Conservation of mass along a moving element

Let a scalar field describe temperature and let a material path describe the element’s position. The chain rule differentiates the composite temperature evaluated on that path. One contribution changes the field at the current fixed position; another comes from crossing the field’s spatial gradients. The result is the derivative following fluid material. A separately propelled sensor has its own velocity, so its sampling derivative need not equal the material derivative of the fluid.

For velocity itself, apply the same chain rule component by component in a fixed Cartesian frame. A steady nozzle can accelerate fluid because faster locations lie downstream. Setting the partial time derivative to zero only removes the local contribution; it does not remove spatial acceleration. Curvilinear coordinates require derivatives of the basis directions as well, which is why a constant speed around a circle still has acceleration.

Choose a fixed volume and orient its surface normal outward. The total mass inside can change only through boundary flux when there are no mass sources. Outward flux decreases storage, establishing the minus sign. The divergence theorem converts surface flux into a volume integral. Since the volume can be arbitrarily small in a smooth continuum region, its integrand must vanish, giving the differential continuity equation.

Expand the divergence of density times velocity using the product rule. The density gradient contribution combines with the partial density derivative to make the material derivative. What remains is density times velocity divergence. Positive divergence describes local volumetric expansion and therefore decreases the density of a conserved material element. Steady density at a fixed point can coexist with changing density along the flow because fresh material keeps arriving there.

An incompressible motion preserves each infinitesimal material volume and has zero velocity divergence. Conservation of mass then gives constant density along each element’s trajectory. The density need not be the same for every element: an idealized stratified incompressible flow can transport a nonuniform density field. Uniform density that is fixed everywhere and in time is a stronger condition, often useful when thermal expansion and composition changes are negligible.

A low-speed gas is not automatically incompressible under strong heating, and a liquid may require compressibility to describe sound. State the processes and accuracy being retained before dropping density changes. At a discontinuity, the differential form must be interpreted through a conservative integral balance rather than derivatives of an invented smooth interpolation. For a steady one-dimensional duct with uniform section averages and impermeable walls, constant mass flow gives density times mean speed times area constant.