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The force $F$ to carry a plate of area $A$ with velocity $v$ in a fluid of depth $d$ is given by

$$\frac{F}{A}=\eta\frac{v}{d}.$$

Hence if the depth is $kd$, the force becomes $F/k$.

Do this relations hold for a ship in water?

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No. This is for a thin film sheared over a distance 'd'. –  ja72 Feb 15 '12 at 20:43

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The formula you quote is not the relation you should use if you want to calculate the drag on a ship in water. Your formula tells you the force required to move one plate at a speed v relative to a fixed plate on the other side of the fluid, and it assumes that viscosity provides the dominant drag force.

Your formula is closely linked to the Stokes' drag formula for the drag force of a submerged object in a viscous fluid, which brings you closer to a formula for the drag force on a boat, but still does not quite apply. First, a boat is only partially submerged. Second, water has a low enough viscosity such that the drag comes primarily from the force required to push the water out of the way (inertial drag), rather than from the water's viscosity.

So one improvement (but also not the complete solution) would be to use a formula that applies to a situation in which inertial drag dominates:

$F = 1/2 \rho v^2 A c_d$

Here $\rho$ is the density the fluid, $v$ is the velocity of the boat, $A$ can be thought of as the area of the portion of the boat encountering the water (although the angles that elements of this surface make with respect to the direction of motion of the boat matter), and $c_d$ is the so-called drag coefficient, which accounts for some extra complications, including the viscosity of the water. The drag coefficient will vary depending on the speed of the boat, its level of submersion, and its shape - you'll probably need to come up with it experimentally.

Note that the first part of this expression looks like the formula for kinetic energy (per volume), which should make sense because the drag is related to the energy required to push the water out of the way.

EDIT: I've now realized that the effects of partial submersion of the boat will be quite important, and the relationship between drag force and speed will be highly dependent on the speed of the boat in relation to the speed of the waves it creates at the surface, which in turn depend on the length of the boat. This is an aspect of the problem that I'm not qualified to address, although it is critical to answering the question.

The cop-out would be to refer to the relevant wiki page, http://en.wikipedia.org/wiki/Wave_making_resistance, but it would be nice to hear more from someone with more experience

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