Magneto-Rotatory Convection in Couple-Stress Fluid

PARDEEP KUMAR

Department of Mathematics, ICDEOL, Himachal Pradesh University,

Summer-Hill, Shimla-171005 (HP),

INDIA

Abstract: - Background: Thermal convection is the most convective instability when crystals are produced

from a single element like silicon and the thermal instability of a fluid layer heated from below plays an

important role in geophysics, oceanography, atmospheric physics, etc. The flow through porous media is of

considerable interest for petroleum engineers, for geophysical fluid dynamicists and has importance in chemical

technology and industry. Many of the flow problems in fluids with couple-stresses indicate some possible

experiments, that could be used for determining the material constants, and the results are found to differ from

those of Newtonian fluid. Keeping this in view, the present work was to study the effect of a uniform vertical

magnetic field on the couple-stress fluid heated from below in the presence of a uniform vertical rotation

through permeable media. Methodology: The present problem is studied using the linearized stability theory,

Boussinesq approximation, normal mode analysis, and the dispersion relation is obtained. Results: The

stationary convection, stability of the system, and oscillatory modes are discussed. In the case of stationary

convection, the rotation postpones the onset of convection. The magnetic field and couple-stress may hasten the

onset of convection in the presence of rotation while in the absence of rotation; they always postpone the onset

of convection. The medium permeability hastens the onset of convection in the absence of rotation while in the

presence of rotation, it may postpone the onset of convection. The rotation and magnetic field are found to

introduce oscillatory modes in the system which was non-existent in their absence. A sufficient condition for

the non-existence of overstability is also obtained.

Key-Words: - Couple-stress fluid, Porous medium, Thermal convection, Uniform vertical magnetic field,

Uniform vertical rotation

Received: October 22, 2022. Revised: August 11, 2023. Accepted: September 21, 2023. Published: October 6, 2023.

1 Introduction

A comprehensive account of thermal convection

(Be'nard convection) in a fluid layer, in the absence

and presence of rotation and magnetic field has been

summarized in the celebrated monograph by, [1]. The

use of the Boussinesq approximation has been made

throughout, which states that the variations of density

in the equations of motion can safely be ignored

everywhere except in its association with the external

force. The approximation is well justified in the case

of incompressible fluids. The study, [2], has studied

the influence of Rayleigh-number in the turbulent

and laminar region in parallel-plate vertical channels.

The influence of radiation on the unsteady free

convection flow of a viscous incompressible fluid

past a moving vertical plate with Newtonian heating

has been investigated theoretically by, [3]. The study,

[4], has considered the unsteady free convection flow

near the stagnation point of a three-dimensional

body.

The flow through porous media is of

considerable interest for petroleum engineers, for

geophysical fluid dynamicists and has importance in

chemical technology and industry. An example in the

geophysical context is the recovery of crude oil from

the pores of reservoir rocks. The derivation of the

basic equations of a layer of fluid heated from below

in a porous medium, using Boussinesq

approximation, has been given by, [5]. The study of a

layer of fluid heated from below in porous media is

motivated both theoretically and by its practical

applications in engineering disciplines. Among the

applications in engineering disciplines, one can find

the food process industry, chemical process industry,

solidification, and centrifugal casting of metals. The

development of geothermal power resources has

increased general interest in the properties of

convection in porous medium. The study, [6], has

studied the stability of convective flow in a porous

medium using Rayleigh’s procedure. The Rayleigh

instability of a thermal boundary layer in flow

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through a porous medium has been considered by,

[7]. When the fluid slowly percolates through the

pores of the rock, the gross effect is represented by

the well-known Darcy’s law. An extensive and

updated account of convection in porous media has

been given by, [8]. The forced convection in the

fluid-saturated porous medium channel has been

studied by, [9]. The studies, [10], [11], have

remarked that the length scales characteristic of

double-diffusive convecting layers in the ocean may

be sufficiently large so that the Earth’s rotation might

be important in their formation. Moreover, the

rotation of the Earth distorts the boundaries of a

hexagonal convection cell in a fluid through a porous

medium and the distortion plays an important role in

the extraction of energy in the geothermal regions.

The study, [12], explained a double-diffusive

instability that occurs when a solution of a slowly

diffusing protein is layered over a denser solution of

more rapidly diffusing sucrose. The study, [13],

found that this instability, which is deleterious to

certain biochemical separations, can be suppressed

by rotation in the ultracentrifuge. The effect of a

magnetic field on the stability of flow is of interest in

geophysics, particularly in the study of Earth’s core

where the Earth’s mantle, which consists of

conducting fluid, behaves like a porous medium that

can become convectively unstable as a result of

differential diffusion. The other application of the

results of flow through a porous medium in the

presence of a magnetic field is in the study of the

stability of a convective flow in the geothermal

region. The fluid has been considered to be

Newtonian in all the above studies.

The theory of couple-stress fluid has been

formulated by, [14]. One of the applications of

couple-stress fluid is its use in the study of the

mechanisms of lubrication of synovial joints, which

has become the object of scientific research. A

human joint is a dynamically loaded bearing that has

articular cartilage as the bearing and synovial fluid as

the lubricant. When a fluid is generated, squeeze-film

action is capable of providing considerable protection

to the cartilage surface. The shoulder, ankle, knee,

and hip joints are the loaded–bearing synovial joints

of the human body and these joints have a low

friction coefficient and negligible wear. Normal

synovial fluid is a viscous, non-Newtonian fluid and

is generally clear or yellowish. According to the

theory of, [14], couple-stresses appear in noticeable

magnitudes in fluids with very large molecules.

Many of the flow problems in fluids with couple-

stresses, discussed by Stokes, indicate some possible

experiments, that could be used for determining the

material constants, and the results are found to differ

from those of Newtonian fluid. Couple-stresses are

found to appear in noticeable magnitudes in polymer

solutions for force and couple-stresses. This theory is

developed in an effort to examine the simplest

generalization of the classical theory, which would

allow polar effects. The constitutive equations

proposed by, [14] are:

and ,

where

and .

Here , , , , , , V,

and , , , , are stress tensor, symmetric part of

anti-symmetric part of the couple-stress

tensor, the deformation tensor, the vorticity tensor,

the vorticity vector, the body couple, the alternating

unit tensor, the velocity field, and the density and

material constants respectively. The dimensions of

and are those of viscosity whereas the dimensions

of and are those of momentum.

Since the long-chain hyaluronic acid molecules

are found as additives in synovial fluids, [15],

modeled the synovial fluid as a couple-stress fluid.

The synovial fluid is the natural lubricant of the

joints of the vertebrates. The detailed description of

the joint lubrication has very important practical

implications. Practically all diseases of joints are

caused by or connected with a malfunction of the

lubrication. The efficiency of the physiological joint

lubrication is caused by several mechanisms. The

synovial fluid is, due to its content of the hyaluronic

acid, a fluid of high viscosity, near to a gel. The

study, [16], has studied the hydromagnetic stability

of an unbounded couple-stress binary fluid mixture

under rotation with vertical temperature and

concentration gradients. The study, [17], have

considered a couple-stress fluid with suspended

particles heated from below. They have found that

for stationary convection, couple-stress has a

stabilizing effect whereas suspended particles have a

destabilizing effect. The study, [18], has considered

the thermal instability of a layer of a couple–stress

fluid acted on by a uniform rotation, and has found

that for stationary convection, the rotation has a

stabilizing effect whereas couple-stress has both

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stabilizing and destabilizing effects. The study, [19],

have investigated the transport of vorticity in couple-

stress fluid in the presence of suspended particles.

Another study, [20], studied the thermosolutal

convection in a couple-stress fluid in the presence of

uniform rotation.

Darcy’s law governs the flow of Newtonian fluid

through isotropic and homogeneous porous medium.

However, to be mathematically compatible and

physically consistent with the Navier-Stokes

equations, [21], heuristically proposed the

introduction of the term (now known as

Brinkman term) in addition to Darcian term

. But the main effect is through the Darcian

term and the Brinkman term contributes a very little

effect, for flow through porous medium. Therefore,

Darcy’s law is proposed to govern the flow of this

non-Newtonian couple-stress fluid through porous

medium heuristically.

Keeping in mind the importance of geophysics, soil

sciences, groundwater hydrology, astrophysics,

chemical technology, industry, and various

applications mentioned above, the present paper,

therefore, deals with the combined effect of uniform

vertical magnetic field and uniform rotation on the

couple-stress fluid heated from below in porous

medium.

2 Structure of the Problem and Basic

Equations

Consider an infinite, horizontal, incompressible,

electrically conducting couple-stress fluid layer of

thickness , heated from below so that, the

temperatures and densities at the bottom surface

are and and at the upper surface

are and respectively and that a uniform

temperature gradient is maintained.

The gravity field , a uniform vertical

magnetic field and a uniform vertical

rotation pervade the system. This fluid

layer is assumed to be flowing through an isotropic

and homogeneous porous medium of porosity and

of medium permeability .

Let and denote,

respectively, the fluid density, pressure, temperature,

resistivity, magnetic permeability, and filter velocity.

Then the momentum balance, mass balance, and

energy balance equations of couple-stress fluid

through a porous medium [1], [5], [14], are

where

stands for the convective derivative.

The equation of state is

where the suffix zero refers to values at the reference

level . In writing equation (1), use has been

made of the Boussinesq approximation, which states

that the density variations are ignored in all terms in

the equation of motion except the external force term.

The kinematic viscosity , couple-stress viscosity ,

magnetic permeability , thermal diffusivity ,

electrical resistivity and coefficient of thermal

expansion are all assumed to be constants. Here

, is a constant, while

and stand for density and heat capacity of the

solid (porous matrix) material and the fluid,

respectively and .

The steady-state solution is;

Here we use the linearized stability theory and the

normal mode analysis method. Consider a small

perturbation on the steady state solution, let

and denote,

respectively, the perturbations in fluid density ,

pressure , temperature , velocity and the

magnetic field . The change in density ,

caused mainly by the perturbation in temperature,

is given by

Then the linearized perturbation equations of the

couple-stress fluid become

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3 The Dispersion Relation

For obtaining the dispersion relation, we now analyze

the disturbances into normal modes, assuming that

the perturbation quantities are of the form

where are the wave numbers along the

and directions respectively, is

the resultant wave number and is the growth rate

which is, in general, a complex constant.

and stand for the

components of vorticity and current density,

respectively.

Using expression (14), equations (9)-(13) in non-

dimensional form transform to

where we have expressed the coordinates in

the new unit of length and put

and is the Prandtl number,

is the dimensionless medium permeability, is

the magnetic Prandtl number and is

the dimensionless couple-stress viscosity.

Consider the case where both the boundaries are free

as well as perfect conductors of heat, while the

adjoining medium is also perfectly conducting. The

case of two free boundaries, though a little artificial

(realistic in stellar atmospheres), enables us to find

analytical solutions and to make some qualitative

conclusions. The appropriate boundary conditions,

with respect to which equations (15)-(19) must be

solved are

on a perfectly conducting boundary.

(20)

Using the above boundary conditions, it can be

shown that all the even order derivatives of must

vanish for and and hence the proper

solution of characterizing the lowest mode is

where is a constant.

Eliminating and between equations (15)-

(19) and substituting the proper solution (21) in the

resultant equation, we obtain the dispersion relation

where

Equation (22) is the required dispersion relation

including the effects of magnetic field, rotation,

couple-stress, and medium permeability on the

couple-stress fluid heated from below in a porous

medium in the presence of uniform vertical magnetic

field and uniform vertical rotation. In the absence of

rotation , the dispersion relation (22)

reduces to that by, [22].

4 Important Theorems and Discussion

Theorem 1: For stationary convection case:

(i) Rotation postpones the onset of convection i.e.

rotation has a stabilizing effect on the system.

(ii) In the absence of rotation, the magnetic field and

couple-stress parameter postpone the onset of

convection i.e. has a stabilizing effect and in the

presence of rotation, the magnetic field and

couple-stress parameter has both stabilizing and

destabilizing effects on the system.

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(iii) In the absence of rotation, the medium

permeability hastens the onset of convection i.e.

has a destabilizing effect and in the presence of

rotation, the medium permeability has both

stabilizing and destabilizing effects on the

system.

Proof: When the instability sets in as stationary

convection, the marginal state will be characterized

by Putting , the dispersion relation (22)

reduces to

To investigate the effects of rotation, magnetic field,

couple-stress parameter, and medium permeability,

we examine the nature of and

analytically.

(i) Equation (23) yields

It is clear from equation (24) that for stationary

convection, the rotation postpones the onset of

convection in a couple-stress fluid heated from below

in a porous medium in the presence of a magnetic

field.

(ii) It is evident from equation (23) that

It is evident from equation (25) and equation (26)

that for stationary convection, the magnetic field and

the couple-stress postpone the onset of convection in

the absence of rotation and also postpone the onset of

convection in the presence of rotation if

whereas the magnetic field and the couple stress

hastens the onset of convection if

(iii) Equation (23) yields

It is evident from equation (29) that for stationary

convection, the medium permeability hastens the

onset of convection in the absence of rotation and

also hastens the onset of convection in the presence

of rotation if

whereas the medium permeability postpones the

onset of convection if

Theorem 2: The system is stable or unstable.

Proof: Multiplying equation (15) by , the

complex conjugate of , integrating over the range

of z, and making use of equations (16)-(19) together

with the boundary conditions (20), we obtain

where

The integrals are all positive definite.

Substituting , where are real and

then equating the real and imaginary parts of

equation (30), we obtain

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It is evident from equation (32) that is either

positive or negative. The system is, therefore, either

stable or unstable.

Theorem 3: The modes may be either oscillatory or

non-oscillatory in contrast to the non-magneto-

rotatory case.

Proof: Equation (33) yields that may be either

zero or non-zero, which means that the modes may

be either non-oscillatory or oscillatory. In the

absence of rotation and magnetic field, equation (33)

reduces to

and the terms in brackets are positive definite. Thus

which means that oscillatory modes are not

allowed and the principle of exchange of stabilities is

satisfied for the couple-stress fluid heated from

below in a porous medium in the absence of rotation

and magnetic field. It is clear from equation (33) that

the oscillatory modes are introduced due to the

presence of rotation and the magnetic field, which

were non-existent in their absence.

Theorem 4: The system is stable for and

under the condition , the system

becomes unstable.

Proof: From equation (33), it is clear that is zero

when the quantity is multiplied it is not zero and

arbitrary when this quantity is zero.

If , equation (32) upon utilizing equation (33)

and the Rayleigh-Ritz inequality gives

since the minimum value of with respect to

is .

Now, let we necessarily have from

inequality (34) that

Hence, if

then . Therefore, the system is stable.

Thus, under condition (36), the system is stable and

under condition (35) the system becomes unstable.

Theorem 5: The sufficient condition for the non-

existence of overstability is

Proof: Here we discuss the possibility of whether

instability may occur as overstability. Since we wish

to determine the critical Rayleigh number for the

onset of instability via a state of pure oscillations, it

suffices to find conditions for which (22) will admit

solutions with real. Equating real and imaginary

parts of equation (22) and eliminating between

them, we obtain

where we have put and

Since is real for overstability, the values of

of equation (37) are positive. So the

product of the roots of equation (37) is positive but

this is impossible if (since the product of the

roots of equation (37) is and .

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is, therefore, a sufficient condition for the non-

existence of overstability.

It is clear from (38) that is always positive if

i.e.

which imply that

The condition (42) is, therefore, a sufficient condition

for the non-existence of overstability, the violation of

which does not necessarily imply the occurrence of

overstability.

5 Conclusions

With the growing importance of non-Newtonian

fluids in modern technology and industries,

investigation of such fluids is desirable. The study,

[14], formulated the theory of couple-stress fluid.

One of the applications of couple-stress fluid is its

use in the study of mechanisms of lubrication of

synovial joints, which has become the object of

scientific research. A human joint is a dynamically

loaded bearing that has articular cartilage as the

bearing and synovial fluid as the lubricant. The

shoulder, ankle, knee, and hip joints are the loaded-

bearing synovial joints of the human body. Since

long-chain hyaluronic acid molecules are found as

additives in synovial fluids, Walicki and Walicka

modeled the synovial fluid as a couple-stress fluid.

Therefore, an attempt has been made to investigate

the combined effects of uniform vertical magnetic

field and uniform vertical rotation on a layer of

couple-stress fluid heated from below in a porous

medium. The main conclusions from the analysis of

this paper are as follows:

 In the case of stationary convection, the rotation

postpones the onset of convection.

 It is also observed for the case of stationary

convection that in the absence of rotation, the

magnetic field and couple-stress parameter

postpone the onset of convection i.e. has a

stabilizing effect and in the presence of rotation,

the magnetic field and couple-stress parameter

has both stabilizing and destabilizing effects on

the system. Also in the absence of rotation, the

medium permeability hastens the onset of

convection i.e. has a destabilizing effect and in

the presence of rotation, the medium

permeability has both stabilizing and

destabilizing effects on the system.

 It is found that magnetic field and rotation

introduce oscillatory modes in the system which

were non-existent in their absence.

 It is observed that the system is stable for

and under the condition ,

the system becomes unstable.

 The case of overstability is also considered. The

condition

is the sufficient condition for the non-existence of

overstability, the violation of which does not

necessarily imply the occurrence of overstability.

Acknowledgements:

The author is grateful to all three learned referees for

their useful technical comments and valuable

suggestions, which led to a significant improvement

of the paper.

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Contribution of Individual Authors to the

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Policy)

The author contributed to the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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