A Microscopic Generalization of Bernoulli's Equation to Include Low-Order Density Gradients

A microscopic generalization of Bernoulli's equation is established by appealing to low-order density gradient theory of an inhomogeneous liquid. This theory, used earlier to relate surface energy, bulk compressibility and the thickness of the liquid surface, is here generalized to embrace the case in which the inhomogeneous fluid is subjected to a velocity gradient to simulate the case of steady flow. Finally the theory is extended to include non-steady flow and contact is again established with Bernoulli's equation.