑟₇𝑃𝜂sin𝜁𝜂𝐷+1/𝐿+𝜂²²𝑑𝜂, (29) where 𝑟₇,𝑟₈=−(1+𝑃𝐷)±(1+𝑃𝐷)²−4𝑃𝐷+1/𝐿+𝜂²,𝑟2𝑃₉,𝑟₁₀=(1−𝑃𝐷)±(1+𝑃𝐷)²−4𝑃𝐷+1/𝐿+𝜂².2𝑃 (30)

The previous expression for hydrodynamic fluid in a nonrotating 𝐷=0 and nonporous space 1/𝐿=0 is 𝑆(𝜁,𝛿)=𝛿²−4𝜋()+4𝛿−𝑃𝜋^∞₀𝜂sin𝜁𝜂𝜂³+4𝑑𝜂𝜋^∞₀𝜂sin𝜁𝜂𝜂⁵+4𝑑𝜂𝜋^∞₀𝑟₁₁𝑟₁₄𝑒^𝑟12𝛿−𝑟₁₂𝑟₁₃𝑒^𝑟11𝛿+𝑟₁₁𝑒^𝑟12𝛿−𝑟₁₂𝑒^𝑟11𝛿𝑟₁₂−𝑟₁₁𝑃×𝜂sin𝜁𝜂𝜂³𝑑𝜂, (31) where 𝑟₁₁,𝑟₁₂=−1±1−4𝑃𝜂²,𝑟2𝑃₁₃,𝑟₁₄=1±1−4𝑃𝜂².2𝑃 (32)

For a magnetohydrodynamic viscous fluid 𝜆=0→𝑃=0 in a porous space, (31) takes the form 𝑆(𝜁,𝛿)=𝛿²𝑒^{−(1/𝐿+𝐷)𝜁}+4𝜋^∞₀1−𝑒^{(𝐷+1/𝐿+𝜂2)𝛿}𝜂sin𝜉𝜂𝐷+1/𝐿+𝜂²³−𝑑𝜂4𝛿𝜋^∞₀𝜂sin𝜉𝜂𝐷+1/𝐿+𝜂²²𝑑𝜂. (33)

In order to see the variations of embedded flow parameters in the solution expressions, Figures 6, 7, 8, 9, and 10 have been displayed in order to illustrate such variations for the variable accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.

5. Results and Discussion

This section concerns the variations of embedded flow parameters in the solution expressions. Hence, Figures 1–10 have been displayed in order to illustrate such variations. We note that Figures 1–5 have been sketched for the constant accelerated flow, whereas Figures 6–10 are shown for the variable accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.

(a)

(b)

Figure 2 

Velocity profiles for different values of 𝑀.

(a)

(b)

Figure 3 

Velocity profiles for different values of 𝐵.

(a)

(b)

Figure 4 

Velocity profiles for different values of 𝜔.

(a)

(b)

Figure 5 

Velocity profiles for different values of τ.

(a)

(b)

Figure 6 

Velocity profiles for different values of 𝑃.

(a)

(b)

Figure 7 

Velocity profiles for different values of 𝐻.

(a)

(b)

Figure 8 

Velocity profiles for different values of 𝜓.

(a)

(b)

Figure 9 

Velocity profiles for different values of 𝐿.

(a)

(b)

Figure 10 

Velocity profiles for different values of 𝛿.

The effect of dimensionless relaxation time parameter 𝛽 on the velocity profile is shown in Figure 1. It is noticed too that the magnitudes of an imaginary component of velocity are decreasing function of 𝛽, and thus 𝑟₁, 𝑟₂, 𝑟₃, and 𝑟₄ in 𝐺(𝜉,𝜏). Figure 2 elucidates the behaviour of 𝑀 on the velocity components. It is observed that the role of 𝑀 on the magnitude of velocity component is qualitatively similar to that of 𝛽. This is in accordance with the fact that the Lorentz force acts as a resistance force to the flow under consideration. On the other hand, the boundary layer thickness decreases when both 𝛽 and 𝑀 increase. The effects of 𝐵 and the velocity components are sketched in Figure 3. Clearly the magnitude of velocity decreases when 𝐵 is increased; that is, when the medium becomes less porous (porosity 𝜙 decreases). Figure 4 plots the effects of the solid body rotation in terms of 𝜔. Figure 4(a) indicates that the real part of velocity is reduced when 𝜔 increases. However, Figure 4(b) shows correspondingly a different behaviour, whereby the velocity first increases and then decreases with respect to 𝜔. It is seen that the variation here is more oscillatory in character. The variation of 𝜏 on the magnitude of velocity components is sketched in Figure 5. It is found that the magnitude of velocity components increases with dimensionless time. Further, it is observed that the variations of parameters in constant accelerated flow (i.e., Figures 1–5) are qualitatively similar (i.e., in Figure 6, the magnitudes of an imaginary component of velocity are decreasing function of 𝛽) to the effects plotted for variable accelerated flow (i.e., Figures 6–10). However, the velocity profiles in constant accelerated flow and variable accelerated flow are not similar quantitatively. Comparison shows that the velocity profiles in variable accelerated flow are larger when compared to those of constant accelerated flow.

6. Conclusions

In this research, the constant accelerated flow and variable accelerated flow of non-Newtonian fluid in a rotating frame are examined. The exact solutions are first established via Fourier sine transform, and then the following observations have been noted through the graphs.(i)The magnitude of velocity components in viscous fluid is greater when compared with Maxwell fluid in case of constant accelerated flow and variable accelerated flow.(ii)The magnitude of velocity components for constant accelerated flow and variable accelerated flow and MHD fluid is less than that of hydrodynamic fluid.(iii)The behaviours of 𝐵 on the magnitude of velocity components are quite opposite to those of 𝑀 and 𝛽.(iv)The salient features of the velocity profile with respect to variations of various embedded parameters in constant and variable accelerated cases are similar in a qualitative sense.

Acknowledgments

This research is partially funded by UTM RUG Vot. no. PY/2011/02418. Dr. Faisal is thankful to UTM for the postdoctoral fellowship.