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Translatory accelerating motion of a circular disk in a viscous fluid

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Abstract

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.

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Abbreviations

a :

radius of the circular disk

b :

one half of the thickness of the circular disk

C :

dimensionless form of C 1

C 1 :

magnitude of the constant force

D :

fluid drag force

f(t) :

externally applied force

F(τ):

dimensionaless form of applied force

F 0 :

initial value of F

g :

gravitational acceleration

H(τ):

Heaviside step function

k :

magnitude of impulsive force

K :

dimensionless form of k

M :

a dimensionless parameter equals to (1+37#x03C0;ερs/4ρf)

S :

displacement of disk

t :

time

t 1 :

time of application of impulsive force

u :

velocity of the disk

V :

dimensionless velocity

V 0 :

initial velocity of V

V t :

terminal velocity

α :

parameter in (13)

β :

parameter in (13)

δ(t):

Dirac delta function

ε :

ratio of b/a

Φ(τ):

function given in (5)

μ :

dynamical viscosity of the fluid

ν :

kinematic viscosity of the fluid

ρ f :

fluid density

ρ s :

mass density of the circular disk

τ :

dimensionless time

τ i :

dimensionless form of t i

ξ :

dummy variable

η :

dummy variable

References

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  2. Lai, R. Y. S. and L. F. Mockros, The Stokes-flow Drag on Prolate and Oblate Spheroids during Axial Translating Accelerations, J. Fluid Mech. 52, part 1 (1972) 1.

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  3. Basset, A. B., A Treatise on Hydrodynamics 2, Dover, 1888.

  4. Lai, R. Y. S., Stokes-flow Solution for an Accelerating Spheroid, Ph.D. dissertation, Northwestern University, Evanston (Ill.), U.S.A., 1969.

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  5. Squires, L. and W. Squires, Jr., Trans. AIChE 33 (1937) 1.

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  6. Happel, J. and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Inc., 1965, 149.

  7. Basset, A. B., On the Decent of a Sphere in a Viscous Fluid, Quart. J. Math. 41 (1910) 369.

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Authors and Affiliations

  1. Departments of Energetics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA

    R. Y. S. Lai

Authors
  1. R. Y. S. Lai
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Lai, R.Y.S. Translatory accelerating motion of a circular disk in a viscous fluid. Appl. Sci. Res. 27, 440–450 (1973). https://doi.org/10.1007/BF00382506

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  • DOI: https://doi.org/10.1007/BF00382506

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