The heat induced by viscous dissipation in a microchannel fluid, due to a small oscillating motion of the lower plate, is investigated for the first time. The methodology is by applying the momentum and energy equations and solving them for three cases of standard thermal boundary conditions. The first two cases involve symmetric boundary conditions of constant surface temperature on both plates and both plates insulated, respectively. The third case has the asymmetric conditions that the lower plate is insulated while the upper plate is maintained at constant temperature. Results reveal that, although the fluid velocity is only depending on the oscillation rate of the plate, the temperature field for all three cases show that the induced heating is dependent on the oscillation rate of the plate, but strongly dependent on the parameters Brinkman number and Prandtl number. All three cases prove that the increasing oscillation rate or Brinkman number and decreasing Prandtl number, when it is less than unity, will significantly increase the temperature field. The present model is applied to the synovial fluid motion in artificial hip implant and results in heat induced by viscous dissipation for the second case shows remarkably close agreement with the experimental literature.

The synovial fluid motion in an artificial hip joint is important in understanding the thermo-fluids effects that can affect the reliability of the joint, although it is difficult to be studied theoretically, as the modelling involves the viscous fluid interacting with a moving surface. A new analytical solution has been derived for the maximum induced fluid motion within a spherical gap with an oscillating lower surface and a stationary upper surface, assuming one-dimensional incompressible laminar Newtonian flow with constant properties, and using the Navier-Stokes equation. The resulting time-dependent motion is analysed in terms of two dimensionless parameters R and β, which are functions of geometry, fluid properties and the oscillation rate. The model is then applied to the conditions of the synovial fluid enclosed in the artificial hip joint and it is found that the motion may be described by a simpler velocity variation, whereby laying the foundation to thermal studies in the joint.

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