The Steady Flow of a Non-Newtonian Liquid through a Conical Nozzle

Abstract

The purpose of this paper is to present a theory of the steady flow of a non-Newtonian liquid through a conical nozzle. Then the apparent viscosity μ is a function of the velocity gradient. Considering that μ is a function of the coordinates of the liquid particle, we have obtained the exact equations of motion of a non-Newtonian liquid in the conical nozzle. We have also derived the differential equation of the velocity distribution for a special non-Newtonian liquid with a power law flow curve. The equation of motion has been obtained on the following assumptions; i) the liquid is incompressible; ii) the motion of the liquid is not turbulent; iii) the motion is steady; iv) no body force acts on the liquid; v) the motion has an axial symmetry; vi) there is no slip at the wall; vii) the stream lines are the straight lines passing through the vertex of the cone, that is, the end effect is neglected; viii) the motion is so slow that the inertia term can be neglected.We have taken a spherical coordinate system r, θ, and φ whose origin is at the vertex of the cone. Then the stress components are given byτrr=2μ∂vr/∂r (1) τrθ=μ/r∂vr/∂θ (4)τθθ=2μvr/r (2) τθφ=0 (5)τφφ=2μvr/r (3) τφr=0, (6)where vr is the velocity component.From the assumptions i)∼v), vii), and viii), the equations of motion are0=-∂p/∂r+{1/r2∂/∂r(r2τrr)+1/rsinθ∂/∂θ(τrθsinθ)-τθθ+τφφ/r} (7)0=-1/r∂p/∂θ+{1/r2∂/∂r(r2τrθ)+1/rsinθ∂/∂θ(τθθsinθ)+τrθ/r-cotθ/rτφφ}, (8)where p is the pressure and the equation of continuity is1/r2∂/∂r(r2vr)=0 (9)Substitution of Eqs. (1), (2), (3) and (4) into Eqs. (7) and (8), with the help of Eq. (9), yields∂p/∂r=2∂μ/∂r∂vr/∂r+1/r2∂μ/∂θ∂vr/∂θ+μ/r2∂2vr/∂θ2+μ/r2∂vr/∂θcotθ (10)∂p/∂θ=2μ/r∂vr/∂θ+∂μ/∂r∂vr/∂θ+2∂μ/∂θvr/r (11)The velocity gradient D is given byD=-1/r∂vr/∂θ (12)We shall treat the special case of a non-Newtonian liquid obeying the power lawD=kτn, (13)where k and n are constants. Then the apparent viscosity μ is given byμ=τ/D=KD-(1-1/n), (14)whereK=k-1/n (15)From Eq. (9), we getvr=f(θ)/r2, (16)where f(θ) is a function of θ alone.From Eqs. (12) and (16) we obtainD=r-3{-f'(θ)} (17)Thus the apparent viscosity μ is expressed as a function of r and θ as follows:μ=Kr3(1-1/n){-f'(θ)}-(1-1/n) (18)Substitution Eq. (18) into Eqs. (10) and (11) yields∂p/∂r=-12K(1-1/n)r-1-3/nf(θ){-f'(θ)}-(1-1/n)+K1/nr-1-3/n{-f'(θ)}-(1-1/n)f"(θ)-Kr-1-3/n{-f'(θ)}1/ncotθ (19)