The Steady Two-Dimensional Flow of Non-Newtonian Liquid through Convergent Duct

Abstract

It is the aim of the present paper to consider the steady two-dimensional flow of purely viscous fluid through V-shaped convergent duct. In this study the cylindrical coordinates r, θ and z have been assumed, and the following Reiner-Rivlin Equation has been used as the rheological equation of state for an incompressible purely viscous fluid.
p^ij=α₁(II, III)e^*ij+α₂(II, III)e^*ime^*_m^j (1)
For the above-mentioned purpose the physical components and the invariants of the rate of strain tensor, and the physical components of stress tensor were calculated for the two dimentional flow of Reiner Rivlin fluid through this duct. The results were integrated into the equation of the motion in terms of stress tensor. From these calculations the following conclusions have been deduced.
(1) For the flow through the convengent ducts the only non-vanishing invariant of the rate of strain tensor is II, and this invariant depends not only on the non-diagonal components of the rate of strain tensor but also on its diagonal components.
(2) The flow pattern of the Refiner-Rivlin liquid depends only on the apparent viscosity α₁ and not on the cross viscosity α₂.
Based on these conclusions the flow behavior of the two kinds of non-Newtonian liquid has been examined through these ducts. The relations between II and α₁ for these liquids are shown by the following equations.
α₁=m(-II)^n-1/2 (2) α₁=a-c(-II) (3)
As the results of theoretical examinations having been made the following conclusions have been reached.
(1) It is tenable that the stream lines of non-Newtonian fluid which obey the Eq. (2) are a group of straight lines that pass through the vertex of V-shape, though it does not follow that the stream lines of the fluid which obey the Eq. (3) are in straight lines.
(2) For the latter case the θ-component of the fluid velocity must be taken into consideration as well as its γ-component. The following differential equation has been obtained which gives the velocity distribution of non-Newtonian fluid that obeys the Eq. (2) in this duct.
64n²g⁴g'+16g⁴g"'-32(1-n²)g³g'g"+16(1+n²)g²(g')³+4(1+n)g²(g')²g'"-12(1-n)g²g'(g")²
+16(1-n)²g(g')³g"-4(2-4n+n²)(g')⁵+n(g')⁴g"'-n(1-n)(g')³(g")²=0 (4)
The numerical solution of the Eq. (4) has been obtained by the Runge-Kutta method for some values of n. When the inclination between the two planes of V-shaped duct becomes comparatively small the equation given above will be simplified as given in the following equation.
4(2n-1)(n-1)gg"-4(1-3n+n²)(g')²+n(n-1)(g")²+ng'g"'=0 (5)
The analytical solution of the Eq. (5)in the case of n=1/2 is found to run as follows
g(θ)=A/8[tan(±β)/1+tan²β-tanθ/1+tan²θ+θ-(±β)] (6)
where A>0 when θ>0, A<0 when θ<0.