### An Introduction to Fluid Dynamics

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Even for homogeneous mixture, there isn't silver bullet to estimate the viscosity. In this book, only the mixture of low density gases is discussed for analytical expression. For most cases, the following Wilke's correlation for gas at low density provides a result in a reasonable range. The mixture viscosity is highly nonlinear function of the fractions of the components. Solution The following table summarizes the known details Table summary 1. For gases with very long molecular structure or complexity structure these formulas cannot be applied. For some mixtures of two liquids it was observed that at a low shear stress, the viscosity is dominated by a liquid with high viscosity and at high shear stress to be dominated by a liquid with the low viscosity liquid.

The higher viscosity is more dominate at low shear stress.

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Estimate the liquid viscosity used between the cylinders. Solution The moment or the torque is transmitted through the liquid to the outer cylinder. Control volume around the inner cylinder shows that moment is a function of the area and shear stress.

## An Introduction to Fluid Dynamics

The shear stress calculations can be estimated as a linear between the two concentric cylinders. The surface is covered with oil film. What is the speed of the block at steady state? Assuming a linear velocity profile in the oil and that the whole oil is under steady state. The edge effects can be neglected. The shear stress can be assumed to be linear. A discussion of viscosity and surface tension should be part of this section but because special importance these topics have separate sections.

The rest of the properties lumped into this section. This figure illustrates the typical situations like the one that appear in Example 1. The density is related to the other state properties such temperature and pressure through the equation of state or similar. Examples to describe the usage of property are provided. The initial pressure can be assumed to atmospheric. Calculate the pressure at the end of the process. Assume that the Young modulus of the water is State your assumptions.

Solution The expansion of the steel tank will be due to two contributions: one due to the thermal expansion and one due to the pressure increase in the tank. For this example, it is assumed that the expansion due to pressure change is negligible. The tank volume change under the assumptions state here but in the same time the tank walls remain straight.

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The basic assumption falls and the water evaporates. If the expansion of the water is taken into account then the change increase of water volume has to be taken into account. The water pressure at the end of the process is unknown but the volume is known. It can be noted that this property is not the result of the equation of state but related to it. Bulk modulus is usually obtained from experimental or theoretical or semi theoretical theory with experimental work to fit energy—volume data.

Most theoretical studies are obtained by uniformly changing the unit cells in global energy variations especially for isotropic systems where the molecules has a structure with cubic symmetries. The bulk modulus is a measure of the energy can be stored in the liquid. This coefficient is analogous to the coefficient of spring. The reason that liquid has different coefficient is because it is three dimensional verse one dimension that appear in regular spring. In the literature, additional expansions for similar parameters are defined.

These definitions are related to each other. The increase of the pressure increases the bulk modulus due to the molecules increase of the rejecting forces between each other when they are closer. In contrast, the temperature increase results in reduction of the bulk of modulus because the molecular are further away.

The liquids are compressed due to the pressure increases. Estimate the change of the heights of the liquids depicted in the Figure 1. State your assumptions. The vessel is entirely filled with water, then a piston at one end of the cylinder is pushed in until the pressure inside the vessel has increased by kPa. Suddenly, a safety plug on the top bursts. How many liters of water come out? Anything else would give a wrong answer in the system. So with this bulk modulus, is 0. In this post several assumptions were made. What is a better way to solve this problem.

## Fluid Dynamics: An Introduction - Michel Rieutord - Google книги

Solution It is assumed that this process can be between two extremes: one isothermal and one isentropic. The assumption of isentropic process is applicable after a shock wave that travel in the tank. If the shock wave is ignored too advance material for this book. The process involve some thermodynamics identities to be connected.

In this short section a discussion about the bulk modulus averaged is presented. When more than one liquid are exposed to pressure the value of these two or more liquids can have to be added in special way. Even in some cases of emulsion a suspension of small globules of one liquid in a second liquid with which the first will not mix the total change is the summation of the individuals change.

In case the total change isn't, in special mixture, another approach with taking into account the energy-volume is needed. There are only several situations in which the bulk modulus is important. These situations include hydraulic systems, deep ocean on several occasions , geology system like the Earth, Cosmology. The Pushka equation normally can address the situations in deep ocean and geological system.

This author is not aware of any special issues that involve in Cosmology as opposed to geological system. The only issue that was not addressed is the effect on hydraulic systems. The hydraulic system normally refers to systems in which a liquid is used to transmit forces pressure for surface of moving object normally piston to another object.

In theoretical or hypothetical liquids the moving one object surface results in movement of the other object under the condition that liquid volume is fix. The movement of the responsive object is unpredictable when the liquid volume or density is a function of the pressure and temperature due to the friction. In very rapid systems the temperature and pressure varies during the operation significantly. The change of the bulk modulus by this amount can change the response time significantly.

Hence the analysis has to take into account the above effects. The surface tension manifested itself by a rise or depression of the liquid at the free surface edge. Surface tension is also responsible for the creation of the drops and bubbles. Surface tension results from a sharp change in the density between two adjoined phases or materials.

There is a common misconception for the source of the surface tension.