Simple mathematical models were developed to analyze the mechanics of feeding on fluids. These models combine the mechanics of the muscular pump whose contractions drive fluid flow, and the fluid mechanics of flow through the insect’s mouthparts. These models were applied to two examples, nectar feeding in butterflies and blood feeding in mosquitoes. Model simulations predict that the rate of energy intake during nectar feeding is maximized at nectar concentrations of 35-45% sucrose; high energy intake rates require that the relative sizes of the muscular pump and the food canal show certain relationships. Similarly, the model predicts that the rate of protein intake during blood feeding is maximized at blood hematocrits (red blood cell [RBC] concentrations) of 60-65%. For both nectar and blood feeding, these predictions result from the exponential relationship between the concentration of dissolved or suspended nutrients and viscosity, as well as the interaction between the mechanical power produced by the muscular pump and the mechanical resistance to flow through the food canal. We discuss several aspects of the mechanical properties of blood that may affect blood intake rates, including the complex relationship of food canal size, RBC size, and blood viscosity, and the changes in blood viscosity resulting from blood parasites.