Abstract
The problem of the longitudinal oscillations of a circular cylinder along its axis of symmetry in an incompressible micro-polar fluid and the flow generated due to these oscillations in the fluid is considered. The Stokes flow is considered by neglecting nonlinear convective terms in the equations of motion on the assumption that the flow is so slow that oscillations’ Reynolds number is less than unity. Here we get a rare, but distinct special case referred to as resonance in which material constants are interrelated in a particular way. In non-resonance case, all material constants are independent and are not related. The solution in this case cannot be obtained as limiting case of a non-resonance problem. The velocity and micro-rotation components of the flow for the case of resonance and non-resonance are obtained. The skin friction acting on the cylinder is evaluated and the effect of physical parameters like micro-polarity and couple stress parameter on the skin friction due to oscillations is shown through graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \( \bar{Q} \) :
-
Fluid velocity vector (ms−1)
- \( \bar{l} \) :
-
Micro-rotation vector
- \( \rho \) :
-
Density of the fluid (kg m−3)
- \( \tau \) :
-
Time (s)
- \( P \) :
-
Fluid pressure at any point (kg m−1 s−2)
- \( W \) :
-
Velocity component (m s−1)
- \( {\mathcal{B}} \) :
-
Micro-rotation component
- \( \sigma \) :
-
Frequency parameter (s−1)
- \( J \) :
-
Gyration coefficient (kg m s−1)
- \( \bar{q} \) :
-
Non dimensional Fluid velocity vector
- \( \bar{\upsilon } \) :
-
Non dimensional Micro-rotation vector
- \( t \) :
-
Non dimensional time
- \( p \) :
-
Non dimensional Fluid pressure at any point
- \( w \) :
-
Non dimensional Velocity component
- \( B \) :
-
Non dimensional Micro-rotation components
- \( \varpi \) :
-
Non dimensional Frequency parameter
- \( j \) :
-
Non dimensional Gyration coefficient
- \( \mu \) :
-
Viscosity coefficient (kg m−1 s−1)
- \( k \) :
-
Micro-viscosity coefficient (kg m−1 s−1)
- \( \alpha , \beta , \gamma \) :
-
Couple-stress viscosity coefficients (kg m−1 s−1)
- \( T_{ij} \) :
-
Stress components
- \( M_{ij} \) :
-
Couple-stress components
- \( s \) :
-
Couple-stress parameter for micro-polar fluid
- \( c \) :
-
Cross viscosity coefficient or micro-polarity parameter
- \( R_{0} \) :
-
Oscillations Reynolds number
References
Eringen A C 1966 Theory of micropolar fluids. J. Math. Mech. 16: 1–18
Lakshmana Rao S K and Bhuganga Rao P 1971 The oscillations of a sphere in a micro-polar fluid. I.J.E.S. 9: 651–672
Lakshmana Rao S K and Bhuganga Rao P 1972 Circular cylinder oscillating about a mean position in an incompressible micro-polar fluid. I.J.E.S. 10: 185–191
Lakshmana Rao S K and Iyengar T K V 1983 Rotary oscillations of a spheroid in an incompressible micro-polar fluid. I.J.E.S. 21: 973–987
Lakshmana Rao S K, Iyengar T K V and Venkatapathi Raju K 1987 The rectilinear oscillations of an elliptic cylinder in an incompressible micro-polar fluid. I.J.E.S 25: 531–548
Ramana Murthy J V, Sai K S and Bahali N K 2011 Steady flow of micropolar fluid in a rectangular channel under transverse magnetic field with suction. In: AIP ADVANCES1, 032123, pp. 1–10
Aparna P and Ramana Murthy J V 2012 Rotary oscillations of a permeable sphere in an incompressible micropolar fluid. Int. J. Appl. Math. Mech. 8: 79–91
Nagaraju G and Ramana Murthy J V 2014 Unsteady flow of a micropolar fluid generated by a circular cylinder subject to longitudinal and torsional oscillations. Theor. Appl. Mech. TEOPM7 41: 71–91
Ramana Murthy J V, Bhaskara Rao G S and Govinda Rao T 2014 Resonance flow due to rectilinear oscillations of a circular cylinder in a micropolar fluid. In: 59th Proceedings of ISTAM, pp. 1–7
Ramana Murthy J V, Bhaskara Rao G S and Govinda Rao T 2015 Resonance flow due to rectilinear oscillations of a sphere in a micropolar fluid. J. Phys. Conf. Ser. 662: 1–8
Ramana Murthy J V, Bhaskara Rao G S and Govinda Rao T 2015 Resonance flow due to rotary oscillations of a sphere in a micropolar fluid. Proc. Eng. 127: 1323–1329
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Govinda Rao, T., Ramana Murthy, J.V. & Bhaskara Rao, G.S. Longitudinal oscillations of a circular cylinder in a micro-polar fluid: case of resonance. Sādhanā 44, 66 (2019). https://doi.org/10.1007/s12046-018-1004-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12046-018-1004-x