There is a common perception that displacing mud in horizontal wellbores is made ineffective by large density differences due to stratification of the fluid layers along the annulus. Hence the industry commonly employs methods such as casing rotation to move fluids around the annulus. Whilst such methods may be effective, they are not always necessary. Using a mathematical model of the process, we show that even with large density differences it is still possible to have a steadily advancing displacement front that displaces mud all around the annulus without casing rotation. The conditions when this is possible depend on the fluid rheologies, flow rate and hole geometry, but not the density difference. These conditions are easily quantified. Although density difference does not enter into the criteria for an effective displacement, the length of the interface along the wellbore does scale with the density difference. This means that, provided one is able to tolerate additional fluid volumes, we are able to effectively displace the mud even using fluids with large density differences. This new understanding of the displacement fluid mechanics opens up new opportunities for cementing difficult horizontal wells, (e.g. slimhole), where casing rotation is not possible.