7.6.1. Navier-Stokes in Cylindrical Coordinates

Let us go to the next slides as you look it from Cauchy equations what we get it just in vector forms. In vector form if you remember it which is the linear momentum equations deriving with the components like rho by rho t rho v is velocity plus is equal to del dot product of rho v is equal to rho g is a vectors plus we have the del product or divergence of the stress tensor.

The Navier-Stokes equations describe the motion of fluid substances. This particular introduction discusses how to express these equations in vector form. The equation essentially balances forces acting on a fluid element, such as pressure forces, viscous forces, and body forces like gravity. In mathematical terms, the equation includes terms for the fluid's density (rho), velocity (v), gravitational forces (g), and the divergence of the stress tensor. This sets the groundwork for analyzing fluid flow mathematically.

Imagine trying to push a rubber duck through water. The forces acting on it include the force of the water pushing back against it (pressure), the force due to the duck's weight (gravity), and the internal friction of the water (viscosity) that affects how quickly it moves. Understanding these forces helps us predict the duck's movement, similar to how the Navier-Stokes equations help predict fluid motion.

If you go back very basic things, the the vectors divergence okay. If you look at that del and V the velocity divergence it is talking about the net volume outflow from a control volume. So if you derive a simple equations as find out the velocity divergence what it says that what you will understand from that if you physical interpretations of that is a volume outflow per unit volume per unit control volume.

The concept of divergence in fluid mechanics is crucial for understanding how fluid moves in and out of a control volume (a specified region in space). The divergence of velocity (denoted as 'del dot V') indicates how much fluid is leaving or entering that region over time. A positive divergence means more fluid is leaving than entering, possibly indicating a leak, while a negative divergence suggests fluid is accumulating in the volume.

Think of a balloon being inflated. The air going into the balloon represents a negative divergence (fluid entering the control volume), while a hole in the balloon would create a positive divergence (fluid escaping the control volume). Understanding divergence helps in situations like predicting how quickly a buoyant object will rise in water.

These two can indicating for us two force component one is body force another is the surface force that is what body force and the surface force component.

In fluid mechanics, forces acting on a fluid can be categorized into two types: body forces and surface forces. Body forces, like gravity, act throughout the volume of the fluid. Surface forces, such as pressure and viscous forces, act on the boundaries of the fluid. Recognizing these distinctions allows for a clearer understanding of how different forces impact fluid movement and behavior.

Imagine a leaf floating on a pond. The weight of the leaf pulling it downwards is a body force (gravity), while the water pushing up against the leaf's surface is a surface force (buoyancy). Considering both types of forces helps us understand why the leaf floats or sinks.

If you look at the basic equations is this part whether you can have this. Just to demonstrate it that we are discussing the velocity field, we are discussing about density field, we are discussing about the pressure field.

Cauchy’s equations describe the motion of fluids in terms of velocity, density, and pressure fields. These fields help visualize how fluid moves and changes within a control volume. By analyzing these equations, we can describe how quantities like velocity and pressure vary in space and time, leading to a better understanding of fluid dynamics.

Consider pouring syrup into a jar. The syrup's movement represents the velocity field, while the syrup's thickness indicates the pressure field. Observing how the syrup interacts with the jar (density) helps us predict how quickly it will pour out. Understanding these fields is crucial in various applications, from designing kitchen gadgets to predicting weather patterns.

First is incompressible flow okay. That means density more or less fairly okay does not change with space as well as time.

A core assumption when using the Navier-Stokes equations is that the fluid is incompressible, meaning its density remains approximately constant regardless of changes in pressure or temperature. This simplifies the equations, making them easier to solve, since we do not need to account for density variations that could complicate the fluid behavior.

Think about drinking a thick milkshake through a straw. As you sip, the density of the milkshake doesn’t substantially change, despite the pressure you create by sucking on the straw. This approximation holds true for many common fluids, like water, which allows for simplified calculations in fluid dynamics.

If you look at that the same way we can always put a relationship between sigma the stress and the strain again I am hypo strain rate okay that is the reasons basic difference between the fluid mechanics and the solid mechanics.

In fluid mechanics, the relationship between shear stress and shear strain rate is foundational. This relationship, represented by Newton's law of viscosity, states that shear stress is proportional to the rate of deformation (strain rate). This contrasts with solid mechanics, where the relationship is between stress and strain. This distinction is essential for analyzing how fluids respond to applied forces.

Imagine spreading butter on toast. The stress you apply (the pressure of the knife on the butter) translates into strain (the butter changing shape). In fluids, we often see this relationship in motion. For example, honey flows slowly out of a jar due to its viscosity, which describes how it resists flow under stress.

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