Isotopic evidence of the chemical variability of basalts implies the existence of distinct, long-lived chemical reservoirs in the mantle. This has commonly been interpreted as evidence that mantle convection is layered, since known models of whole-mantle convection have too much shearing to allow preseration of these reservoirs. However, the geochemical evidence requires at least three reservoirs and models of layered convection have dynamical difficulties, casting serious doubt on the viability of these models. This difficulty has been resolved in the present paper by the development of a simple model of whole-mantle convection which allows the preservation of chemically distinct reservoirs. In this model, heating of the mantle from below produces axisymmetric plumes which carry heat upward through the mantle without causing over-turning. Hence the lower mantle is much less sheared than in the usual two-dimensional model. Mantle convection is driven by the penetration of cold lithospheric slabs from above. It is argued that the pattern of convection is governed primarily by the dependence of viscosity on temperature and this dependence drives the mantle toward an isoviscous state. These ideas are quantified in a simple one-dimensional model of mantle convection in which the slab mass flux as a function of radius is governed by the requirement that the temperature profile remains isoviscous. The resulting convection cycles material through the upper mantle more rapidly than through the lower mantle. This provides a mechanism for an initially homogeneous mantle to become chemically stratified with a depleted upper mantle and primitive lower mantle.