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Oscillating Flows of Fractionalized Second Grade Fluid

  • Jamil M
  • Khan N
  • Rauf A
ISRN Mathematical Physics (2012) 2012 1-23
DOI: 10.5402/2012/908386
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Abstract

New exact solutions for the motion of a fractionalized (this word is suitable when fractional derivative is used in constitutive or governing equations) second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of Laplace and finite Hankel transforms. These solutions are presented in series form in term of generalized G a , b , c ( ⋅ , t ) functions and satisfy all imposed initial and boundary conditions. In special cases, solutions for ordinary second grade and Newtonian fluids are obtained. Furthermore, other equivalent forms of solutions for ordinary second grade and Newtonian fluids are presented and written as sum of steady-state and transient solutions. The solutions for Newtonian fluid coincide with the well-known classical solutions. Finally, by means of graphical illustrations, the influence of pertinent parameters on fluid motion as well as comparison among different models is discussed.

Figures

  • Figure 1: Geometry of the problem for oscillating flows of fractionalized second grade fluid through a cylinder.
  • Figure 2: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, w1 w2 1, ν 0.5566, μ 33, α 0.5, β 0.5, and different values of t.
  • Figure 3: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, ν 0.5566, μ 33, α 0.5, β 0.5, and different values of w1 and w2, respectively.
  • Figure 4: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, w1 w2 1, ν 0.5566, μ 33, β 0.5, t 2.5 s, and different values of α.
  • Figure 5: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, w1 w2 1, ν 0.5566, μ 33, α 0.5, t 2.5 s, and different values of β.
  • Figure 6: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, w1 w2 1, ρ 59.289, α 0.1, β 0.8, t 2.5 s, and different values of ν.
  • Figure 7: Profiles of the velocity components w r, t and v r, t given by 3.18 and 3.19 , for R 3, W1 W2 V1 V2 1, w1 w2 1, ν 0.5566, μ 33, α 0.5, β 0.5, and different values of r.
  • Figure 8: Profiles of the velocity components w r, t given by 3.18 , for R 3, w1 1, ν 0.5566, μ 33, α 0.5, β 0.5, t 2.5 s, and different values ofW1 and W2.

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APA

Jamil, M., Khan, N. A., & Rauf, A. (2012). Oscillating Flows of Fractionalized Second Grade Fluid. ISRN Mathematical Physics, 2012, 1–23. https://doi.org/10.5402/2012/908386

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