FLUIDS: Pascal’s Principle, Conservation of Mass, and Conservation of Energy

Like all other systems, fluids that travel within closed systems abide by all of the laws of physics. This claim can be validated via mathematical derivations that begin with Pascal’s principle. In short, Pascal’s principle states that a change in pressure within a fluid is equally distributed throughout a system provided that the fluid is at rest. Of interest to us is a system in which a relatively small force ( F₁ ) that is incident to an area ( A₁ ) gives rise to a multiple of force ( F₂ ) at a surface area ( A₂ ):

The pressure ( P ) within a system is defined by the amount of force imparted per unit area of the system at hand. At all points within such a system, forces are perpendicular to its boundaries, as is the case with the system of fluid above. Since area ( A₂ ) is 50 times larger than ( A₁ ), the amount of force that is incident upon ( A₂ ) is 50 times greater than the force input at ( A₁ ). Nonetheless, the overall pressure within the system is constant:

P₁ = P₂

( F₁ / A₁ ) = ( F₂ / A₂ )

A₂ = A₁ x 50

( F₁ )[ ( A₁ x 50 ) / A₁ ] = F₂

F₂ = F₁ x 50

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Since a relatively small force is essentially multiplied across a larger area, does this force multiplication occur at the expense of anything??? The answer to this question is a resounding “ yes “. To the extent that the cylinder to the left is pushed downward by some distance ( d₁ ), the cylinder on the right will move upward by only ¹/₅₀ of ( d₁ ). This observation is in perfect agreement with the Principle of Conservation of Energy. When the system’s cylinders are in motion, equal amounts of work ( W ) are performed on them. Since work is a product of force times distance, the smaller force acts through a relatively longer distance, while the larger force works over a shorter distance. This relationship gives rise to an expression that relates these forces to the distances over which fluids travel within each cylinder:

W₁ = W₂

F₁d₁ = F₂d₂

( F₁ / F₂ ) = ( d₂ / d₁ )

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A similar relationship between the ratio of forces ( F₁ ) and ( F₂ ) associated with areas ( A₁ ) and ( A₂ ) can be derived as well:

P₁ = P₂

( F₁ / A₁ ) = ( F₂ / A₂ )

( F₁ / F₂ ) = ( A₁ / A₂ )

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We are now ready to establish a relationship between the areas and distances within the system. This relationship will enable us to mathematically prove that the volume ( V ) of fluid pumped out of the leftmost cylinder is equal to that pumped into the cylinder to the right:

( A₁ / A₂ ) = ( d₂ / d₁ )

A₁d₁ = A₂d₂

V₁ = V₂ 

Density ( ρ ) is defined as mass ( kg ) per unit volume ( V ). Since the fluid’s density is uniform throughout the system, and the volume of fluid pumped into and out of each cylinder is the same, the quantity of mass transferred within the system is constant as well:

ρ = ( mass / Volume )

m = ρV

This relationship verifies that the system operates according to the Principle of Conservation of Mass and the Principle of Conservation of Energy as well. Yet another subtle relationship between pressure and energy ( J ) can be established via usage of the definition of work:

W = Fd

P = ( F / A )

P = ( F / A )( d / d )

P = ( Energy / Volume )

Thus, each unit volume of fluid transferred possesses a quantum of energy that has the ability to perform work, which is what we’d expect.