Resistive force

When fluid flow in the reservoir is considered, it is necessary to estimate the viscosity of the fluid, since viscosity represents an internal resistance force to flow given a pressure drop across the fluid. Unlike liquids, when the temperature and pressure of a gas is increased the viscosity increases as the molecules move closer together and collide more frequently. 

Most instmments make use of a probe geometry which gives an increasing area of contact as penetration proceeds. In this way, at some depth of penetration, the resisting force can become sufficient to balance the appHed force on the indentor. Unfortunately, many geometries, eg, diamonds, pyramids, and cones, do not permit the calculation of basic viscoelastic quantities from the results. Penetrometers of this type include the Pfund, Rockwell, Tukon, and Buchholz testers, used to measure indentation hardness which is dependent on modulus. 

F is zero at the equilibrium point r = ro) however, if the atoms are pulled apart by distance (r - Tq) a resisting force appears. For small (r - Tq) the resisting force is proportional to (r - rg) for all materials, in both tension and compression. 

We now have all the information necessary to develop some working expressions for particle settling. Look back at equation 3 (the resistance force exerted by the water), and the expressions for the drag coefficient (sidebar discussion on page 261). The important factor for us to realize is that the settling velocity of a particle is that velocity when accelerating and resisting forces are equal ... 

If force P is greater than zero, the particle will be in motion relative to the continuous phase at a certain velocity, w. At the beginning of the particle s motion, a resistance force develops in the continuous phase, R, directed at the opposite side of the particle motion. At low particle velocity (relative to the continuous phase), fluid layers running against the particle are moved apart smoothly in front of it and then come together smoothly behind the particle (Figure 14). The fluid layer does not intermix (a system analogous to laminar fluid flow in smoothly bent pipes). The particles of fluid nearest the solid surface will take the same time to pass the body as those at some distance away. 

The total resistance is the sum of friction and eddy resistances. Both factors act simultaneously, but their contribution in the total resistance depends on the conditions of the flow in the vicinity of the particle. Hence, for the most general case the resistance force is a function of velocity, w, density, p, viscosity, the linear size of a particle, C, and its shape, ijr. Thus,... 

Coefficient A and exponent a must be evaluated experimentally. Experiments have shown that A and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a... 

Consider again the simple motion of a sphere. In this case, the equivalent diameter of a sphere, d, is equal to its geometric diameter, d. Equating the above expressions and replacing 5 by d (and denoting the Euler umber, Eu, by Y), we obtain an expression for the resistance force ... 

Substituting the resistance force into equation 51 and expressing F and V in terms of d, the basic equation of sedimentation theory is obtained ... 

Lyachshenko number, dimensionless left hand side, dimensionless particle mass, kg pressure, N/m or force, N mass feed rate, kg/s or volumetric flowrate in mVhr drag or resistance force, N physical properties correction factor for slurries Reynolds number, dimensionless right hand side hydraulic radius, m... 

From Darcy s equation we can determine a formula for the counterforce produced by the porous material to the flowing or diffusing component A, If this counterforce is found, it can be added to the diffusion resistance force caused by component B to component A hence the sum of these two forces represents the total diffusion resistance. 

Equation (4.303) is valid but it is lacking something. The resistance force that applies to the component A has to be found, and not that for the whole mixture. The force applying to the whole mixture is the sum of the... 

We now consider the resistance force caused by the diffusion. This force resists the diffusion flow in a porous material together with Writing the linear momentum equation for component A in accordance with Eq. (4.302),... 

This gives a model for Eq. (4,305b), but not a model for force While force gives the flow force caused by the material, it is normal to represent this fact so that gives the pure diffusion resistance force that is not caused by the material. This requires treating independently from the material or porosity. For (f) = 1 or E = °°, where 0 EQ- (4.306) gives... 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

Unlike the preceding discussion, hydraulic systems can provide mechanical advantage or a multiplication of input force. Figure 40.12 illustrates an example of an increase in output force. Assume that the area of the input piston is 2 square inches. With a resistant force on the output piston, a downward force of 20 pounds acting on the input... 

This pressure of lOpsi acts on all parts of the fluid container, including the bottom of the output piston. The upward force on the output piston is 200pounds (lOpsi x piston area). In this case, the original force has been multiplied tenfold while using the same pressure in the fluid as before. In any system with these dimensions, the ratio of output force to input force is always 10 to 1, regardless of the applied force. For example, if the applied force of the input piston is 50 pounds, the pressure in the system will be 25 psi. This will support a resistant force of 500 pounds on the output piston. The system works the same in reverse. 

You have learned that if a force is applied to a system and the cross-sectional areas of the input and output are equal, the force on the input piston will support an equal resistant force on the output piston. The pressure of the liquid at this point is equal to the force applied to the input piston divided by the piston s area. Let us now look at what happens when a force greater than the resistance is applied to the input piston. 

Stiffness Stiffness is a spring-like property that describes the level of resisting force that results when a body undergoes a change in length. Units of stiffness are often given as pounds per inch (Ibf/in). Machine-trains have more than one stiffness property that must be considered in vibration analysis shaft stiffness, vertical stiffness, and horizontal stiffness. 

For turbulent flow, Rmjpit is almost independent of velocity although it is a function of the surface roughness of the channel. Thus the resistance force is proportional to the square of the velocity. Rm/pu2 is found experimentally to be proportional to the one-third power of the relative roughness of the channel surface and may be conveniently written as ... 

The fluid resistance force acting on the droplet should be taken as that given by Stokes law, that is 3ntidu where /< is the viscosity of the continuous phase, velocity relative to the continuous phase. 

Figure 4. When a muscle contracts isotonically or a constant resisting force is imposed on it during a contraction, the velocity at which it shortens quickly comes to a constant. The force-velocity curve shows the relationship between the force applied to a muscle and the steady-state velocity of shortening. As in all other muscles, the force-velocity curve of smooth muscle is a rectangular hyperbola for all positive shortening velocities. In order to compare the behavior of muscles of different lengths and diameters, it is common to normalize force and velocity by dividing each by its maximum value and expressing the result as a percentage, nd...

A particle falling freely in vacuum is subjected to a constant acceleration, and its velocity increases continuously. The velocity at any point depends only on the distance from the starting point, and is independent of the size and the density of the particle. Thus a heavy stone and a feather fall at exactly the same rate in an evacuated system. However, in the event of a particle falling in a fluid medium, there is resistance to this fall or movement. The resistance increases as the velocity of the particle increases, and this continues until the forces tending to accelerate the particle and the fluid resistance forces become equal. The particle is then said to have attained its terminal velocity it continues to fall, but with a uniform velocity. 

Thus, in an isothermal system, the mass flow rate depends on the difference in pressures of the gas across the orifice and does not depend upon the thickness of the plate. One may define an area-normalized resistance, R, for mass transfer through the orifice using a generalization of Ohm s law, i.e., Resistance = force/ flux. For Knudsen flow, the force is the pressure difference (analogous to voltage difference in Ohm s law) and the flux is the mass flow per unit area of the hole (analogous to the electrical current density in Ohm s law). Thus, we have... 

Friction is the resistive force that we experience when we try to slide one object over the surface of another. The coefficient of friction is the ratio of the lateral force required to slide the surfaces past one another relative to the force holding them in contact. Polymers exhibit two coefficients of friction the static coefficient of friction is a measure of the force required to initiate movement, the dynamic coefficient of friction is a measure of the force required to sustain movement at a constant rate. In general, the force required to initiate sliding is somewhat greater than that required to maintain a constant rate of movement. 

Neglecting the elastic forces, lumping the geometric factors into a constant, b, and assuming the plastic shear deformation is x/r, yields the plastic resistive force ... 

Similarly, for the frictional resistive force, Ff, lumping the geometry factors into a constant, c, and letting a = friction coefficient ... 

The stress needed to move a dislocation line in a glassy medium is expected to be the amount needed to overcome the maximum barrier to the motion less a stress concentration factor that depends on the shape of the line. The macro-scopic behavior suggests that this factor is not large, so it will be assumed to be unity. The barrier is quasi-periodic where the quasi-period is the average mesh size, A of the glassy structure. The resistive stress, initially zero, rises with displacement to a maximum and then declines to zero. Since this happens at a dislocation line, the maximum lies at about A/4. The initial rise can be described by means of a shear modulus, G, which starts at its maximum value, G0, and then declines to zero at A/4. A simple function that describes this is, G = G0 cos (4jix/A) where x is the displacement of the dislocation line. The resistive force is then approximately G(x) A2, and the resistive energy, U, is ... 

It may be assumed that the resistance force may be calculated from Stokes Law and is equal to 3np,du, where u is the velocity of the particle relative to the liquid. 

---