A solid moving relative to afluid, a liquid or a gas, is subject to a force, different from friction, but as friction opposing the relative motion of the body and the medium. It is called viscous drag or viscous resistance. Differently from friction, there is no drag when the relative velocity is zero, and an increasing function of the relative velocity. The direction of the drag force is always equal and opposite to the relative velocity.
The magnitude of a force depends on the magnitude of the relative velocity, on the shape of the body and on the fluid. Moving relative to the fluid, the body induces a number of effects that may perturb substantially its flow. Think for examples of vortices. Consequently, the dependence of the drag force on velocity is complicated. We shall study it in the second volume of this course, together with fluid dynamics. Here we anticipate only a few elements that are needed in our study of the motions of bodies.
The force depends on the shape of the body, for example it is different for a cylinder or a sphere, on its orientation, for example the case of a disc is different for its orientation parallel or perpendicular to the flow, and, for a given geometrical shape, on its size. We shall limit the discussion here to a spherical body, of radius a. The force depends on two characteristics of thefluid, its density ρ (mass per unit volume) and the viscosityη. The latter will be discussed in the second volume. It suffices to know here that it characterizes the difficulty with which the fluid flows, so, for example, oil has larger viscosity, is more viscous, than water, but is less viscous than honey. For a givenfluid, the viscosity depends on the temperature.
The physical units of viscosity are g
½ ¼ ML 1T1¼ FL 2T; ð3:24Þ where, in the third member we have taken into account that the dimensions of the force are [F] = [MLT−2]. Pressure has the dimensions of a force per unit surface (FL−2) and its unit is the pascal (Pa), from Blaise Pascal (1623–1662). The unit for viscosity is then the pascal second (Pa s). For example, for some everydayfluids at
ambient temperature, their viscosities are for oils η ≈ 0.5–1.5 Pa s, for water η ≈ 10−3Pa s, and for air η ≈ 1.8 × 10−5Pa s.
The Reynolds number is a parameter that gives relevant information on the regime of the motion, named after Osborne Reynolds (1842–1912). It is dimen- sionless, namely a pure number. The four quantities of the problem have the physical dimensions ½ ¼ MLq ½ 3, g½ ¼ ML½ 1T1, a½ ¼ L½ and t½ ¼ LT½ 1. They can be arranged in a dimensionless quantity as
Re¼ q=gð Þta; ð3:25Þ
which is the Reynolds number for a sphere. Its expressions for other shapes are similar.
Figure3.17shows schematically how the drag force on a body can be measured. The body isfixed to a thin bar and to the pointer of a dynamometer fixed on a support and is immersed in the fluid under study, which is moving at a known velocityυ, that we can vary in a known manner. Experiments of this type show that at small velocities the drag force can be written as the sum of a term proportional to the velocity and one proportional to its square
R¼ At þ Bt2; ð3:26Þ
where the coefficients A and B depend on the body and the fluid but, for not too large velocities, are independent of velocity. As the ratio between the second and the first term is proportional to the velocity, the first term dominates at small velocities, the second at larger ones. We define as critical velocity υcthe velocity at which the two terms are equal. It corresponds to a quite small value of the Reynolds number 0 1 2 3 4 5 Fig. 3.17 Measuring the drag
Rec 20 30: ð3:27Þ Consider now the sphere moving in air, as pendulums or free falling bodies, at normal temperature and pressure conditions. The air density in these conditions is ρ = 1.2 kg/m3
. With the value for viscosity already given, the Reynolds number is
Reðair) ¼ 1:5 105ta ð3:28Þ
and the critical velocity, in a round number
tc 4 104=a m/s: ð3:29Þ
If for example a = 1 cm, the critical velocity isυc= 4 cm/s. The time taken to reach it by a body freely (in a vacuum) falling from rest is t =υ/g = 4 ms, which is very short indeed. In this time it would travel in vacuum d = gt2/2 = 80 µm. For larger dimensions bodies moving in the air the critical velocities are even smaller. We conclude that only for very small velocities, smaller thanυc, is the viscous drag proportional to the velocity. However, it becomes proportional to the square velocity very gradually, reaching that regime only at Reynolds numbers two orders of magnitude larger than in Eq. (3.27), corresponding to velocities of a few meters per second for a sphere of 1 cm radius.
As a second example consider the same sphere moving in water. With ρ = 103kg/m3and the viscosity given above,η/ρ = 10−6kg/m3, which is a value, notice, smaller than for air. The Reynolds number at velocity υ for a = 1 cm is Re = 104υ. The critical velocity is only υc* 2.5 mm/s.
In the elementary study of free fall, of the motion on an incline and of the pendulum, the viscous drag of air is usually neglected. Is this a good approxima- tion? Let us control on a few typical cases. Consider a bronze (density ρ = 8 × 103kg/m3) ball of a = 2 cm radius and three cases: free fall from a h = 20 m tall tower, descent of an incline of elevation h = 1 m and oscillation of a pendulum abandoned at the height from the position at rest h = 0.5 m. The weight of the ball is Fp= 2.7 N. Neglecting the presence of the air, and the energy of the rotation in the second case, the velocities at the end of the fall would be in any caset ¼pffiffiffiffiffiffiffiffi2gh, hence υ1 = 20 m/s, υ2= 4.5 m/s, υ3= 2 m/s in the three cases respectively. In presence of air all velocities would be somewhat smaller, but larger than the critical velocity. The drag force is approximately proportional to the square velocity, but is not very large. For the just mentioned velocities its values are approximately R1= 2.4× 10−2N, R2= 1.2× 10−2N, R1= 2.4× 10−3N, which are in any case small compared to the weight. Neglecting the drag in these cases is not a bad approximation. However, the effect will be noticeable on much longer times.
Finally notice that, whatever its expression, the viscous drag is a dissipative force. As it is always directed opposite to velocity, its work is negative for any displacement of the application point.