非ニュートン液体に対する二重円すい型レオメータの理論
1968 年 17 巻 175 号 p. 314-317
詳細
1968 年 17 巻 175 号 p. 314-317
The double cone viscometer consists of two coaxial cones. The generator of the external cone makes an angle γ with the horizontal plane, and the generator of the internal cone makes an angle γ+α with the horizontal plane. We shall consider that the angle α is very small. The wedge-like spacc between the two cones is filled with a liquid to be investigated. Either of the cones, internal or external, is rotated. In the present treatment, the external cone has been rotated with a constant angular velocity Ω around the axis of the cone.
We shall first find general relationship between the torque M and the angular velocity Ω for a time-independent non-Newtonian liquid specified by an arbitrary flow curve. Then we shall show how to determine the flow curve from the experimental relationship between M and Ω for some special cases.
With regard to the motion of the liquid, the following assumptions are made: (1) the liquid is incompressible; (2) the motion of the liquid is laminar; (3) the motion is steady; (4) there is no force acting on the liquid; (5) the motion has an axial symmetry; (6) each liquid particle moves in a circle on the horizontal plane perpendicular to the axis of rotation; (7) there is no relative motion between the walls and the liquid in immediate contact with the walls; (8) the edge-effect is negligible. The assumption (6) corresponds to neglect of centrifugal forces. For small values of Ω, this assumption as well as the assumption (2) may be allowed.
We shall take a spherical coordinate system r, θ, and φ whose origin is at the vertex of the cone. If we assume that the angular velocity ω of a liquid particle around the axis of the cone is a function of θ alone, then the shear stress τθφ is given by τθφ=c/sinsin2θ, where c is a constant. For a non-Newtonian liquid specified by an arbitrary flow curve f(τ), we get
The constant c is related to the torque M on the internal cone by the relationship M=c·2πa3/3. For a non-Newtonian liquid obeying a power law flow curve f(τ)=kτn the following formula has been obtained:
For the special case where n is equal to unity, the above formula is reduced to the well-known formula for a Newtonian liquid:
Here η=1/k is the coefficient of viscosity. We have also examined other special cases: (1) non-Newtonian liquid whose flow curve is expanded into power series, (2) Bingham body and (3) non-Newtonian liquid obeying Casson's equation.
