On unsteady hydromagnetic flows of a dusty viscous fluid between two oscillating plates

A study is made of the unsteady motion of an incompressible viscous conducting fluid with embedded small spherical particles bounded by two infinite rigid non-conducting plates. The operational method derives exact solutions for the fluid and the particle velocities and the wall shear stress. The quantitative evaluation of these results is considered when the two plates oscillate in phase but with different frequencies. The results are shown graphically for different values of the time period of oscillations of the plates which represent the cases: (i) the lower plate oscillates with time period less than the upper, (ii) both the plates oscillate with the same time period, (iii) the lower plate oscillates with time period greater than the upper. The magnetic field damps the fluid motion for all values of the time period of oscillations of the plates. When the time periods are small, i.e., when the plates oscillate with high frequency, the fluid motion is retarded by the particles. However, when the plates oscillate with larger time periods (smaller frequencies), the fluid velocity is increased by the presence of the particles at the early stage of the motion, and this effect persists until the equilibrium is reached when the particles exert their influence to resist the flow. The drag on the plate, which is evaluated numerically for the lower plate oscillating with large time period, depends on the ratio of the time periods of the oscillating plates. If the ratio of the time periods is not equal to unity, the drag on the plate, irrespective of the values of the magnetic field, oscillates with larger amplitude compared to its value when the ratio of the time periods is equal to unity. Further, for the ratio of the time periods less than or equal to unity and for any fixed values of the magnetic field, the drag increases by the presence of the particles after a time t ≈ 1.2 which is the upper time limit for the non-equilibrium stress-value to exist. In a similar situation, a reverse effect, i.e., the decrease of the drag with increasing particle concentration, is found for the ratio of the time periods being greater than unity.