Introduction to Fluid Mechanics

Fluids and Vector Calculus

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This clip explains why, in continuum mechanics, we use partial derivatives rather than ordinary derivatives.

Fluid mechanics usually involves partial derivatives rather than ordinary derivatives. This is because we model fluids as continuous materials in three-dimensional space. This clip also mentions the Knudsen number and the point at which the continuum model will break down.

The del operator (also called nabla) is a multi-function mathematical operator in vector calculus. It is particularly powerful because its meaning is independent of the coordinate system.

The Del Operator (the upside-down triangle) is one of the most useful operators in fluid mechanics. This clip shows how the Del operator can be used to find the gradient of a scalar field and the divergence of a vector field. Both Cartesian and radial polar coordinates are used and the clip shows why you have to be particularly careful in polar coordinates. (There is a small mistake at 18:00 minutes. 1/r d(a_theta)/d(theta) should be 1/r^2 d(a_theta)/d(theta).)

When del acts directly on a scalar field, it gives the gradient of that field. This is a vector that points in the direction of maximum steepness with magnitude equal to the maximum steepness.

The law of conservation of mass is derived in two dimensions and expressed using the del operator.

The dot product of del with a vector field gives the divergence of that vector field. In Fluid Mechanics this has a clear physical interpretation: the divergence of streamlines away from a point.

The equation of state of a fluid determines the relationship between three thermodynamic properties: density, pressure, and temperature. Setting the density to be uniform and constant is a useful simplification in Fluid Mechanics but precludes us from using any other equation of state.

In fluid mechanics, the equation of state is usually expressed as an equation that relates the density to the pressure and the temperature of a material. There is no universal equation of state so we have to use one of the many possible equations of state, such as the ideal gas equation of state or the Van der Waal's equation of state or the incompressible equation of state. This clip introduces these equations of state and poses the question; 'what do you do with the other equations of state if you assume that a flow is incompressible'?

The vector product of del with a vector field gives the curl of that vector field. In Fluid Mechamics this has a clear physical interpretation: twice the angular velocity of the fluid at that point, with a pseudo-vector pointing along the local axis of rotation.

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