Summary
The problem of the creeping flow field induced by a spinning sphere including couple stress effects is studied analytically. The solution turns out to be somewhat more complex, in an algebraic sense, than the problems of a uniformly translating sphere or a sphere in a shear flow. By the same token, the spinning sphere problem should provide more information regarding rheological parameters and boundary conditions employed in couple stress theories. The solution for the boundary condition of no slip of rotation is written out in full detail. An interesting feature of the solution is that it is necessary to employ a non solenoidal (in fact, irrotational) component in the “inherent angular velocity” or “spin velocity” field to properly satisfy the boundary conditions.
Zusammenfassung
Das im Titel genannte Problem wird analytisch untersucht. Es zeigt sich, daß die Lösung algebraisch komplizierter ist als die der Probleme einer translatorisch gleichförmig bewegten Kugel oder einer Kugel in einer Scherströmung. Dieses Problem sollte aber auch mehr Information über die rheologischen Größen und die in der Momentenspannungstheorie verwendeten Randbedingungen liefern. Die Lösung für die Rand-bedingung verschwindender Differenzgeschwindigkeit der Rotation wird im Detail angegeben. Eine interessante Eigenschaft der Lösung ist, daß es sich als notwendig erweist, eine nichtdivergenzfreie Komponente des “Springeschwindigkeitsfeldes” zu verwenden, um die Rand-begingungen erfüllen zu können.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a :
-
constant
- A 2 :
-
non dimensional parameter, 2μR 2/θ
- b :
-
constant
- c :
-
constant
- e :
-
unit alternating tensor
- F :
-
constant defined below
- F 1 :
-
function defined below
- F 2 :
-
function defined below
- \(\bar M\) :
-
moment per unit area on a spherical surface centred at centre of sphere
- k 1 :
-
parameter {−(η+θ+τ)/2γ}1/2
- k 2 :
-
parameter\(\left\{ {\frac{\theta }{2}\left[ {\frac{1}{u} - \frac{1}{\gamma }} \right]} \right\}^{1/2} \)
- k :
-
non dimensional parameter,R/k 2
- k′ :
-
non dimensional parameter,R/k 1
- P :
-
pressure
- R :
-
radius of sphere
- r :
-
distance of field point from centre of sphere
- \(\dot r\) :
-
rotational strain rate
- \(\bar T\) :
-
force per unit area on a spherical surface centred at centre of sphere
- \(\bar \upsilon \) :
-
translational velocity field
- \(\bar \upsilon ^{\left( 0 \right)} \) :
-
portion of\(\bar \upsilon \) satisfying classical equations
- \(\bar V\) :
-
velocity of\(\bar \omega \times \bar r\)
- x :
-
cartesian coordinate
- α:
-
constant appearing in boundary condition (15)
- β:
-
constant appearing in boundary condition (14)
- \(\dot \varepsilon \) :
-
strain rate
- γ:
-
rheological parameter
- δ:
-
Kronecker delta
- φ:
-
scalar function used in representing\(\bar \Omega \)
- λ:
-
rheological parameter
- μ (unsubscripted):
-
rheological parameter
- μ (subscripted):
-
couple stress tensor
- η:
-
rotational viscosity parameter
- \(\bar \omega \) :
-
angular velocity of sphere
- Ω:
-
“intrinsic angular velocity” or “spin velocity” field of fluid
- \(\bar \Omega _1 \) :
-
vector function used in representing\(\bar \upsilon , \bar \Omega _1 \)
- σ:
-
force stress tensor
- τ:
-
rheological (rotational viscosity) parameter
- θ:
-
rheological (rotational viscosity) parameter
- \(\bar v\) :
-
vector normal to solid fluid interface (pointing into fluid)
- i, j, k, l :
-
integers 1, 2 or 3
- n :
-
normal tangential
References
Aero, E. L., A. N. Bulygin andE. V. Kuvshinskii: Asymmetric Hydromechanis. J. Appl. Math. Mech.29, 333–346 (1965).
Brenner, H.: Rheology of Two-Phase Systems, pp. 137–176. Annual Reviews of Fluid Mechanics, Vol.II (1970).
Landau, L. D., andE. M. Lifshitz: Fluid Mechanics. Addison Wesley. 1959.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mani, R. Creeping flow induced by a spinning sphere including couple stresses. Acta Mechanica 18, 81–88 (1973). https://doi.org/10.1007/BF01173459
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01173459