Nonlinear Taylor stability of viscoelastic fluids期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Nonlinear Taylor stability of viscoelastic fluids

Authors:	C. F. Chan Man Fong

Affiliation:	(1) Dept. of Mathematics, University of Reading, Reading, England

Abstract:	Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.Nomenclature a amplitude of the fundamentals - A_ij⁽¹⁾, A_ij⁽²⁾ first and second Rivlin-Ericksen tensors - d r₂–r₁ - D d/dx - E $frac{{2pi r_1 r_o eta _0^2 h}}{{rho d^2 }}left( {1 - frac{{I_1 I_4 }}{{I_3 }}} right)$ - F $frac{{2pi r_1^2 r_o eta _0^2 T_c h}}{{rho d^3 }}frac{{I_1 I_4 }}{{I_3 }}$ - g_ij metric tensor - G torque on the inner cylinder in the supercritical regime - h height of the cylinders - k₀ /d² - k₁ /d² - I₁ $mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} - uupsilon dx + left( {k_o + 2k_1 } right)r_1 /dmathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} upsilon (D^2 - lambda ^2 ) u dx$ - I₂ $mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} [(D^2 - lambda ^2 ) u]^2 dx$ - I₃ $begin{gathered} (1 + 2k_0 lambda ^2 )(mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} - uv dx)^2 - (1 + k_0 lambda ^2 )mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} u^2 v^2 dx + hfill + left( {k_0 + 3k_1 } right)mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} - uv dxmathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} u^2 v dx + hfill + k_0 mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} (2u^3 v + 4uv Du Dv + 2u^2 (Dv)^2 + hfill + v^2 (Du)^2 )dx + k_1 mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} (u^2 (Dv)^2 + v^2 (Du)^2 + 2uv Dv Dv + 3u^3 v) dx hfill end{gathered}$ - I₄ $(1 - 2k_0 lambda ^2 )mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} - uv dx + (k_0 - 2k_1 )mathop smallint limits_{ - tfrac{1}{2}}^{tfrac{1}{2}} u^2 dx$ - r₁, r₂ radii of inner and outer cylinders respectively - r₀ 1/2(r₁+r₂) - R Reynolds number ₁r₁d/₀ - R_c critical Reynolds number - T Taylor number r₁₁²d³²/₀²) - T_c critical Taylor number - u₁, v₁, w₁ Fundamentals of the disturbance - u_i, v_i, w_i, (i>1) harmonics - mean velocity (not laminar velocity) - u –u₁/ar₁₁ - v v₁/Rar₁₁ - x (r–r*₀)/d - , material constants - 0 viscosity - wave number d - density - ₁ angular velocity of inner cylinder - tilde denotes complex conjugate

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