An apparatus has a valve body defining a first passageway, a valve seat, and a flow aperture extending through the valve body and coupled in fluid communication with the first passageway. A valve cover of the apparatus is formed of an elastic material and includes a cover base mounted on the valve body and fixedly secured against movement relative thereto, and a valve portion overlying the valve seat. The valve portion defines a predetermined radial thickness and forms an interference fit with the valve seat. The valve portion and the valve seat define a normally closed, axially-extending valve opening therebetween. The valve portion is movable between a normally closed position with the valve portion engaging the valve seat, and an open position with at least a segment of the valve portion spaced away from the valve seat to connect the valve opening in fluid communication with the flow aperture to thereby allow the passage of fluid from the flow aperture through the valve opening. A hermetically sealed variable-volume storage chamber stores therein multiple portions of the fluid and is connectible in fluid communication with the one-way valve assembly. A pump is coupled between the variable-volume storage chamber and the one-way valve assembly, and is configured to pump discrete portions of fluid from the variable-volume storage chamber, through the flow aperture, and through the valve opening to dispense the portions of fluid through the valve assembly.