Force on a moving surface

Force on a moving surface     Fig 11.5   Impingement of liquid jet on a moving flat plate If the plate in the above problem moves with a uniform velocity u in the direction of jet velocity V (Fig. 11.5). The volume of the liquid striking the plate per unit time will be Q = a(V − u)     (11.10) Physically, when the plate recedes away from the jet it receives a less quantity of liquid per unit time than the actual mass flow rate of liquid delivered, say by any nozzle. When u = V, Q = 0 and when u > V, Q becomes negative. This implies physically that when the plate moves away from the jet with a velocity being equal to or greater than that of the jet, the jet can never strike the plate. The control volume ABCDEFA in the case has to move with the velocity u of the plate. Therefore we have to apply Eq. (10.18d) to calculate the forces acting on the control volume. Hence the velocities relative to the control volume will come into picture. The velocity of jet relative to the control volume at its inlet becomes VR1 = V − u Since the pressure remains same throughout, the magnitudes of the relative velocities of liquid at outlets become equal to that at inlet, provided the friction between the plate and the liquid is neglected. Moreover, for a smooth shockless flow, the liquid has to glide along the plate and hence the direction of VR0, the relative velocity of the liquid at the outlets, will be along the plate. The absolute velocities of the liquid at the outlets can be found out by adding vectorially the plate velocity u and the relative velocity of the jet V - u with respect to the plate. This is shown by the velocity triangles at the outlets (Fig. 11.5). Coordinate axes fixed to the control volume ABCDEFA are chosen as ”0s” and ”0n” along and perpendicular to the plate respectively. The force acting on the control volume along the direction ”0s” will be zero for a frictionless flow. The net force acting on the control volume will be along ”0n” only. To calculate this force Fn, the momentum theorem with respect to the control volume ABCDEFA can be written as Substituting Q from Eq (11.10), Hence the force acting on the plate becomes (11.11) If the plate moves with a velocity u in a direction opposite to that of V (plate moving towards the jet), the volume of liquid striking the plate per unit time will be Q = a(V + u) and, finally, the force acting on the plate would be (11.12) From the comparison of the Eq. (11.9) with Eqs (11.11) and (11.12), conclusion can be drawn that for a given value of jet velocity V, the force exerted on a moving plate by the jet is either greater or lower than that exerted on a stationary plate depending upon whether the plate moves towards the jet or-away from it respectively. The power developed due to the motion of the plate can be written (in case of the plate moving in the same direction as that of the jet) as P = Fp . U   (11.13)