Analytical study of Bingham fluid flow through a conical tube

Da Darcy number; L length of conical tube, mm; Q Darcy velocity, ms -1 ; R0 entrance radius, mm; RL exit radius, mm; Rp radius of the plug flow region, mm; p pressure, N/m 2 ; r radius, mm; uB slip velocity, ms -1 ; uav average velocity, ms -1 ; vz velocity component in z direction, ms -1 ; k permeability; z flow axis;  slip parameter; μ0 coefficient of viscosity at the interface, kgm -1 s -1 ; τ0 yield stress, N/m 2 ; τrz shear stress, N/m 2 .


Introduction
The blood vessel can be idealized as a tube with tissue space as circular porous bed (Vide Guha and Chuadhury, 1985).The rotating viscometer data of Rand et al. [1], Bugliarello et al. [2] and Chien et al. [3] suggest the non-Newtonian behaviour of blood.Lew et al. [4] suggested chyme as a non-Newtonian material having plastic-like properties.In view of this the biofluid flow in a living body can possess the non-Newtonian behaviour in general.In order to have a better understanding of blood flow in arteries and veins and chyme flow in stomach.It is necessary to consider the biofluid to be an yield stress fluid.One of the models for yield stress fluids is Bingham model.Hence the study of Bingham fluid flow through a conical tube with permeable wall is of considerable importance in medicine.
Bird et al [5] investigated the Bingham fluid flow in a rigid circular tube.Rathy [6] studied the flow of a Bingham fluid in a channel and in an annulus with impermeable walls.Vajravelu et al. [7] made a study on the Bingham fluid in a circular tube with permeable wall.The velocity field is obtained using Beavers and Joseph [8] slip condition at the permeable wall.Buckingham-Reiner equation for the flow is obtained.
The Bingham fluid flow between two permeable beds is discussed by Goverdhan et al. [9].The flow in the channel is assumed to be governed by Bingham model.The flow in the permeable beds is governed by Darcy's law.The velocity distribution is obtained.Some results are deduced and discussed.Comparini [10] discussed a onedimensional model for the time dependent flow of a Bingham fluid between two parallel plates.The global existence and uniqueness of classical solution to the problem is proved.
Ravana et al. [11] studied the free surface flow of a Bingham fluid in an inclined channel over a permeable bed.The flow in the channel is described by Bingham model, whereas the flow in the permeable bed is according to Darcy's law.The velocity field, the shear stress, the mass flow rate and its fractional increase are obtained.
The problem of rotational motion of a Bingham fluid in the gap between two coaxial cylinders, the outer one being at rest and the inner one moving at given angular velocity is solved by Comparini [12].
Narahari [13] discussed unsteady flow of a Bingham fluid between two permeable beds having different permeabilities.The velocity distribution in the porous and non-porous regions are obtained.Some deductions are made and the results are discussed.
Sankara Reddy et al. [14] made a detailed study on the Bingham fluid flow in an inclined channel bounded by two permeable beds.The velocity distribution in the porous and non-porous regions are obtained.The temperature variation with Ec.Pr is discussed.
Viswanatha Reddy et al. [15] made a study on Bingham fluid flow in an annulus.The velocity field, the mass flow rate and its fractional increase are obtained.The results are deduced and discussed.Helical flow of a powerlaw fluid in a thin annulus with permeable walls is investigated by Vajravelu et al. [16].It is observed that velocity increases due to permeable nature of the annulus.
In this paper, Bingham fluid flow through a conical tube with permeable wall is investigated.The velocity distribution, the volume rate of flow and its fractional increase are obtained.The results are deduced and discussed through graphs.

Mathematical formulation
Consider the flow of a Bingham fluid through a conical tube of length L with permeable wall.The flow takes place due to pressure gradient, and the porous medium is homogeneous with permeability k.The flow surrounded by the porous medium is governed by the Bingham model, and the flow in the porous medium is governed by the Darcy's law.The tube has a radius R 0 at the entrance and radius R L at the exit (Fig. 1).The tube radius at any distance z from the inlet is given by The flow is axi-symmetric.Cylindrical polar coordinate system is used.The following assumptions are made in deriving the governing equations: -the flow is steady and incompressible; -the flow is in axial direction; -all physical quantities except the pressure are function of z only; -the body forces are negligible.
In view of the above assumptions, the governing equations and boundary conditions of the flow take the following form: where = zB vu at rR  ; where By integrating equation Eq. 2 and using the boundary condition (4), we obtain: Substituting ( 9) in (3), we get: where , p and R p are the yield stress, Shear stress, slip parameter, permeability, slip velocity, Darcy velocity, coefficient of viscosity at the interface r = 0, average velocity, velocity component in zdirection, pressure and radius of the plug flow region.

Dimensionless equations
The flow physical parameters are made dimensionless as follows:

Governing equation
Eq. ( 10) takes the form (* are omitted here after): where = dp P dz  .

Solution of problem
Solving (12) and using the boundary condition ( 14) we obtain the velocity field around the plug flow region as: Using (15) in (16), we get the slip velocity at the porous wall as: Using ( 13) in (16), we obtain the relation between p  and p R as: Taking = p rR in Eq. ( 16) and using the relation (18), we get the velocity field in the plug flow region as: when 0 The flow in the porous region ( p rR  ) is gov- erned by the Darcy law which is given by:

Deductions
) and ( 19) we get the velocity field outside the plug flow region and that in the plug flow region which are in agreement with those of Bird et al (1960).
Case (b): when   =1 Rz , the Eqs.( 16), ( 17) and (19) we get the velocity field for the flow of a Bingham fluid flow through a circular pipe with permeable wall as follows: p Rr ; These results are in a good agreement with those of Sreenadh and Arunachalam (1983).

Volume rate flow
The Volume rate of flow for the flow of a Bingham fluid flow through a conical tube is: where When the permeability parameter k 0  (i.e., 0 a D  ) (28) reduces to: where

Fractional increase
The fractional increase in the volume rate of flow of the Bingham fluid through a conical tube with permeable wall over what it would be if the wall of the tube were impermeable is: where

Discussion of the results
The velocity profiles are shown in Figs. 2 -12 for different values of τ 0 , Darcy number D a , P = 10 and  = 1.
It is observed that the velocity attains maximum value at r = 0 and decreases with the increment in r.
For fixed r and D a , the velocity decreases due to increase in z.The slip velocity at the permeable wall decreases with the increment in the value of D a .On comparing the velocity profiles for permeable and impermeable conical tubes, it is found that the velocity is enhanced due to the permeability of the wall of the conical tube.It is also observed that the velocity remains constant from the axis (i.e., r = 0) up to some value of r (which is the Plug flow region) and then decreases to a value at the permeable wall (which is the non Plug flow region).
For fixed r and τ 0 , the velocity increases with decrement in D a .As D a decreases the gap between the velocity curves becomes smaller for any fixed τ 0 .For larger τ 0 , there is an increase in plug flow region.From Fig. 13 it is observed that, for the fixed Darcy number D a , the velocity decreases along the axis with the increase in τ 0 .It is the other way at the permeable wall.

Conclusion
In this work, we have investigated a Bingham fluid flow through a conical tube with permeable wall.The velocity distribution, the volume rate of flow and its fractional increase are obtained.The Velocity profiles obtained for different values of the yield stress, and a fixed Darcy number, pressure gradient along the flow equal to 10 and slip parameter equal to 1, show that the velocity attains maximum value on the tube axis and decreases with the increment of the rayon.
For a fixed rayon and fixed pressure gradient along the flow, the velocity decreases along the z-axis (when z increase).The slip velocity at the permeable wall decreases with the increment in the value of the Darcy number.The comparison between the velocity profiles for permeable and impermeable conical tubes shows that the velocity is enhanced due to the permeability of the wall of the conical tube.The velocity remains constant in the Plug flow region and decreases in the non Plug flow region.
For fixed radius and fixed yield stress, the velocity increases with decrement in the Darcy number.As Darcy number decreases the gap between the velocity curves becomes smaller for any fixed yield stress.For larger yield stress, there is an increase in plug flow region.For fixed Darcy number, the velocity decreases along the axis flow with the increase in yield stress.It is shown that for a fixed Darcy number, the fractional increases with yield stress.

Fig. 1
Fig. 1 Physical model: conical tube field for the flow with permeable wall is given by: