        Introduction to Fluid Mechanics   Overview Picturing Fluids Fluids and Vector Calculus Inviscid Flow and Bernoulli Viscous Flow Boundary Layers Laminar/Turbulent Pipe Flow Pipe Flow Networks Boundary Layers External Flow and Drag Dimensional Analysis/Scaling Compressible Flow   # Inviscid Flow and Bernoulli's equation   Blank handout  2.1 Changes due to motion through a field (08:21)

As a blob of fluid moves, it experience changes in its properties (such as pressure and velocity) due to its motion through the fluid. This clip describes why and also introduces the material derivative, D/Dt.  Aside: The material derivative (11:58)

Although we model fluids as if they are continuous materials, we can still identify blobs of fluid within that field. For example, we could drop a blob of ink into water and follow the ink as it moves around. If we knew the velocity field in the fluid, we could use the material derivative to calculate the velocity of that blob of fluid. This clip explains the material derivative by using the example of a punt drifting down the river next to the Mill pub, opposite Queen's College in Cambridge.  2.2 Euler's equation: Newton's second law in an inviscid fluid (03:46)

Euler equation is Newton's second law (f=ma) applied to an inviscid fluid. This clip derives Euler's equation by applying f=ma to a cube of fluid and calculating its acceleration from the material derivative of the velocity field  Aside: Euler's equation (06:56)

Euler's equation is simily f=ma written for an inviscid fluid. In this clip, Euler's equation is derived by considering the forces on a fluid blob and its resultant acceleration. At the end of the clip, Euler's equation is linked to Bernoulli's equation and to streamline curvature.  2.3 Euler's equation along a straight streamline (03:26)

Euler's equation can be integrated along a straight streamline to give Bernoulli's equation.  2.4 Benoulli's equation and streamline curvature (01:29)

Euler's equation can be integrated along a curved streamline to give (i) Bernoulli's equation and (ii) a relationship between the streamline curvature and the cross-stream pressure gradient.  2.5 Determining the pressure field from the streamlines (01:18)

By examining the streamlines in the steady flow around an object, we can determine the pressure field around the object.  Worked example: flow over a river bed (10:35)

This worked example concerns the flow over a wavy river bed. In the first part of the example, the velocity field is sketched. This worked example also contains step-by-step instructions on how to handle the Del operator (the upside-down triangle).  Worked example: point source in 2D (10:43)

This worked example uses the material derivative to find the acceleration of a fluid blob in a velocity field. The worked example is in polar coordinates, which means that we have to be careful about the directions of the unit vectors when using the Del operator.  Aside: Including gravity in Euler's equation (04:37)  Aside: Applying Bernoulli across streamlines (mathematical) (07:09)  Aside: Applying Bernoulli across streamlines (physical) (03:41)  Completed handout

Check your notes against this completed handout   © Matthew Juniper matthewjuniper@learnfluidmechanics.org