Steady Motion of Viscous Fluid Due to A Slowly Motion Under Sphere Through Porous Medium

In this Paper we have investigate the steady motion of viscous liquid due to a slowly rotating sphere through porous medium. We have investigate the angular velocity, couple and rate of dissipation energy. When a solid sphere rotates inside a viscous fluid, the fluid near the surface of the sphere is dragged along due to viscous forces. If this motion happens inside a porous medium, the fluid flow becomes more complex because the pores offer resistance to the movement of the liquid,Oil recovery from porous rocks, Groundwater flow, and Filtration and chemical processes involving slow motion in porous materials. By analyzing the motion mathematically, we can determine: How the fluid velocity changes with distance from the sphere, How much torque is needed to keep the sphere rotating, and How much energy is lost due to viscous friction. These results help to better understand and predict the behavior of viscous fluids in porous materials under rotational motion." A rigid sphere is embedded in, or surrounded by, a porous medium. The porous medium is saturated by a viscous incompressible fluid. The sphere rotates slowly with a constant angular velocity about some axis. Far from the sphere (in the porous medium) the fluid is essentially at rest (or some known boundary condition). The problem is steady (time = constant, no transient effects). We are interested in how the rotation of the sphere drives flow in the porous medium + fluid, and in particular, the angular velocity distribution of the fluid (how the fluid "spins" as a function of radius/position)the torque (or couple) required to maintain rotation of the sphere the rate of viscous (and porous?medium) dissipation of energy.