A positive-displacement rotary pump comprising: a casing (4, 104, 204) defining a pumping cavity (6, 106, 206); an inlet port (14, 114, 214) connected to the pumping cavity (6, 106, 206); a discharge port (16, 116, 216) connected to the pumping cavity; a positive-displacement auxiliary pumping port (18, 118, 218) connected to the pumping cavity (6, 106, 206) and being smaller in comparison to the discharge port (16, 116, 216); pumping elements (20, 120, 220) that move within the pumping cavity (6, 106, 206) of the casing (4, 104, 204) and define a collapsing pocket (42, 194, 230) that maintains fluid communication with the positive-displacement auxiliary pumping port (18, 118, 218) after the collapsing pocket (42, 194, 230) is no longer in fluid communication with the discharge port (16, 116, 216); wherein the casing configuration includes having the positive-displacement auxiliary pumping port (18, 118, 218) positioned sufficiently proximate the discharge port (16, 116, 216) to permit the positive-displacement auxiliary pumping port (18, 118, 218) to remain in fluid communication with the collapsing pocket (42, 194, 230) immediately following discontinuation of fluid communication between the discharge port (16, 116, 216) and the collapsing pocket (42, 194, 230); wherein fluid is positively-displaced through the positive-displacement auxiliary pumping port (18, 118, 218) at a flowrate that is substantially independent of the differential pressure of the pump (2, 102, 202) and/or fluid viscosity; and further comprising a passage that is in fluid communication with the positive-displacement auxiliary pumping port (18, 118, 218), wherein the passage is configured to direct positively displaced fluid to at least one dynamic seal positioned within the casing (4, 104, 204), to bearings (28, 134) positioned within the casing (4, 104, 204) and/or to an interior of an annular separation canister positioned within the casing (4, 104, 204). a casing (4, 104, 204) defining a pumping cavity (6, 106, 206); an inlet port (14, 114, 214) connected to the pumping cavity (6, 106, 206); a discharge port (16, 116, 216) connected to the pumping cavity; a positive-displacement auxiliary pumping port (18, 118, 218) connected to the pumping cavity (6, 106, 206) and being smaller in comparison to the discharge port (16, 116, 216); pumping elements (20, 120, 220) that move within the pumping cavity (6, 106, 206) of the casing (4, 104, 204) and define a collapsing pocket (42, 194, 230) that maintains fluid communication with the positive-displacement auxiliary pumping port (18, 118, 218) after the collapsing pocket (42, 194, 230) is no longer in fluid communication with the discharge port (16, 116, 216); wherein the casing configuration includes having the positive-displacement auxiliary pumping port (18, 118, 218) positioned sufficiently proximate the discharge port (16, 116, 216) to permit the positive-displacement auxiliary pumping port (18, 118, 218) to remain in fluid communication with the collapsing pocket (42, 194, 230) immediately following discontinuation of fluid communication between the discharge port (16, 116, 216) and the collapsing pocket (42, 194, 230); wherein fluid is positively-displaced through the positive-displacement auxiliary pumping port (18, 118, 218) at a flowrate that is substantially independent of the differential pressure of the pump (2, 102, 202) and/or fluid viscosity; and further comprising a passage that is in fluid communication with the positive-displacement auxiliary pumping port (18, 118, 218), wherein the passage is configured to direct positively displaced fluid to at least one dynamic seal positioned within the casing (4, 104, 204), to bearings (28, 134) positioned within the casing (4, 104, 204) and/or to an interior of an annular separation canister positioned within the casing (4, 104, 204).