JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN
CIVIL ENGINEERING
FLOW OF A SECOND ORDER FLUID OVER
AN INCLINED RIGID PLANE
1CH.V. RAMANA MURTHY, 2K.GOWTHAMI
3K.R. KAVITHA*, 4P. SUMATHI KUMARI
1Professor, K.L.University, Vaddeswaram, Guntur District, Pin : 522 502 (INDIA)
2, 4Lecturer, K.L.University, Vaddeswaram, Guntur District, Pin : 522 502 (INDIA)
Asst. Professor,N.R.I. Institute of Technology, Pothavarappadu , Pin : 521 212
,
Keywords: Visco elastic fluids, second order fluid, visco elasticity, steady state flow
1. NOMENCLATURE
Ai:Acceleration component in i th direction
ai:Non dimensional acceleration in i th direction
:Given history
:Retarded History
H:Characteristic length
P:Indeterminate hydrostatic pressure
p: Non-dimensional indeterminate pressure
:Coefficient of viscosity
:Coefficient of elastico viscosity
:Coefficient of cross viscosity
:Stress tensor
T:Dimensional time parameter
t:Non-dimensional time parameter
Ui:Non dimensional velocity component in i th direction
:Visco elasticity parameter
:Density of fluid
:Frequency of excitation
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ISSN: 0975 –6744| NOV 09 TO OCT 10| Volume 1, Issue 1 Page 1
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN
CIVIL ENGINEERING
2. INTRODUCTION
Due to wide ranging applications in the fields of Physics, Chemistry, Chemical Technology and in situations demanding efficient transfer of mass over inclined beds, the viscous drainage over an inclined rigid plane has been the subject of considerable interest to theoretical and experimental investigators during the last several years.
In all experiments, where the transfer of viscous liquid from one container to another is involved, the rate at which the transfer takes place and the thin film adhering to the surfaces of the container is to be taken into account for the purpose of chemical calculations. Failure to do so leads to experimental error. Hence, the need for such analysis.
The problem of flow of viscous incompressible fluid moving under gravity down a fixed inclined plane with the assumption that the velocity of the fluid at the free surface is given has been examined earlier by Sneddon. Later Bhattacharya [1] examined the problem when uniform tangential force ‘S’ acts on the upper surface for a finite interval of time. Jeffreys [2]initiated the problem of steady state profile over a vertical flat plate which was further examined by Green [3]. Later, Gutfinger and Tallmadge[4]investigated steady state drainage over a vertical cylinder. These authors examined the problem in the absence of fluid inertia.
Noll [5] defined a simple material as a substance for which stress can be determined with the entire knowledge of the history of the strain. This is called a simple fluid, if it has the property that at all local states, with the same mass density, are intrinsically equal in response, with all observable differences in response being due to definite differences in the history. For any given history, a retarded historycan be defined as
(1) being termed as a retardation factor. Assuming that the stress is more sensitive to recent deformation that to the deformations at distant past, Coleman and Noll [6] proved that the theory of simple fluids yields the theory of perfect fluids as and that of Newtonian Fluids as a correction (up to the order of ) to the theory of the perfect fluids. Neglecting all the terms of the order of higher than two in , we have incompressible elastic viscous fluid of second order type whose constitutive relation is governed by
(2)
where (3)
and (4)
In the above equations, is the stress-tensor, are the components of velocity and acceleration in the direction of the i th coordinate , P is indeterminate hydrostatic pressure and the coefficients and are material constants.
The constitutive relation for general Rivlin-Ericksen [7] fluids also reduces to equation (2) when the squares and higher orders of are neglected, the coefficients being constants. Also the non-Newtonian models considered by Reiner [8] could be obtained from equation (2) when, naming as the coefficient of cross viscosity. With reference to the Rivlin – Ericksen fluids, may be called as the coefficient of viscosity. It has been reported that a solution of poly-iso-butylene in cetane behaves as a second order fluid and Markovitz [9] determined the constants and.
Viscous fluid flow over wavy wall had attracted the attention of relatively few researchers although the analysis of such flows finds application in different areas such as transpiration cooling of re-entry vehicles and rocket boosters, cross hatching ablative surfaces and film vaporization in combustion chambers, especially the stream, where the heat and mass transfer takes place in the chemical processing industry. The problem by considering the permeability of the bounding surface in the reactors assumes greater significance.
In view of several industrial and technological importance, Ramacharyulu [10] studied the problem of the exact solutions of two dimensional flows of a second order incompressible fluid by considering the rigid boundaries. Later, Lekoudis et al [11] presented a linear analysis of the compressible boundary layer flow over a wall. Subsequently, Shankar and Sinha [12] studied the problem of Rayleigh for wavy wall. The effect of small amplitude wall waviness upon the stability of the laminar boundary layer had been studied by Lessen and Gangwani [13]. Further, the problem of free convective heat transfer in a viscous incompressible fluid confined between vertical wavy wall and a vertical flat wall was examined by Vajravelu and Shastri [14] and thereafter by Das and Ahmed [15]. The free convective flow of a viscous incompressible fluid in porous medium between two long vertical wavy walls was investigated by Patidar and Purohit [16]. Rajeev Taneja and Jain [17] had examined the problem of MHD flow with slip effects and temperature dependent heat in a viscous incompressible fluid confined between a long vertical wall and a parallel flat plate.
The aim of the present note is to examine the unsteady state flow of a visco elastic fluid of a second order type over an inclined rigid plane when a uniform tangential force F acts on the free surface for a finite interval of time.
3. MATHEMATICAL FORMULATION OF THE PROBLEM
The constitutive relation for a visco elastic fluid of second order type as stated in eqn (2) is considered.The problem is examined hereunder with reference to the rectangular Cartesian co-ordinate system with the x-axis along the plane in the direction of the motion and y - axis into the fluid perpendicular to this direction. The motion is assumed to be unidirectional i.e., along x - axis and hence the components of the velocity can be regarded as .
The motion is now governed by the equation
(5)
where is the angle of inclination of the plane with the horizontal, is the fluid density which is assumed to be constant throughout and g is the acceleration due to gravity.
The condition of no slip on the boundary would yield when . Further, the condition of uniform tangential force F on the free surface for a finite interval of time is
,
(6)
together, with the initial condition
(7)
Introducing the following non-dimensionalization scheme
, ,
, (8)
,
where is the non-dimensional visco elastic parameter.
The governing equation for the fluid motion together with the required conditions reduces to
(9) (9)
where (10) (10)
and when (11)
(12) (12)
in eqn (6) represents the Heavisides Unit step function given by
for (13)
for
Taking Laplace Transforms for eqn (6), we now have
(14) (14)
together with the conditions on the boundary as when (15) (15)
at (16)
The solution of eqn (14) satisfying eqn (15) and eqn (16) is given by
(17)
where (18) (18)
Taking inverse Laplacetransform of eqn (17) the velocity field is obtained as
(19)
for and
.
. for (20)
4. RESULTS AND CONCLUSIONS:
1. As , the problem reduces to as that of Bhattacharya.
2. As , the problem reduces to the steady state flow over an inclined rigid plane. The effect of the inertial term in the governing equation of motion is only to introduce exponentially decaying unsteady terms in the governing equation of motion do not qualitatively alter the shapes of the velocity profiles.
3. Fig(1) shows the nature of fluid flow over an inclined rigid plane for various values of visco elasticity parameter. It is observed that as the visco elasticity of the fluid increases, the draining of the fluid along the rigid plane is found to be slow.
.
Fig1: Velocity Profiles along the plate for different visco elasticity parameter for T = 3.5
4. The draining of the fluid for different visco elasticity parametersis illustrated in fig(2). The phenomena as stated above is noticed in fig(1) also. However the initial draining of fluid is found to be delayed on observing fig (1) and fig(2).
Fig 2:Velocity Profiles along the plate for different visco elasticityparameter for T =3.0
5. For constant visco-elasticity parameter of the fluid, the velocity profiles at different intervals of time are illustrated in fig (3). It is observed that as T increases, it is noticed that the velocity also increases which is in agreement with observable phenomena.
Fig 3 : Velocity Profiles along the platefor different time and for visco elasticity parameter1.0
6. The velocity profiles for are illustrated for different time parameters in fig(4). The profiles are found to be slightly distinct at the lower bottom when compared to fig(3) stated earlier. This can be attributed to the fact that the lower the visco elasticity, the intra molecular forces are less dominant and are weak in nature. Hence, the variation could be seen for smaller values of visco elasticity parameter of the fluid under consideration.
Fig 4 : Velocity Profiles along the platefor different time and for visco elasticity
parameter 0.5
5. REFERENCES
[1].P. Bhattacharya: Flow of viscous incompressible fluid down and inclined plate, I.J.M.M. III
Pp 65-69 (1965).
[2]. H. Jeffreys: The drainage at a vertical plate. Proc. Comb. Phil. Soc. 26, 204 (1930).
[3]. G.Green:Viscous motion under gravity in a liquid film adhering to vertical plate. Phil Mag. 22, Pp 730-736 (1936).
[4]. C. Gutfinger and J.A. Tallmadge: Some remarks on the problem of drainage of fluids on vertical surfaces. A.I.Ch. E.J., 10, 774-780 (1964).
[5].W.Noll: A mathematical theory of mechanical behavior of continuous media. Arch. Ratl.Mech. Anal. 2, Pp 197 – 226 (1958).
[6]. B.D. Coleman and W. Noll: An approximate theorem for functionals with applications in continuum mechanics, Arch. Ratl. Mech. Anal., 6 Pp 355-370 (1960).
[7].R.S.Rivlin and J.L.Ericksen: Stress relaxation for isotropic materials. J.Ratl. Mech. Anal.4,
Pp 350 – 362 (1955).
[8]. Reiner: A mathematical theory of diletancy. Am.J. Maths. 64, Pp 350 -362 (1945).
[9].H.Markovitz and B.D. Coleman: Non-steady helical flows of second order flows. Physics of fluids.7, Pp 833 – 841 (1964).
[10].N.Ch. Pattabhi Ramacharyulu: Exact solutions of two dimensional flows of second order fluid. App. Sc. Res. Sec - A, 15, Pp 41 – 50 (1964).
[11].S. G. Lekoudis, A. H. Nayef and Saric: Compressible boundary layers over wavy walls.Physics of fluids.19, Pp 514 – 519 (1976).
[12].P. N. Shankar and U. N. Shina: The Rayeigh problem for wavy wall. J. fluid Mech. 77,
Pp 243-256 (1978)
[13].M. Lessen and S. T. Gangwani: Effects of small amplitude wall waviness upon the stability of thelaminar boundary layer, Physics of fluids. 19, Pp 510- 513 (1976).
[14]. K.Vajravelu and K.S.Shastri: Free convective heat transfer in a viscous incompressiblefluid confined between along vertical wavy wall and a parallel flat plate. J. Fluid Mech. 86, Pp 365 – 383(1978).
[15].U. N. Das and N. Ahmed: Free convective MHD flow and heat transfer in a viscous
incompressible fluid confined between a long vertical wavy wall and a parallel flat wall.
Indian J. Pure Appl. Math. 23, Pp 295 –304 (1992).
[16]. R. P. Patidar and G. N. Purohit: Free convection flow of a viscous incompressible fluid in a porous medium between two long vertical wavy walls. Indian J. Math40,Pp 76 -78 (1998).
[17]. R. Taneja and N. C. Jain: MHD flow with slips effects and temperature dependent heat source in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flatwall. Def. Sc. J., Jan Pp 21- 29 (2004).
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ISSN: 0975 –6744| NOV 09 TO OCT 10| Volume 1, Issue 1 Page 1