Effect of rotation and non-uniform temperature gradients on the onset of Marangoni convection in a fluid with suspended particles under microgravity condition

Introduction

The fluids with suspended particles which have microstructure are known as the micropolar fluids. Eringen [1] was the first person to develop the theory of such fluids. The particles or the elements which are present in the micropolar fluid possess both translation and rotational motion. Physically, these fluids may be thought of as fluids with dumb-bell shaped particles or molecules. Muddy fluids like crude oils, paint, toothpaste, blood, polymeric fluid suspension etc are the examples for micropolar fluids. So far there are very less researches done on the effect of rotation and non-uniform temperature gradients in a micropolar fluid under microgravity condition. Microgravity condition is the condition where the effect of gravity is neglected as it is very small [2] . Marangoni convection plays a vital role in the field of science and technology because of its applications. The existence of spontaneous interfacial convection was first reposited by Thomson [3] and Yong et al. [4] . By observing the spreading of an oil droplet on a water surface, Marangoni was able to conclude that the fluid with lower surface tension spreads on a fluid with higher surface tension and so the effect was named after him. Due to wide applications of Marangoni effect in day to day life people have been studying the science of Marangoni convection for a few decades; there are still many characteristics and properties of Marangoni convection that have not been fully understood. As space exploration became a more popular topic, more attention has been paid to the problems encountered in space where drops of immiscible fluid were found to have a great influence on numerous phenomenon such as heat and mass transfer, reaction rates and physical behaviour of fluids [5] . The most effective mechanism to control convection is by rotating the fluid along vertical axis or by maintaining a suitable non-uniform temperature profile. The literature incorporating this particular problem is actually concerned with the non-uniform basic temperature profiles, microrotation and a corresponding study for micropolar fluids. The effect of rotation on the onset of convection is missing in spite of understanding its importance in controlling the convection in many problems in the areas like Science and Technology.

The effect of non-uniform temperature gradients in micopolar fluids is analyzed by Siddheshwar and Pranesh [6] , Pranesh and Riya [7] . Marangoni convection in Micropolar fluids and conducting fluids has been investigated by many authors (Rudraiah [8] , Hashim [9] , Murthy and Ramana Rao [10] and others). The effect of non-uniform temperature gradients on the onset of Marangoni convection has been studied in detail by Rudraiah and Siddheshwar [11] and Siddheshwar and Pranesh [2] . Also the effect of coriolis force on the onset of Marangoni convection has been studied by Hashim and Sarma [12] and Nanjundappa et al. [13] for various Newtonian and Non-newtonian fluids. The aim of this work is to study the effect of rotation and different non-uniform basic temperature gradients on the onset of Marangoni convection in a layer of Non-newtonian fluid. The critical Marangoni number is determined by matrix method using the Galerkin Technique.

Mathematical Formulation

Consider a layer of a micropolar fluid confined between two, infinite horizontal boundaries of depth 'h'. A Cartesian coordinate system is taken with the origin in the lower boundary and z-axis vertically up-wards. Let ∆T be the temperature difference between lower and upper boundaries of the fluid.

Governing Equations

The governing equations of this particular problem are as follows:

Equation of Continuity

Equation of Conservation of Angular Momentum

Equation of Energy

where, 𝑞 ⃗ is the velocity, 𝜔 ⃗ ⃗⃗ is the spin, T is the temperature, p is the pressure, 𝜌 0 is the reference density of the fluid at reference temperature T=T 0 , 𝜁 is the coupling viscosity co-efficient or vortex viscosity, 𝜂 is the shear kinematic viscosity co-efficient, I is the moment of inertia, 𝜆 ′ and 𝜂 ′ are the bulk and shear spin viscosity coefficients respectively, 𝛽 is the Micropolar heat conduction co-efficient, 𝐶 𝑣 is the specific heat, 𝜒 is the thermal conductivity, Ω ⃗⃗⃗ is the angular velocity and t is the time.

The boundary condition on temperature is given as 𝑇 𝑏 = 𝑇 0 + ∆T at z = 0 and 𝑇 𝑏 =𝑇 0 at z = h. The interface at the upper boundary has a temperature dependent surface tension σ(T) ( Pearson [14] ) given by,

Basic State

The basic steady state is assumed to be quiescent and we consider the solution of the form:

where, f(z) is the non-dimensional basic temperature gradient satisfying the condition that ∫ 𝑓(𝑧). 𝑑𝑧

The different non-uniform temperature gradients are shown in the below table: Heating from below

Inverted Parabolic 2(1-z) 6.

where 𝜖 is the thermal depth and 𝛿 is the Dirac function. This type of basic temperature gradients arise due to radiation, sudden heating or cooling etc. [2] .

Linear Stability Analysis

Let the basic state be disturbed by an infinitesimal thermal perturbation. We now have,

where primes denote perturbed quantities and subscript 'b' indicates basic state value. On substituting equation ( 6) in equations ( 1) to (4) and using basic state equations, we get the perturbed state equations as:

Operating curl once on equations ( 8) and ( 9) we get

Operating curl twice on equations ( 8) and ( 9) we get,

Non-dimensionalizing the perturbed state equations using the following definitions,

where, * is neglected for simplicity and six non-dimensional parameters obtained are as follows:

Normal Mode Analysis

The infinitesimal perturbations 𝑊, Ω 𝑧 , 𝜑 𝑟 , 𝜓 𝑟 , 𝜔 𝑧 𝑎𝑛𝑑 𝑇 are assumed to be periodic waves and hence these permit a normal mode solution in the form,

[𝑊, Ω 𝑧 , 𝜑 𝑟 , 𝜓 𝑟 , 𝜔 𝑧 , 𝑇] = [𝑊(𝑧), Ω 𝑧 (𝑧), 𝜑 𝑟 (𝑧), 𝜓 𝑟 (𝑧), 𝜔 𝑧 (𝑧), 𝑇(𝑧)] exp [𝑖(𝑙𝑥 + 𝑚𝑦)]

where, l and m are the dimensionless wave number in the x and y directions respectively and 𝜎 is the growth rate. Substituting (21) in equations ( 15) to (20), noting that the principle of exchange of stabilities is valid and hence assume only the stationary convection. We arrive at the following stability equations.

(1 + 𝑁 1 )(D 2 -a 2 ) 𝜑 𝑟 + 𝑁 1 (𝐷𝜓 𝑟 ) + 𝑁 1 a 2 (𝜔 𝑧 ) + √𝑇𝑎(𝐷𝑊) = 0 (22)

) √𝑇𝑎𝑁 2 𝜓 𝑟 -𝑁 1 (D 2 -a 2 ) 𝑊 -2𝑁 1 Ω 𝑧 = 0 (24) The condition on Ω z is the spin-vanishing at the boundaries.

On multiplying equations from ( 22) to ( 27) by 𝜑 𝑟 , 𝑊, Ω 𝑧 , 𝜓 𝑟 , 𝜔 𝑧 and T respectively and by integrating with respect to z from 0 to 1, and substituting 𝜑 𝑟 = A𝜑 𝑟1 , W = B𝑊 1 , Ω 𝑧 = CΩ 𝑧1 , 𝜓 𝑟 = E 𝜓 𝑟1 , 𝜔 𝑧 = F𝜔 𝑧1 and T = G𝑇 1 , where A, B, C, E, F, G are the trial functions which satisfy the boundary conditions, we get the following equations.

(1 + 𝑁 1 ) < 𝜑 𝑟1 (D 2 -a 2 ) 𝜑 𝑟1 > 𝐴 + 𝑁 1 < 𝜑 𝑟1 (𝐷𝜓 𝑟1 ) > 𝐸 + 𝑁 1 a 2 < 𝜑 𝑟1 𝜔 𝑧1 > 𝐹 +√𝑇𝑎 < 𝜑 𝑟1 (𝐷𝑊 1 ) > 𝐵 = 0 (29)

where, <--> denotes integration with respect to z between 0 and 1.

Equations ( 27) to (32) can be written as YS = 0 (

The trial functions satisfying the equation (35) are:

They satisfy all the boundary conditions except the one given by 𝐷 2 𝑊 + 𝑎 2 𝑀𝑇 = 0 at z = 1, but the residual from this is included in the residual from the differential equations. Performing the integration, we can calculate the critical Marangoni number M c which attains its minimum at 𝑎 𝑐 2 .

Results and Discussion

We consider the effect of rotation, micropolar fluid parameters and non-uniform basic temperature gradients on the onset of Marangoni convection in a micropolar fluid which are represented by Ta, N 1 , N 2, N 3, N 4, N 5 and f(z) respectively. The results obtained in the study are depicted by the Figures 2345and Table-II.

In the case of the temperature profiles heating from below, cooling from above and step function, the least value of M c is attained at thermal depth, 𝜖 = 0.93, 𝜖 = 0.43 and 𝜖 = 0.74 respectively. Fig. 2 is the plot of critical Marangoni number, M c versus coupling parameter, log N 1 for various non-uniform temperature profiles. With increase in N 1 , M c also increases. Increase in N 1 increases the concentration of microelements in the fluid. These microelements consume most of the energy of the system in order to develop the gyrational velocity. This stabilizes the system. Therefore, onset of convection is delayed with increase in N 1 . Fig. 3 is the plot of critical Marangoni number, M c versus couple stress parameter, log N 3 for different non-uniform temperature gradients. Clearly M c decreases with increase in N 3 . Increase in N 3 indicates an increase in the couple stress of the fluid. This increase of couple stress causes the microrotation to be decreased and so the energy required for microrotation will be retained in the system making the system more unstable. It advances the convection. Fig. 4 is the plot of critical Marangoni number, M c versus micropolar heat conduction parameter log N 5 for different non uniform temperature profiles. It is clear that M c increases with increase in N 5 . As N 5 increases, the amount of heat induced into the system due to these microelements will be more. This reduces the heat transfer from bottom to top which stabilizes the system. Hence the convection is delayed with increase in N 5 . Fig. 5 is the plot of critical Marangoni number, M c versus Taylor number, Ta 1/2 for different non-uniform temperature profiles.

From the graph it is evident that M c increases with Ta 1/2 . When rotation is introduced into the system along vertical axis, the motion of the fluid in the horizontal direction will be predominant. This causes a decrease in the vertical motion which in turn makes the system more stable. Hence with the increase in Ta 1/2 the onset of convection is delayed. and N 4 stabilise the system. This shows that the onset of convection is delayed.

Conclusion

It is observed that for the critical Marangoni number, M c the following inequality holds for the six models chosen in the study: M c4 < M c3 < M c6 < M c2 < M c1 < M c5 . i.e., the inverted parabolic temperature is the most stabilizing temperature profile and step function is the most destabilizing temperature profile. Hence by creating conditions for a suitable non-uniform basic temperature gradient, we can also make an appropriate decision in delaying or advancing the convection. It also indicates that the convection in Non-newtonian fluids may be delayed by rotation and by adding micron sized suspended particles.

Acknowledgement

AcknowledgementsThe authors would like to thank the Management and the Department of Mathematics of CHRIST (Deemed to be University) for their support in completing this work.

References

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