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

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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+αα with the horizontal plane. we shall consider that the angle α is very small. The wedge-like space 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 Ωaround 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 θalone, then the shear stress τ_<θφ> is given by τ_<θφ>=c/sin^2θ, where c is a constant. For a non-Newtonian liquid specified by an arbitrary flow curve f(τ), we get [numerical formula] The constant c is related to the torque M on the internal cone by the relationship M=c・2πa^3/3. For a non-Newtonian liquid obeying a power law flow curve f(τ)=kτ^n the following formula has been obtained: [numerical formula] For the special case where n is equal to unity, the above formula is reduced to the well-known formula for a Newtonian liquid: [numerical formula] 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 f(τ)=Σ__<n=1> A_nτ^n, (2) Bingham body and (3) non-Newtonian liquid obeying Casson's equation.
社団法人日本材料学会の論文
1968-04-15

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

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