caa a22 4500 465814379 CHVBK 20180323112123.0 cr unu---uuuuu 170327e19900901xx s 000 0 eng 10.1007/BF01376795 doi (NATIONALLICENCE)springer-10.1007/BF01376795 The stability of the helical flow of pseudoplastic liquids in a narrow annular gap with a rotating inner cylinder [Elektronische Daten] [S. Wroęski, M. Jastrzębski] 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. Steinkopff, 1990 Flow s tability nationallicence h elical flow nationallicence v ortices nationallicence power-law v iscosity nationallicence C ouette flow nationallicence a : frequency number (Eq. (27)), 1/s nationallicence b : wave number (Eq. (27)), 1/m nationallicence B = M/N : parameter nationallicence d = R 2 − R 1 : gap width, m nationallicence f(y, B, k) : function of viscosity distribution (Eq. (7)) nationallicence f 0 (x) : function of viscosity distribution (narrow gap Eq. (35)) nationallicence F(x) = V(x)/V m : dimensionless distribution of axial flow velocity nationallicence G(x) = U(x) i : dimensionless distribution of angular flow velocity nationallicence K : consistency coefficient, N sn/m2 nationallicence M = (ΔP/L)R 2 : parameter of the stress field (Eq. (1)), N/m2 nationallicence M 0 : torque per unit length, N nationallicence n : flow index nationallicence N = M 0/(2πR22) : parameter of the stress field (Eq. (1)), N/m2 nationallicence p = 1/2 n -1/2 : parameter nationallicence $$\hat p$$ : pressure disturbance amplitude, N/m2 nationallicence p′ : pressure disturbance, N/m2 nationallicence ( ΔP/L ) : pressure drop per unit length of the gap, N/m2 nationallicence r : radial coordinate, m nationallicence r m : location of the maximum value of the axial velocity, m nationallicence R 1, R 2 : inner, outer radius of the annulus, m nationallicence Re = V m ϱ2d/ε m : Reynolds number nationallicence S = (ΔP/L · d/N) : parameteer of the stress field (narrow gap) nationallicence t : time, s nationallicence Ta = ω i d 3/2 R 11/2ϱ/ε m : Taylor number nationallicence U : tangential velocity, m/s nationallicence U i : tangential velocity at the surface of the inner cylinder, m/s nationallicence V : axial velocity, m/s nationallicence V m : mean axial velocity, m/s nationallicence V′ : disturbance vector of velocity field, m/s nationallicence $$\hat V_k$$ : amplitude of theV k ′ -disturbance, m/s nationallicence X, Y, Z : functions in Eqs. (36-38) nationallicence y = r/R 2 : dimensionless radial coordinate nationallicence x = (r—(R 1+ R 2)/2) d : radial coordinate (narrow gap) nationallicence L 1 ⋯L 4 : linear operators in Eqs. (36-38) nationallicence α = ad/V m : dimensionless frequency number nationallicence β = b·d : dimensionless wave number nationallicence $$\dot \gamma _{ij}$$ : component of the rate of strain tensor, 1/s nationallicence $$\dot \gamma '_{ij}$$ : component of the rate of strain tensor corresponding to the disturbance, 1/s nationallicence η = R 1/ R 2 : radius ratio nationallicence ε : apparent viscosity, Ns/m2 nationallicence ε 0 : apparent viscosity in the main flow, Ns/m2 nationallicence µ′ : disturbance of the apparent viscosity, Ns/m2 nationallicence µ m : mean apparent viscosity, Ns/m2 nationallicence ϱ : density, kg/m3 nationallicence τ ij : component of the stress tensor, N/m2 nationallicence ω : angular velocity, rad/s nationallicence ω i : angular velocity of the inner cylinder, rad/s nationallicence Wroęski S. Institute of Chemical and Process Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut Jastrzębski M. Institute of Chemical and Process Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut Rheologica Acta Steinkopff-Verlag 29/5(1990-09-01), 442-452 0035-4511 29:5<442 1990 29 397 https://doi.org/10.1007/BF01376795 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01376795 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Wroęski S. Institute of Chemical and Process Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut NATIONALLICENCE 700 1- Jastrzębski M. Institute of Chemical and Process Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut NATIONALLICENCE 773 0- Rheologica Acta Steinkopff-Verlag 29/5(1990-09-01), 442-452 0035-4511 29:5<442 1990 29 397 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer