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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.

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.

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:

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,

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.

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