The flow pattern away from flow boundaries is accurately described by conservation equations where the effects of viscosity are neglected. Such flows can also be described as being "irrotational". In such a field, it is possible to define a scalar potential, whose gradient is the velocity vector.

When the flow speed is low compared to the speed of sound, we can prove that the fluctuations in density that are caused by the largest possible variations in velocity, are also small. In fact, if the "Mach number", which is the ratio of flow speed to speed of sound, is less than 0.3, the largest density fluctuation caused by speed changes, is less than 5%. Such flows are described as being "incompressible flows" though the fluid itself, if it is a gas, can be compressed.

Here the mass conservation equation becomes simply the statement that the dilatation of the flow is zero, or that the divergence of the velocity vector is zero (the flow cannot expand or contract, since density changes are negligibly small). The concept of velocity potential allows us to reduce the mass conservation equation to the Laplace equation. Thus, velocity fields can be obtained as solutions of the Laplace equations for the specified boundary conditions of the flow problem. The pressure field can be found independently by applying momentum conservation, in the form of Bernoulli's equation, to the velocity field. Please download the notes in pdf here.

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