IMPACT OF PRESENCE OF SUSPENDED PARTICLES IN STRATIFIED FLUID AND STABILITY UNDER INFLUENCE OF TEMPERATURE
N.K. Gaur1 , Vivek Parkash2
1Professor & Head, Department of Mathematics, M.M.College, Modinagar, Ghaziabad (U.P), India
2Assistant Professor of Mathematics, Dyal Singh College, Karnal (Haryana), India
ABSTRACT -
Stratified fluid flows are the flows through a gravitational field whose origin lies in the changes in density within the field. The atmosphere and the oceans are both examples of stratified medium. The stratification of atmosphere is due to thermal reasons and the salinity induces stratification in oceans. If the fluid is not heterogeneous, the gravity has no effect on the flow of the fluid, in turn it is only responsible for hydrostatic pressure. Heterogeneity will also have minute effects even when force of gravity is absent. But the presence of both heterogeneity and the force of gravity is responsible for very strange and complicated things happening. One such situation is reflected in the instability of RayleighTaylor configuration. Moreover presence of suspended particles in fluid flows makes the fluid much more sensitive to variations and deflections in temperature. The study of these flows is very crucial in several industrial applications and in atmospheric sciences. Moreover, the study of thermal instability of fluid layers with suspended particles is very essential for knowing the nature of these flows.
Keywords and Phrases: heterogeneity, relaxation time, kinetic energy, thermohaline convections.
1. CASE OF THERMAL INSTABILITY OF FLUID LAYERS HAVING SUSPENDED PARTICLES
Admirable work in this field was done by Saffman1 and Veronis. Saffman[1] did a considerable good work in this direction.Heinvestigatedtheeffectofthedustparticlesin terms of two parameters, namely the contribution of dust particles in its motion and the relaxation time Relaxation time denotes the rate of adjustment of velocity of dust particles with respect to the gas velocity change and is a function of size of the each particle individually.
2. CASE WHEN DUST PARTICLES ARE SUFFICIENTLY FINE IN SIZE
Saffman[1] discussed the case when the dust particles are sufficientlyfine,sothattherelaxationtimeismuchsmaller than the characteristic-time scale of the disturbances. The dust particles move in the gas roughly with the same velocity as that of the gas, so that the effect of the dust particles is simply to increase the density of the gas. He alsodiscussedthecaseofcoarsedustparticlesi.e.thecase whentherelaxationtimeiscomparableorgreaterthanthe characteristic time of the disturbances. The physical explanation is that the disturbance has to flow round the particles, and the energy is dissipated. The kinetic energy present in the disturbances is responsible for this energy. This results in decrease in the amplitude of the perturbations. Thus the presence of dust particles, makes thesystemstable.Itisalsonotedthatthestabilizingaction of the coarse dust depends only on the parameter
where df isproportionaltothefractionof'dust particlesand s df isthesizeofthedustparticles.
Thus,if df iskeptconstant,thenanincreaseinthesizeof coarse and irregular shaped dust particles decreases the valueandhencereducesthestabilizingeffect.
3. THERMAL INSTABILITY OF A HORIZONTALLY FLOWING FLUID
Sharma & Sharma[2] studied the problem of thermal instability of a horizontal fluid layer through a porous medium in the presence of suspended particles. They proved that the effect of suspended particles as well as medium permeability was to destabilize the fluid layer.
They also investigated the thermal instability of the layer inthepresenceofrotationandsuspendedparticles.Itwas foundthattherotationhasastabilizingeffectbutthenPES isnotvalid.Finally,theyexaminedtheproblemofthermosolutal convection in a layer of fluid heated form below
andsubjectedtoastaticallystablesolutegradientthrough aporous mediumwithuniformrotation.Itwasalsofound that the medium permeability has got both stabilizing as well as destabilizing effects depending on the rotation parameters.
4. THERMO-SOLUTAL CONVECTION IN PRESENCE OF SUSPENDED PARTICLES
Sharma&Rani[3]studiedthethermo-solutalconvectionin porous medium and analyzed the effect of suspended particles. They showed that for thermal Rayleigh number greater than or equal to solute Rayleigh number, PES is validandthattheoscillatorymodes maycome intoplayif the thermal Rayleigh number is less than the solute Rayleigh number. Also destabilization of the fluid layer is caused by the suspended particles. It was also deduced that the permeability of medium and solute gradient in stable mode respectively destabilize and stabilize the system. Moreover, the rotation stabilizes a certain wave number range in thermo-solutal convection in porous medium, which were unstable in the absence of rotation. Gupta et al generalized, their earlier result given in 1983[4,5] and established two separate semi-circle theorems for Vernonis's and Sterns's view point thermohaline convections with a uniform rotation and magnetic field. These results are valid for all wave numbers and for all combinations of rigid, conducting or insulating boundaries. Sharma and Singh[6] discussed the stability of a mixture of fluid and particles having density in variable form and having viscous nature under the impact of horizontal magnetic field which is present horizontallyandhavingvariableintensity.Thebenchmark thatdetermines thatbothstabilityandinstabilitybehaves independently of what viscosity and suspended particles do. The magnetic field has a stabilizing impact on the systembuttheabsenceofmagneticfieldmakesitunstable.
5. EFFECT OF VISCOSITY ON CONDUCTING PLAZMA
Prakash & Manchanda[7] studied the effect of magnetic viscosityandsuspendedparticlesonthermalinstability of an infinitely conducting plasma in porous medium. It was noted that suspended particles, magnetic viscosity and magnetic field instigate the arrival of oscillatory modes in the system though earlier in their absence oscillatory modes were not noticed. As soon as instability in the system happens to be there in the form of stationary convection, the system noticed stabilizing as well as destabilizing behavior. They further showed that the mediumpermeabilityhasdestabilizing(orstabilizing)and
magnetic field has stabilizing (or destabilizing) nature under certain conditions in the presence of magnetic viscosity,whereasintheabsenceofmagneticviscositythe magnetic field succeeds in stabilizing the thermal instability of plasma particle layer and the medium permeability and the suspended particles have destabilizingeffectonthelayer.
6. CASE OF STRATIFIED FLUIDS AND DISCUSSION ON ITS STABILITY
Thehistory of thisproblem goes back toRayleigh in 1900 who studied the instability of heterogeneous and incompressible fluid having almost zero viscosity and showedthatthenecessaryandsufficientforasystemtobe stable is that the density should decrease upwards everywhereinthefluidregionandifthedensityincreased everywhere, the system is unstable. The direction of gravity is in the vertical direction acting downwards Taylor[8] investigated the problem of instability of the plane interface between two fluids. This is called the Rayleigh-Taylor problem of instability. He showed that if thesurfacetensionisneglected,thenthesystemisstableif the upper fluid is lighter one and is unstable otherwise it also follows from the analysis that the surface tension succeeds in stabilizing a statically unstable arrangement for all sufficiently small wavelengths but the arrangement remains unstable for sufficiently long wavelengths. Hence when there is water above and air below, the maximum value of wavelength that is consistent with stability parameter comes out as 0.0173 m. This established a stabilizing role of surface tension. An experimental demonstration of the development of the Rayleigh-Taylor instability is described by Lewis[9]. For a detailed treatmentoftheproblemoneisreferredtoChandrasekhar in 1961. Miles,in 1961, also proved that for ) (z V , 4
everywhere in the flow domain is the sufficient condition for stability of heterogeneous shear flow. Sachdev and Narayanan[10] studied the instabilities of compressible stratified fluid in horizontal sheared motion. The integrated criteria of instability of heterogeneous shear flows given by Banerjee et. al.[11] is further modified by Banerjee and Gupta[12], by relaxing the very rigid condition on the curvature of the basic profile. Howerd's semi-circular region- in which complex wave velocity of any arbitrary unstable mode must be present, is also reduced.
RESULTS
a. It follows that for fine dust particles the critical Rayleigh number is reduced by the factor (1+ df ), where df is the mass concentration of the dust particles.Thusthefinedustdestabilizestheflow.
b. Theperturbationsdiesindustparticlesandthereisno movement of coarse dust with the gas when perturbationisinducedintheflowbutgoesalongwith the speed of initial flow, so that the net effect of the dust added to the gas flow is analogous to an extra frictional force proportional to the relative velocity. Thus coarse dust will increase the critical Rayleigh number for the existence of neutral disturbances of givenwavelength.
c. It was observed that in this case PES is satisfied and thusthemarginalstateisastationarystate.
d. It was deduced that solute gradient is responsible for theintroductionof oscillatorymodes which were not present when it was absent. The critical Rayleigh number was found to increase with the increase in Stablesolutegradientaswellasrotationparameters.
e. It is clear from the condition that the top heavy arrangementofthefluidisunstable.
f. Miles in 1961 and further in 1963 investigated the effectofsmallperturbationsonthestabilityofparallel flowinaninviscid,incompressiblefluidhavingdensity as variable It was further proved that the dynamic instability of statically stable flows cannot be otherwise exponential, in consequence of which it suffices to consider spatially periodic, travelling waves.
CONCLUSION
Theworkdoneinthispaperconcernsthestudyofthermal instability of fluid layers having suspended particles, case of dust particles having sufficiently fine in size, thermal instability of a horizontally flowing fluid, thermo-solutal convection in presence of suspended particles, effect of viscosity on conducting plazma and case of stratified fluid anditsstability.Thestudyshowsthatthekineticenergyof the disturbances makes the disturbances flow around the particlesandtheenergyisreleased.Itisalsoobservedthat the permeability of the medium and the presence of suspended particles induces instability in the fluid.
Moreoveritisnoticedthateffectofrotationistointroduce stability in fluid layer. The application of magnetic field makes the system stable otherwise it remains unstable. The surface tension makes the fluid behave in a stable mannerprovidedthewavelengthsareinfinitesimallysmall insize.
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