A new experimental method for determining the pattern of flow and three‐dimensional progressive finite deformation of fluid elements in low Reynolds number (Re) viscous flow has been applied to creeping flow past a rigid sphere falling in a cylinder. This is intended as a model analogue of natural rock deformation associated with magmatic and solid‐state diapirs rising through ductile crust. The kinematics of finite strain in such analogue experiments may give useful information on the development of natural structures. The experiment and a new method for quantifying the progressive finite deformation of fluid volumes in the transparent Newtonian test fluid used (SGM36) are discussed in detail. The progressive finite deformation of fluid flowing along specific streamlines passing the sphere appears to be complex and largely a function of the initial distance of fluid volumes and the particular streamlines which they follow from the flow axis. The progressive deformation history in the material surrounding a rising spherical diapir is characterized by progressive flattening and rotation of material above and progressive stretching below the sphere. The results compare favorably with the external strain pattern associated with post tectonic, mesozonal diapiric intrusions.