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| Flow of A Dusty Gas Through A Hexagonal Channel With Volume-Fraction | |||||||
| Paper Id :
16199 Submission Date :
19/07/2022 Acceptance Date :
22/07/2022 Publication Date :
25/07/2022
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| Abstract | In this paper, we have considered the unsteady flow of a viscous incompressible dusty gas taking into consideration the volume-fraction through a hexagonal channel of uniform cross-section under the influence of an arbitrary time varying pressure gradient. The change in velocity profiles of the gas and particles with time and volume fraction φ for a constant pressure gradient have been shown graphically and in tabular form. The upper limit of φ is taken as 0.1 in order to maintain the Saffman’s Model. | ||||||
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| Keywords | Flow of A Dusty, Hexagonal Channel, Volume-Fraction. | ||||||
| Introduction |
The flow behaviour of particulate suspensions ( The suspended matter may consist of solid particles, liquid droplets, gas bubbles or some combination of the three) is very important in engineering problems concerned with atmospheric fall out, cooling, sedimentation, batch settling, air craft icing, lunar ash flows, dust collection, acoustics, nuclear reactor, rain erosion of guided missiles aerosol and paint spraying, performance of solid fuel rocket nozzles, fluidization (flow through packed beds), combustion and many others, In the studies of pulsatile flow of blood, it has been considered as binary system of plasma (fluid phase) and blood-cells (particle phase). |
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| Aim of study | The purpose of the present investigation is to extend the analysis of Gupta and Varshney [2] by taking volume–fraction of dust particles into consideration. Explicit expression for both the gas and particle velocities have been obtained in exact form by using integral transform techniques. One special case when the flow takes place from rest due to constant pressure gradient has been discussed in detail and represented graphically and in tabular form for various values of volume fraction φ and time t. It is interesting to note that velocities increase with the increase of φ and the gas moves faster than the particles. It should be noted that upper limit of φ is 0.1 and should be small enough so that Saffman’s model exists. | ||||||
| Review of Literature | Using basic equations for the flow of dusty viscous fluid due to Saffman[1], a number of investigations have been made by many researchers with different initial and boundary conditions neglecting the volume -fraction of the dust particles. In a previous paper, Gupta and Varshney[2] have considered the similar problem with usual simplifying assumptions. Some times these assumptions are not justified because of high fluid densities or a high particle mass-fraction. Under these circumstances the volume fraction of the particles may become significantly large so that it can not be neglected. Rudinger[3] has show that the error introduced in a flow analysis of gas particle mixtures by neglecting the finite particle volume ranges from significant to large. Nayfeh[4] formulated the equations of motion of the fluid and dust particles taking the volume–fraction of the dust particles into account. Due to this reason Gupta[5], Nag and Dutta[6], Usha[7] , Gupta and Agarwal[8], Gupta et.al.[9] etc. have studied the problems on flow of a dusty fluid through channels of various cross-section with volume-fraction of dust particles. As per the researcher's knowledge no latest review has been found for this study. |
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| Main Text | 2. Equations and Formulation of The Problem Using a rectangular Cartesian coordinate system (x,y,z) such that z-axis is along the axis of the hexagonal channel and the cross-section of the channel is formed by the straight lines:
The fluid and dust particle velocities u(x,y,t) and v(x,y,t) respectively, are in z-direction. Following Rudinger [3] and Nayfeh [4] the equations of motion in Cartesian coordinates, taking into consideration the volume-fraction of the dust particles into account can be expressed as
where φ is the volume occupied by the particles per unit volume of the mixture (i.e. volume –fraction of the dust particles), N0 is the number density of the particle, m the mass of each particle, K the Stoke’s resistance coefficient. The other symbols have their usual meaning. Introducing the following non-dimensional quantities:
x=ax*, y=ay*, z=az*, pa2= 2p*, t =a2t*, au= u*, av= v*, the equations (2) and (3) after dropping the asterisks become:
the fluid is independent of z. Non-dimensional initial and boundary conditions are :
Under the transformations (9), equations (4) and (5) transform into :
subject to the initial and boundary conditions, (Gupta and Varshney[2])
Due to symmetry of motion about x1=0, x2=0 and x3=0, we shall consider the motion in the region x1≥0, x2≥0 and x3≥0, so accordingly the boundary conditions(13) become:
3. SOLUTION OF THE PROBLEM: To solve the problem we use the technique of integral transforms, for it, let us define finite Fourier cosine transforms
It can easily be shown that the inversion formulae for the transforms defined by (15) to (17) are given by:
It is to be noted that on account of u and v being an even functions of x1, x2 and x3, the finite Fourier Sine transforms be zero. Multiplying (10) and (11) by
To equations (22) and (23), we apply Laplace transforms under the transformed initial condition U=0=V at t=0, to get
Now first solving the equations (27) and (28), we get
Now invert the Laplace transform by applying convolution theorem (Sneddon [10]) and inversion formulae for cosine transforms, we get:
Equations (33) and (34) represent the most general solution of the velocities of the gas and dust particles respectively under the influence of arbitrary time varying pressure gradient and sufficiently well behaved for its Laplace transform to exist. It can be easily seen that if =0 the solution represented by (33) and (34) is in complete agreement with Gupta and Varshney[2].
1. PARTICULAR CASE- Constant Pressure Gradient: In this case put f(t)=C, where C is an absolute constant, in equation (33) and (34) and on simplifying, we get:
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| Result and Discussion | From these studies we conclude that: (i) The gas moves faster than the particles and as time increases from the start of the motion, the velocities approach their steady state in all the cases. (ii) With the increase of volume fraction, there is a further increase in these velocities. Table – 1 u/C versus y for various values of when:
Figure-1 Velocity profile of gas under a constant pressure gradient C, for various values of t and x, when
Figure-2 Velocity profile of dust particles under a constant pressure gradient C, for various values of t and x,
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| Conclusion | Tables 1-4 give the values of velocities of gas (u/C) and particles (v/C) versus y for different values of φ and x when γ=1.666,f=0.2 and t=1.0 under constant pressure gradient. From tables it is clear that both velocities increase with the increase of φ. In figures 1 and 2, we have plotted the velocity profiles for gas and particles versus y under constant pressure gradient for various values of t and x when γ=1.666,f=0.2 and φ=0.04. It is evident from the figures that the velocities of the gas and particles both increase with the lapse of time. From tables and figures, it is also noted that both the velocities are maximum on the axis of the channel and the gas moves faster than the particles. If the masses of the particles are small, then influence on the gas is reduced and in the limit as m→0, the gas becomes an ordinary viscous. | ||||||
| References | 1. P.G. SAFFMAN, On the stability of laminar flow of a dusty gas, J. Fluid Mech, 13 (1962), 120.
2. M.GUPTA and O.P. VARSHNEY, Flow of a dust gas through hexagonal channel, J.Engg. Appl. Sci. (Printed in USA), 2 (1983),277.
3. G.RUDINGER, Some effects of finite particle volume on the dynamics of gas particle mixtures, A.I.A.A.J, 3 (1965), 1217.
4. A.H. NAYFEH, Oscillating two-phase flow through a rigid tube, A.I.A.A.J., 4 (1966),1868.
5. R.K. GUPTA, Unsteady flow of a dusty fluid through ducts with volume-traction, Acta Ciencia Indica, VII, 2 (1981), 127.
6. S.K. NAG and N.DATTA, Flow of a dusty fluid through a rectangular channel, Ind. Jour. Tech., 27 (1989), 377.
7. USHA, Unsteady MHD flow of a dusty fluid with volume-fraction of fluid particles, Ph.D. Thesis (Agra University, Agra) Chap II (1992), 30.
8. M. GUPTA and R. K. AGARWAL, Hele-shaw flow of dusty gas with volume-fraction, Acta Ciencia Indica, XXVI M, 1 (2000), 055.
9. M. GUPTA, S.P. SINGH and PARVEEN, MHD Hele-shaw flow of a dusty viscous fluid with volume-fraction past an elliptical cylinder, Indian Journal of Biomechanics: Special Issue (NCBM 7-8 March, 2009), 89-94.
10. I.N. SNEDDON, Fourier transform, Mc Graw Hill Book Co., Inc, New York, 1951. |
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