Euler’s Equation

This week we have been looking at Euler’s equation.

We can look at the dynamics of inviscid flows through Euler’s equation. This equation, together with the equation of continuity, specifies the dynamic of inviscid flows. Euler’s equation can be simplified under certain circumstances to give Bernoulli’s equation.

If we use S to denote the surface of a small volume V of a moving fluid, the volume moving with the fluid with velocity u. The mass of the volume of fluid is thus Rho*V, where Rho is the density of the fluid.

If we call deltaS the surface element of Sn is the unit outward-pointing normal to deltaS and p is the fluid pressure on deltaS. The force on the surface element is:  CodeCogsEqn(8)

We can now say that the total surface force acting over S is:  CodeCogsEqn(8)

The total body force is:  CodeCogsEqn(8)

If we use Newton’s Second Law, the equation of motion of the fluid volume V is:  CodeCogsEqn(8)

Now, if we allow the volume V to tend to zero, we get:  CodeCogsEqn(8)

And lastly, if we divide through by Rho*V  and taking the limit as V->0 we get:  CodeCogsEqn(8) which is Euler’s Equation.

This week, I feel like I’m starting to grasp more on fluids, and luckily we went over dimensional analysis so I could refresh my memory as we are about to settle down to learn a new topic!

Example sheet 3, question 1:

IMG_0090[2]

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