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Research Paper

Mathematics

E-ISSN No : 2454-9916 | Volume : 5 | Issue : 8 | Aug 2019

UNSTEADY FLOW OF DUSTY FLUID THROUGH CIRCULAR PIPE: A STUDY Smt. N.G. Siddagangamma Assistant Professor, Department of Mathematics, Siddaganga Institute of Technology, Tumkur, Karnataka-572013, India. ABSTRACT The present paper an attempt to study flow of an unsteady dusty fluid through a circular pipe under the influence of impulsive pressure gradient. We have assumed that the initial fluid velocity to be some average velocity. Analytical expressions for the velocities of fluid and dust particles are obtained by using Laplace transform technique. The influence of mass concentration and relaxation time of the dust on the velocity have been clearly pointed out. Results have been discussed with the help of graphs. KEYWORDS: Pipe flow; Two-phase flow; Non-Newtonian fluid; incompressible flow; volume fraction.

1. INTRODUCTION: The mechanical behaviour of dusty fluids has been a subject of study receiving greater attention during the recent past of several researchers in the field of Fluid Dynamics. Problems dealing with the influence of dust particles on viscous flows find place in several branches of Science and Technology. Some such flows are those of red corpuscles and other bodies in blood, fluidization, combustion, environmental pollution and chemical engineering flow in rocket tubes where small carbon or metallic fuel particles are present and so on.

particles and we have assumed that the initial fluid velocity to be some average velocity. Analytical expressions have been obtained for the velocity components of both of the dust and the fluid. The influence of mass concentration and relaxation time of the dust on the velocity have been clearly pointed out. This problem finds its application in the blood flow through Arteries.

Since a dusty fluid is a mixture of fluid and fine dust particles, its study is of practical importance. Particularly, the study of flow of dusty fluids through circular pipes is of great importance which has applications in various Industries. Considering blood as a two-phase fluid, this study also gives some insight into the blood flow through veins or arteries

2. To know the effect of time on velocity of ﬂuid

Interest in problems of mechanics of systems with more than one phase has developed rapidly in recent years. Much work has already been done on dusty fluid flow. The equation n of motion of fluid carrying small particles were first derived. P.G.Saffman (1962) who investigated the stability of the laminar flow of a suspension in which the particles are uniformly distributed. A.H. Nayfeh (1966) introduced equation of motion of fluid containing dust particles taking volume fraction of dust particles into account. S.Uchida (1956) has made some theoretical analysis of pulsating viscous flow in circular pipe. S.K.Nag et al., (1981) have considered the flow through an elastic tube rather than a rigid tube. Kishore and Pandey (1977) have discussed the same problem but they found the effect of particles size on velocity of sedimentation. Datta and Dalal (1992) have discussed the unsteady flow of dusty fluid through a circular pipe. S.N. Dube and Srivastava (1972) have discussed the unsteady flow of dusty viscous fluid in a channel and pipe under the influence of linear pressure gradient. Sambasiva Rao (1969) has obtained the analytical solutions for the dusty fluid flow through a circular tube under the influence of constant pressure gradient. S. Rashmi et.al. (2007) have discussed the unsteady flow of a dusty fluid between two oscillating plates under varying constant pressure gradient and Siddapasappa et.al. (2008) have discussed the viscous dusty fluid flow with constant velocity magnitude.

2. OBJECTIVES: 1. To know the behavior of blood ﬂow

3. METHODOLOGY: The Laplace transform method is use in this research paper and result discuss through graphs using Origion software. 4. BASIC EQUATIONS OF MOTION AND THEIR SOLUTION: Consider the flow of in incompressible dusty fluid through a circular pipe of radius a. Let the flow be induced by an impulsive pressure gradient along the axis of the pipe taken as x – axis. The equations of motion of unsteady viscous fluid with uniform distribution of dust particles are given by (Saffman, 1962)

r (1 - f )

æ ¶ 2 u * 1 ¶u * ö ¶u * ¶p * = - * + m çç *2 + * * ÷÷ + NK (v * - u * ) (1) * ¶t ¶x r ¶r ø è ¶r

¶v * = NK u * - v * ¶t *

(

)

(2)

Where u , v are the components of the fluid and particle velocity in the direction of x- axis which is taken along the length of the tube. The last term on the right hand side of equation (1) represents the force exerted by the particles on the flow while the term on the right hand side of equation (2) is a similar force term exerted by the fluid on the particles. f = Nm is the ratio of the mass density of the particles

and the fluid, commonly known as mass concentration of the dust In all the above discussed papers the authors considered the zero initial velocity, but in this paper, the authors study the problem of unsteady flow of dusty fluid through a circular pipe induced by an impulsive pressure gradient taking into account the volume fraction of the dust

particles. b =

a m is the relaxation time of the particles. ,t = Ut k

is the ﬂuid density,

m is the viscosity of ﬂuid, N is constant number

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Research Paper

E-ISSN No : 2454-9916 | Volume : 5 | Issue : 8 | Aug 2019

density of the particles, f is volume fraction of particle phase, p is the pressure of ﬂuid. K is stokes resistance coefﬁcient. (i.e. K = 3

m p d, d is particle diameter) and m is mass of each particle, r *

v ( s) =

b s+ b

1 s+ b

is Substituting the value of

the radial distance.

(11)

in (9), we get

d u 1 du R R + - b u = - F1 2 dr 2 r dr 2

Initial and boundary conditions are

u * = U , v* = U

t* £ 0 , 0 £ r* £ a

(3) Where

u* = 0

t * > 0,

r* = a

(4)

(12)

r fb s fb + (1 - f )s , F1 = F + (1 - f ) + s+b s+b

Solving Eq. (12) satisfying boundary condition the regularity condition at r = 0 we get

Introducing the following dimensionless variables:

u = 0 at r = 1 and

p * - p0 u* v* r* t *U x* ,v= ,r = , p = ,t = ,x= 1 U U a a a rU 2 2

Where U being a reference velocity and

constant hydrostatic

æ Rb ö ÷ J 0 çç ir 2 ÷ø F1 F1 è u= b b æ Rb ö ÷ J 0 çç i 2 ÷ø è

(13)

pressure, Equations (1) and (2) reduce to Using the convolution theorem and Cauchy’s residue theorem, inverse transform of Eq. (13) can be written as

æ 2 ö (1 - f ) ¶u = F (t ) + 2 çç ¶ u2 + 1 ¶u ÷÷ + f b (v - u ) ¶t R è ¶r r ¶r ø

u (t ) = å An ò exp( s n t ) F (t - t ) d t +

(5)

¶v = b (u - v ) ¶t

Where

(6)

Where F (t ) =

2aUr 1 ¶p , R= - Reynolds Number, 2 ¶x m

n =1

An =

2 J 0 (ln r ) ln y n J 1 (ln )

An bn exp( s n t) (14) sn (15)

’s are the zeros of the Bessel function J æç i Rb ö÷ i.e 0ç 2 ÷

f =

Nm The mass concentration of dust particle (ratio of the mass

density of the particles and the ﬂuid),

a Ut

mö æ , çt = ÷ - particle kø è

relaxation time

u=0

b n R = -2ln , hence bn (sn + b ) = fb sn + sn (1 - f ) (sn + b ) and

y n = (1 - f )+

f b2 . (s n + b )2

Applying inverse Laplace transform to Eq. (11), we get

The pertinent initial and boundary conditions are

u = v =1

v(t ) = exp(- b t ) + b ò u (t ) exp(- b (t - t ) ) dt

t £ 0 ,0 £ r < 1

(7)

t > 0, r = 1

(8)

Applying Laplace transform with respect to time t and using the initial conditions given in Equation (7), Equations (5) and (6) reduce to

(16)

5. IMPULSIVE PRESSURE GRADIENT CASE: Consider a particular case of the impulsive pressure gradient in the form F(t) = P H ( t ), where P represents the constant adverse pressure gradient and H (t) is Heaviside function of time t. For this case, the ﬂuid velocity given in Eq. (14) can be written as

æ 2 ö (1 - f ) (s u (s) - 1) = F (s) + 2 çç d u2 + 1 du ÷÷ + f b (v - u ) R è dr r dr ø

(9)

u (t ) = å n =1

s v (s) - 1 = b (u - v )

(10)

{u (s) , v (s), F (s)} = ò {u(t ), v(t ), F (t )} e 0

(17) Substituting (17) in (16), we get

Where

An P (exp( s n t ) - 1)+ An bn exp( sn t ) sn sn

- st

v(t ) = exp( - b t ) + å n =1

An Pb sn

æ exp( s n t ) - exp( - bt ) (1 - exp( - bt )) ö bAn bn çç ÷÷ + sn + b b sn è ø

æ exp( s n t ) - exp( - bt ) ö çç ÷÷ sn + b è ø

(18) From (10), we get

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E-ISSN No : 2454-9916 | Volume : 5 | Issue : 8 | Aug 2019 changes in ﬂuid and particle phases and also the discharge rate of ﬂuid and particle phase between 20 £ t £ 100 .

6. INSTANTANEOUS RATE OF DISCHARGE: The instantaneous rate of discharge of ﬂuid is given by 1

Qu =

Q* = 2 ò urdr Qs 0

(19)

Similarly the dimensionless instantaneous rate of discharge for particle is 1

Qv = 2 ò vrdr 0

(20) Using the values of u and v from Equations (17) and (18) in Eqs. (19) and (20) respectively, the time dependent rate of discharge of fluid and particle phase are obtained. 7. RESULT AND DISCUSSION: In order to discuss results, numerical computation for Equations (17) and (18) has been made on using typical values of the dimensionless parameters e.g. volume fraction f, mass concentration f, adverse pressure gradient P, Reynolds number R, and for various values of time t. In order to compute the series solution for u- fluid velocity and v – particle velocity, we have considered the first 275 zeros of Bessel function of Zeroth order and first kind numerically. Considering the density ration r s » O(10) and the parameter b = 1.0, it is noted that

r f f @ . The ﬂuid velocity and rate of discharge at different times 10

have been computed for various values of the parameter f, (i.e. f also) and they are presented in Figures. 1a – 4a. Figure1a shows the velocity profiles of fluid and particle phase for various values of f. From the graphs we find that the velocity of the fluid as well as that of the particle phase decreases with the increase of mass concentration f. i.e. of particle loading and volume fraction f. Comparison of the graphs shown in Fig.1a revealed that the velocity of the particle phase is always less than that of the fluid phase, but near the wall velocities of particle and the fluid phase approach to be equal to each other as f increases. It is also noted that velocities of both fluid and dust particles are maximum on the axis of the tube and fluid particles moves faster than the dust particles everywhere in the tube. Figure 2a shows the distribution of velocity of the fluid and that of the particle phase at the center of the pipe with respect to f for different values of t. The difference between the velocities of fluid and particle phase tends to zero as time increases and they are nearly equal at for t ³ 100 . Figure 3a gives the graphs of the velocity of fluid and particle phase against time t for various values of particle loading f. From figure it is observed that the velocities of fluid and particle phase first increases with time due to acceleration up to a constant value for large time as f increases and the steady state is reached at time t ³ 100 for both fluid and particle phase. Figure 4a shows the growth of discharge rates of fluid and particle phase with time. It is seen that these rates of discharge decreases first with time and increase with increase of f (or f). For large time i.e. t ³ 100 the discharge rates attain the steady state due to the steadiness of velocity for both fluid and particle phase. Further, the time to reach the steady state increases with the increase of f. Also we compare these results with the results obtained by Datta and Dalal [10] and they are presented in Figures. 1 – 4. We observed the

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E-ISSN No : 2454-9916 | Volume : 5 | Issue : 8 | Aug 2019

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E-ISSN No : 2454-9916 | Volume : 5 | Issue : 8 | Aug 2019

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