# MHD instability of pressure-driven flow of a non-Newtonian fluid

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## Abstract

The effect of a uniform vertical magnetic field on the stability of pressure-driven flow of an electrically conducting non-Newtonian fluid in an isothermal channel is numerically investigated using the Chebyshev collocation method. The non-Newtonian fluid is modeled by the couple stress fluid theory, which permits for polar effects and encountered regularly in liquids with very large molecules. It is established that Squire’s theorem is valid and the modified Orr–Sommerfeld equation is derived by considering the fact that the magnetic Prandtl number *P*_{m}, for recognized electrically conducting liquids is too small. The triplets (*R*_{c}, *α*_{c}, *c*_{c}), where *R*_{c} is the critical Reynolds number, *α*_{c} is the critical wave number and *c*_{c} is the critical wave speed, are obtained for different values of couple stress parameter *Λ* and the Hartman number *M*. It is found that increasing *M* has a stabilizing effect on the system while an increase in the couple stress parameter shows twofold deeds. Individual aspects of the kinetic energy spectrum are observed and presented for different parametric values to obtain comprehensive information at the critical state of fluid flow.

## Keywords

Stability Pressure-driven flow Magnetic field Couple stress fluid## List of symbols

*B*_{x},*B*_{y},*B*_{z}Components of magnetic induction \( \vec{B} \)

- \( B_{b} = B_{bx} /R\,P_{m} \)
Basic magnetic induction

- \( c = c_{r} + ic_{i} \)
Wave speed

*c*_{r}Phase velocity

*c*_{i}Growth rate

*E*_{c}Kinetic energy due to couple stress viscosity

*E*_{d}Kinetic energy due to surface drag

*E*_{M}Kinetic energy due to magnetic field

*E*_{s}Kinetic energy due to the shear term

*h*Thickness of the porous channel

- \( M = B_{0} h\sqrt {\sigma /\mu_{d} } \)
Hartmann number

- \( P = p + B^{2} /2\mu \)
Total pressure

- \( P_{m} = \sigma \mu \nu \)
Magnetic Prandtl number

- \( R = u_{0} h/\nu \)
Reynolds number

*u*,*v*,*w*Components of velocity vector \( \vec{q} \)

*u*_{0}Average velocity

*t*Time

- (
*x*,*y*,*z*) Cartesian co-ordinates

## Greek symbols

*α*Horizontal wave number

*β*Vertical wave number

*η*Couple stress viscosity

- \( \varLambda = h\sqrt {\mu_{d} /\eta } \)
Couple stress parameter

*μ*_{d}Dynamic viscosity

*μ*Magnetic permeability

*ν*Kinematic viscosity

*σ*Electrical conductivity

*ϕ*Velocity stream function

*ψ*Magnetic stream function

## Subscripts

*b*Basic state

*c*Critical

## 1 Introduction

The pressure-driven flow in an isothermal channel has preoccupied fluid dynamicists for centuries. The ubiquitous nature of turbulence and its inability to give a universal theory have led many to try conceiving of the transition from laminar state to turbulent as a simple problem. It has many applications in engineering, geophysics, meteorology, oceanography and so on. The studies undertaken by many authors on this subject are very collectible in the book by Drazin and Reid [1].

The stability of hydromagnetic Poiseuille flow has additionally gotten equivalent significance in the literature. Lock [2] initiated this study by assuming *P*_{m} is exceptionally small and showed that the magnetic field has a stabilizing effect on the fluid flow. Later, Potter and Kutchey [3] re-examined the problem without any simplification and observed that the fluid flow becomes more stable as *P*_{m} increases. Takashima [4] re-considered the problem explored by Potter and Kutchey [3] under the appropriate boundary conditions on the magnetic field perturbations. The Lock’s arguments were supported by Lingwood and Alboussiere [5], who solved the problem numerically including all magnetic contributions and their results differed only a few percent from Lock’s results. Using the multi-deck asymptotic approach, the effect of transverse magnetic field on the stability of Poiseuille flow at high Reynolds number was investigated by Makinde [6]. Proskurin and Sagalakov [7] analyzed the effect of longitudinal magnetic field on the stability of Poiseuille flow. Later, Makinde and Mhone [8] extended the work of Takashima [4] to an isotropic porous domain case and subsequently to an anisotropic porous domain by Shankar and Shivakumara [9]. In the past, the stability of conducting fluid flows in a channel in the presence of a uniform magnetic field has attracted the attention of a host of researchers (Takashima [10], Kaddeche et al. [11], Balyaev and Smorodin [12], Adesanya et al. [13], Hayat et al. [14], Hudoba et al. [15], Hudoba and Molokov [16], Bhatti et al. [17], Shankar et al. [18], Ijaz et al. [19], Hassan et al. [20], Shankar et al. [21] and references therein).

The stability of forced convection in a fluid layer has been studied extensively. However, its counterpart for non-Newtonian fluids is in the much-to-be preferred state as these fluids bound to occur in many engineering applications. There exist various types of non-Newtonian fluids and the couple stress fluid, formulated by Stokes [22] based on micro-continuum theories, is one such non-Newtonian fluid which takes into account the size of fluid particles. These fluids are found to be useful in scientific and engineering applications. The effect of couple stresses on the stability of fluid flow was initially studied by Jain and Stokes [23] and its counterpart in a porous medium was examined by Shankar et al. [24]. In a slit channel with metrically astringent walls, the hydromagnetic flow of a couple stress fluid was investigated by Mekheimer [25]. The stability of free (natural) convection in a vertical couple stress fluid layer was investigated by Shankar et al. [18] while later this study was extended by Shankar et al. [26] and Shankar et al. [27] to include a uniform AC electric field and magnetic field, respectively. In addition, many researchers investigated the effect of couple stresses on fluid flows in different contexts (Rudraiah et al. [28], Shivakumar et al. [29], Turkyilmazoglu [30], Shivakumara and Naveen Kumar [31], Srinivasacharya and Rao [32], Ramesh [33], Tripathi et al. [34], Mahabaleshwar et al. [35], Layek and Pati [36], Nandal and Mahajan [37], Reddy et al. [38], Naveen Kumar et al. [39]).

To the best of our knowledge, the simultaneous presence of couple stresses and magnetic field on the pressure-driven stability of fluid flow in a channel is missing in the literature despite the study finds importance in biological fluid flows. The novelty of this paper is to unveil the validity of Squire’s theorem in analyzing the stability characteristics of the couple stress fluid in a channel in the presence of a uniform vertical magnetic field and undertaking energy budget analysis to understand the physical mechanisms involved in the flow transition.

## 2 Mathematical formulation

*h*(Fig. 1). For modeling this flow, a Cartesian coordinate system (

*x*,

*y*,

*z*)is used where the axes

*x*,

*y*and

*z*denote the streamwise, spanwise, and wall-normal coordinates, respectively. The fluid layer is permeated by a uniform vertical magnetic induction \( \vec{B}_{0} = (0,0,B_{0} ) \). Following Stokes [22], Takashima [4] and Shankar et al. [27], the governing linear stability equations in the dimensionless form are

*u*

_{b}and

*B*

_{xb}are real, from Eqs. (13) and (14) it follows that

*Λ*> 2

*M*. We look for normal mode solution of Eqs. (1)–(8) in the form

Substituting Eq. (15) into Eqs. (1)–(8), we obtain

*β*=

*v*=

*B*

_{y}= 0. The transformation involving the Reynolds number suggests that 2D disturbances are more unstable than 3D ones. The velocity stream function

*ϕ*(

*x*,

*z*,

*t*)and the magnetic stream function

*ψ*(

*x*,

*z*,

*t*)are introduced through (neglect the tilde in the subsequent analysis)

*P*from the Eqs. (26) and (27) to obtain

*P*

_{m}for the liquid sodium is of the order 10

^{−5}and for mercury it is of the order 10

^{−7}and hence terms in Eqs. (33) and (34) concerning

*B*

_{b}, and \( \left( {u_{b} - c} \right)\psi \) in Eq. (34) can be neglected, provided that

*R*is not too large. As a result, the above equations lead to a single stability equation

Equation (35) agrees with Takashima [4] for the Newtonian case (*Λ* → ∞).

## 3 Numerical solution

*c*is the eigenvalue and

*X*is the discrete representation of eigenfunctions;

*Δ*

_{1}and

*Δ*

_{2}are matrices of order (

*N*+

*1*). For fastened values of all parameters concerned within, the eigenvalue

*c*which compose a definite non-trivial solution of Eq. (41) is reached. Once

*c*is established, then the method and algorithm explained in Shankar et al. [41] are followed to find the critical parameters.

## 4 Energy analysis

Here, the brackets 〈 〉 represent the volumetric average over the volume of disturbance waves.

## 5 Results and discussion

*Λ*> 2

*M*. We begin the presentation by demonstrating the basic velocity

*u*

_{b}and the basic magnetic induction

*B*

_{b}, which strongly modify the stability of the system. Figures 2a, b and 3a, b display the impact of Hartman number and couple stress parameter on

*u*

_{b}and

*B*

_{b}, respectively. Figure 2a, b, respectively exhibit that raise in the values of

*M*and

*Λ*is to boost the fluid flow. Figure 3a, b show that

*B*

_{b}is concealed with rising

*M*(Fig. 3a) and

*Λ*(Fig. 3b).

*R*,

*α*)—plane showing the division of the plane into stable and unstable zones. The neutral stability curve is obtained when

*M*→ 0 (hydrodynamic case) and \( \varLambda \to \infty \)(absence of couple stresses) (Fig. 4a) from the present code and it can be seen that

*R*

_{c}= 5772.22 and

*α*

_{c}= 1.020, which are the known standard results (Orszag [42]). The computed values of

*R*

_{c}for several values of

*M*as

*Λ*→ ∞ are compared in Fig. 4b with those of Takashima [4] and the outcomes complement with one another. These cross checking of results inspire further assurance in the prophecy of our numerical procedure.

The neutral stability curves for various values of Hartman number and couple stress parameter are exhibited, respectively, in Fig. 4c, d. We observe that increasing *M* is to increase *R*_{c}, indicating that the Hartman number enhances the stability of the system (Fig. 4c), whereas a dual behavior is observed with increasing *Λ* (Fig. 4d). We locate that *R*_{c} move towards higher values of *α* with increasing *M* representing that the cell width diminishes. However, the cell width enlarges with increasing *Λ*.

*R*

_{c}as a function of

*Λ*for various values of

*M*. When

*M*= 0, the curve emanates from

*R*

_{c}= 45695.77 and experiences a steep fall as the value of

*Λ*increases to 6. As the curve approaches to 20, the gradient of the curve changes from negative to positive. The curve progresses with a gradually increasing positive gradient thereafter. The curve of \( M = \sqrt 2 \) has a value of

*R*

_{c}= 11179.47 at

*Λ*= 2.9. The curve faces a negative gradient initially and thereafter progresses with a gradually increasing positive gradient. The curves of

*M*

^{2}= 5, 10, 20 and 25 follow a similar trend throughout the domain of

*Λ*considered. The curve of

*M*= 0 lies well below the curves of

*M*≠ 0 indicating the magnetic field is more effective in stabilizing the system. This is due to the Lorentz force seizes backing the flow and therefore higher Reynolds number is necessary to coerce the flow. It is further seen that

*R*

_{c}decrease with increasing value of

*Λ*initially for each

*M*due to decrease in the couple stress viscosity and thus it has a destabilizing effect on the flow. However,

*R*

_{c}increases at higher values of

*Λ*as it turns to Newtonian case.

Figure 5b establishes a relationship between *α*_{c} and *Λ* for different values of*M*. The curve of *M *= 0 emanates from *α*_{c}= 1.604 when *Λ *= 1 and it progresses with a decreasing value of *α*_{c} for increasing value of*Λ*. The curve of *M*^{2} = 2 runs parallel to the curve of *M *= 0 but originates from (2.9, 1.538). The curve of *M*^{2} = 5 emanates from (4.5, 1.480) and has a constant negative gradient throughout where the value of *α*_{c} decreases with increasing value of*Λ*. The curves of *M*^{2} = 10, 20 and 25 follow a similar trend with a positive gradient at the beginning followed by a gradual change to negative gradient, that is, the value of *α*_{c} increases initially and then gradually decreases as the curve progresses. The corresponding critical wave speeds are illustrated in Fig. 5c as a function of *Λ* for various values of*M*. The curves of *M*^{2} = 0, 2, 5, 10, 20 and 25 follow a similar trend where they experience a gradual fall in their value as they progress. The curves for higher values of *M* originate with a smaller value initially and the curves run almost parallel to each other. All the curves have their maximum value at their respective origin.

*E*

_{kin}/ ∂

*t*= 0. Of all the numerical results produced here for the energy analysis, this phenomenon can be observed at the critical points. It can be seen that for all the values of

*M*considered here,

*E*

_{s}has a propensity to destabilize the flow whereas

*E*

_{d},

*E*

_{c}and

*E*

_{M}stabilize it, throughout the domain of couple stress parameter. However, the contribution of

*E*

_{M}is negligible in the domain. One important finding of this study is that prior to the threshold value of couple stress parameter, the contribution of

*E*

_{c}to stabilize the flow is more than offered by

*E*

_{d}but beyond the threshold value this characteristic reverses. Threshold values of couple stress parameter from where this reversal observed is shown in Fig. 6a–d. Quantitatively, prior to the threshold value of couple stress parameter, contribution of

*E*

_{c}(

*E*

_{d}) in the stability of the fluid flow varies from 78 to 48% (19 to 48%). It is also noted that the contribution of

*E*

_{d}(

*E*

_{c}) is strictly increasing (decreasing) as a function of couple stress parameter.

## 6 Conclusions

The stability of forced convective flow is investigated in a horizontal channel of electrically conducting couple stress fluid under the influence of a uniform vertical magnetic field. The complicated complex stability equation is obtained and solved computationally by using the Chebyshev collocation method after determining that two-dimensional motions are more unstable than three-dimensional motions via an analogue of Squire’s transformation. The existence of a point in the basic flow field, for which the couple stress parameter must be greater than twice the Hartman number, is a necessary condition to be noted in order to exacerbate a disturbance. The Hartman number significantly affects the flow and increase in its value results in the stabilization of the fluid flow while increase in the couple stress parameter shows a twofold behavior. Energy spectrum indicates that kinetic energy due to the shear term (*E*_{s}) has a propensity to destabilize the flow while kinetic energy due to surface drag (*E*_{d}) and couple stress viscosity (*E*_{c}) stabilizes it, throughout the domain of couple stress parameter. Nevertheless, the contribution of kinetic energy due to magnetic field (*E*_{M}) is found to be negligible. Contribution of *E*_{c} on the stability of flow is observed to be more (less) than *E*_{d} prior (beyond) the threshold value of couple stress parameter. Also noted that the contribution of *E*_{d} (*E*_{c}) is strictly increasing (decreasing) as a function of couple stress parameter. Finally, it is observed that the transverse magnetic field has a stabilizing effect on the Poiseuille flow even in the presence of couple stresses.

## Notes

### Acknowledgements

The authors would like to acknowledge the institutions of their work for support. The authors are grateful to the referees for their most valuable comments that improved the paper considerably.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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