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Bernoulli’s Theorem

Bernoulli’s Theorem

1. Isentropic motion

Here we will make recourse to a bit of thermodynamics. We can define the enthalpy as

(15)\[W = E + p V\]

where \(E\) is the inernal energy. Using the first law of thermodynamics, \(dE = T dS - p d V\), where \(S\) is the entropy, we find

(16)\[dW = T dS + V dp = T dS + \frac{1}{\rho} dp\]

For “isentropic” fluids, with no input of hea and no heat exchange between fluid parcels, \(dS = 0\) for each parcel, and \(\frac{dp}{d\rho} = dW\). Thus

\[\frac{\partial}{\partial t} {\vec v} + {\vec v} \cdot {\vec \nabla} {\vec v} = - {\vec \nabla} W\]

2. Constant density

By the same argument, if \(\rho\) is constant, then

\[ \frac{1}{\rho} {\vec \nabla} p = {\vec \nabla} \frac{p}{\rho} \]

3. Bernoulli’s theorem

Let us consider a velocity field \({\vec v}\) which is constant in time. In this case, the trajectories of fluid parcels follow the streamlines of the velocity field. COnsider an isotropic fluid with \(H = W\) or a constant density fluid with \(H = p/\rho\). Bernoulli’s theorem states that in these cases, \(H + \frac{1}{2} {\vec v}^2\) is constant along streamlines.

The proof starts with the vector identity

\[ {\vec A} \times ({\vec \nabla}\times {\vec B}) = A^i {\vec \nabla} B_i - {\vec A} \cdot {\vec \nabla} {\vec B} \]

(where we have used Einstein summation notation). Applying this to \({\vec A} = {\vec B} = {\vec v}\).

\[ {\vec v} \times ({\vec \nabla} \times {\vec v}) = \frac{1}{2} {\vec \nabla} {\vec v}^2 - {\vec v} \cdot{\vec \nabla} {\vec v} \]

Applying this to the Euler equation, and taking the case \(\partial_t {\vec v} = 0\), we find

(17)\[{\vec \nabla} (H + \frac{1}{2} {\vec v}^2) = {\vec v} \times({\vec \nabla}\times{\vec v})\]

Now the RHS is perpendicular to \({\vec v}\) since is a cross product with this vector, and thus

\[ {\vec v} \cdot {\vec \nabla} (H + \frac{1}{2} {\vec v}^2) = 0 \]

But \({\vec v} \cdot {\vec \nabla}\) corresponds precisely to the derivative along streamlines.

In the case of irrotational flow, we can write \({\vec v} = {\vec \nabla \phi}\). In this case, the right hand side of Equation (17) clearly vanishes, and we can state that \((H + \frac{1}{2} {\vec v}^2\) is constant throughout the fluid.

Irrotational flows also have a version of Bernoulli’s theorem when they are not steady. In this case \({\vec v} = {\vec \nabla} \psi\), and we can run the same arguments as above to write

\[ \partial_t \phi + H + \frac{1}{2} {\vec v}^2 = constant \]

and we can set the constant to zero by shifting \(\phi \to \phi + (constant t\) wihout changing \({\vec v}\).

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