The stability of the helical flow of pseudoplastic liquids in a narrow annular gap with a rotating inner cylinder

The stability of a laminar helical flow of pseudoplastic liquids in an annular gap with a rotating inner cylinder is investigated theoretically. The analysis is carried out under the assumption of a torroidal form of the secondary flow (torroidal Taylor vortices) for the narrow gap geometry. The power law model has been applied to describe the pseudoplasticity of liquids. The problem of the stability has been formulated with the aid of the method of small disturbances, and solved using the Galerkin method. In order to describe the stability limit the Reynolds and Taylor numbers defined with the aid of the mean viscosity value have been introduced. It has been found that pseudoplasticity has a considerably destabilizing influence on the Couette motion as well as on the helical flow in the initial range of the Reynolds number values (Re<30). A decrease of the flow index value,n, is accompanied by a decrease of the critical value of the Taylor number. This destabilizing effect of pseudoplasticity vanishes in the range of the larger values of the Reynolds number. In the rangeRe>30, the stability limit of the flow of pseudoplastic liquids can be described by a general dependence of the critical valueTa _c onRe, which is consistent with results obtained for the case of Newtonian fluids. a frequency number (Eq. (27)), 1/s - b wave number (Eq. (27)), 1/m - B = M/N parameter - d = R ₂ –R ₁ gap width, m - f(y, B, k) function of viscosity distribution (Eq. (7)) - f ₀ (x) function of viscosity distribution (narrow gap Eq. (35)) - F(x) = V(x)/V _m dimensionless distribution of axial flow velocity - G(x) = U(x) _i dimensionless distribution of angular flow velocity - K consistency coefficient, N sⁿ/m² - M = (P/L)R ₂ parameter of the stress field (Eq. (1)), N/m² - M ₀ torque per unit length, N - n flow index - N = M ₀/(2R ₂ ² ) parameter of the stress field (Eq. (1)), N/m² - p = 1/2n–1/2 parameter - pressure disturbance amplitude, N/m² - p pressure disturbance, N/m² - (P/L) pressure drop per unit length of the gap, N/m² - r radial coordinate, m - r _m location of the maximum value of the axial velocity, m - R ₁,R ₂ inner, outer radius of the annulus, m - Re = V _m 2d/ _m Reynolds number - S = (P/L · d/N) parameteer of the stress field (narrow gap) - t time, s - Ta = _i d ^3/2 R ₁ ^1/2 / _m Taylor number - U tangential velocity, m/s - U _i tangential velocity at the surface of the inner cylinder, m/s - V axial velocity, m/s - V _m mean axial velocity, m/s - V disturbance vector of velocity field, m/s - amplitude of theV _k -disturbance, m/s - X, Y, Z functions in Eqs. (36–38) - y = r/R ₂ dimensionless radial coordinate - x = (r—(R ₁+R ₂)/2)d radial coordinate (narrow gap) - L ₁ L ₄ linear operators in Eqs. (36–38) - = ad/V _m dimensionless frequency number - = b·d dimensionless wave number - component of the rate of strain tensor, 1/s - component of the rate of strain tensor corresponding to the disturbance, 1/s - = R ₁/R ₂ radius ratio - apparent viscosity, Ns/m² - ₀ apparent viscosity in the main flow, Ns/m² - µ disturbance of the apparent viscosity, Ns/m² - µ _m mean apparent viscosity, Ns/m² - density, kg/m³ - _ij component of the stress tensor, N/m² - angular velocity, rad/s - _i angular velocity of the inner cylinder, rad/s