Effect of Ratio of Visco-Elastic Material Viscosity to Fluid Viscosity on Stability of Flexible Pipe Flow

In the present study, a flexible pipe has been considered to study the effect of ratio of visco-elastic material viscosity to fluid viscosity on the stability of flexible laminar pipe flow with axi-symmetric disturbances. The effect of thickness of visco-elastic material on the stability of flexible pipe flow with outer rigid shroud has also been studied. The stability curves are drawn for various values of the ratio of visco-elastic material viscosity to fluid viscosity. It is observed that stability of flow is increasing by decreasing the ratio of visco-elastic material viscosity to fluid viscosity.


I. INTRODUCTION
Flexible pipe flow is normally seen in nature such as human body, industrial applications for reactors and membranes. Two types of flexible pipe flow are found in nature. First type is one in which shape and size of pipe is changed due to internal pressure of the fluid. Second type is the one in which shape and size are not deforming, which means pressure gradient is constant along the direction of flow.
Kramer's [1] - [4] studied the flow over flexible surface on a flat plate and it was found that flexibility of flat surface delay the transition and drag is reduced that means flow remains laminar for a longer period. Carpenter [5], [6], Davies, Carpenter and Lucey [7], [8] also obtained the same result.
Reynolds [9] performed the experiment in rigid circular pipe and classified the flow as laminar, transition and turbulent based on the Reynolds number. Davey and Drazin [10] shown by numerical analysis that flow is stable to very small disturbances at all Reynolds number R and axial wavenumbers. Rouleau and Garg [11] and Grosch and Salwen [12] have also confirmed this result by numerical studies. Salwen and Grosch [12] showed that center line modes (waves) are more unstable as compared to wall modes. Venkateswarlu et.al [13] have studied that non-linear terms increased the instability of pipe flow. Hamadiche and Gad-el-Hak [14] also studied the flexible tube flow with 2-Dimension and 3-Dimemnsion disturbances problems. They found that flexible tube is unstable at all Reynolds number R and all axial wavenumbers.which indicates flexible tube is unstable at very small, medium and higher Reynolds  considered a non-collapsible pipe flow problem. The flow have been chosen in such a way that the interface of the visco-elastic material and the fluid is getting deformed in the normal (N) as well as on the tangential (T) direction to flow and hence such a flow is defined by (N+T). , where V is maximum center line velocity and  is the kinematic fluid viscosity. The ˆa nd ˆare the deformation of flexible material along r-direction and x -direction and s p is the pressure in the visco-elastic material. All details of mathematical terms are given in figure-1. In this paper, the stability is studied for flexible laminar pipe flow and outer surface of pipe is considered as rigid. The Fourier waves (modes) are inserted into pipe as a disturbance as given below: Here the disturbance velocity is given by The Navier Stokes and continuity equations may be expressed as follows: x-momentum: y-momentum: Continuity: Here prime ) ( represent the differentiation with r. The Fourier waves are inserted into the linearized Navier-Stokes and continuity equations (2)(3)(4). After doing some mathematical exercise, the 4 th order differential equation in terms of v for axi-symmetric disturbances (2-D waves) in the fluid-side is obtained as follows:

The flexible material equations
The Navier and continuity equations for displacements in the flexible materials may be expressed as follows:  -Displacement:  -Displacement: Continuity: Where, K is the flexible term and is given as Kumaran (1995) gave a flexible parameter  as , where G is shear modulus of flexible material. The K is given by Kumaran as For the visco-elastic material and fluid sides. Introducing The flexible material deformations may written as follows: The following differential equation for the visco-elastic material can be obtained in terms of  displacement as follows: C. Boundary conditions for visco-elastic and fluid sides The boundary conditions for combined fluid-solid problem for axisymmetric case, n = 0, are given as below: Pipe center at, 0  r : Surface of the pipe At the surface of the visco-elastic pipe, viz. at r = H: (16)

D. Interface between the fluid and the visco-elastic material
The boundary conditions at the interface between the solid and the fluid-sides may be written as below: The continuity of velocities, linearised with respect to r = 1, is given as follows: Particularly, eq. (22a) is the tangential no-slip boundary condition. The above interface velocities can also be written as follows: The expressions for the stresses from the fluid-side and solid-side are given as below, where generically  represents to the fluid-side stresses and  represents to the visco-elastic material -side stresses. Fluid-side: Solid-side: The stress matching conditions respectively for the normal and axial directions, are given as

III. Numerical Methods
The above differential equations can be solved by using the basic concept of the finite difference techniques. The differential equations can be written for the Fourier modes in the matrix forms as follows: Here, N is the number of intervals for above matrix. The above matrix can be solved for visco-elastic material side and fluid-side by using the finite difference method. The numerical equation for the fluid side and visco-elastic side are solved by using the FORTRAN programming and these programs are compiled by the Fortran Lahey Fitjtsu.     viscosity of visco-elastic material as per the definition of r  . The higher values of r  makes the flexible pipe more rigid. As we know that the rigid pipe flow is more stable than the flexible pipe flow. Hence, final conclusion can be drawn that the more viscosity of visco-elastic material makes the flow more stable.

V. CONCLUSIONS
The following conclusions have been drawn on the basis of the results given above.
(1) It is found that the flexible pipe is unstable for all ranges of Reynolds number (R) and azimuthal wave number (). (2) The increase in the visco-elastic material thickness (H) increases the instability of flexible pipe flow, which means higher visco-elastic material thickness makes the pipe more   flexible and hence more flexibility of pipe increases the instability. As it is know that the rigid pipe is more stable as compare to flexible pipe.