Dimensional Formula of Velocity Gradient

When studying fluid mechanics, it is important to know about velocity gradients as well. Fluid mechanics is a section of physics that deals with the mechanics of liquids, gases and the forces imposed on them. Two properties associated with fluids on how they behave are density and viscosity. 

Viscosity is not seen unless the fluid is set in motion. It also establishes the application of velocity gradient throughout the layers of fluid. We will discuss all the aspects associated with viscosity, velocity gradient, and its dimensions.

What Is Velocity Gradient?

According to the definition of velocity gradient, the difference in velocity between the layers of the fluid is known as the velocity gradient. It is represented by v/x, where v stands for velocity and x stands for the distance between the adjacent layers of the fluid.

Example

When water flows through a pipe or any solid object, the layer of water in contact with the pipe of the solid surface is in the same state of motion. However, the layer in the centre is of a different state of motion. This difference between the layers of fluid flowing through an object is defined as velocity gradient.

Another example would be when someone paints a wall. The whole paint force should be applied using a paintbrush to spread across the wall. The lower layer of paint adheres to the solid surface, whereas the surface layers of the paint adhere to the paintbrush. The force applied by the brush makes a difference in the different layers of paint. 

The molecules of paint in contact with the surface are in the stationary phase. But the motion increases through the layers when it reaches towards the layer in contact with the brush. This shows that velocity gradient is imposed throughout the layers of paint.

Equation of Velocity Gradient

The velocity gradient is proportional to the amount of force applied on fluid layers per unit.

T α dV/dx

Where, 

• T stands for the force.
• V is velocity.
• x is the distance perpendicular to the surface.

To establish a relationship in the equation, a constant of proportionality is introduced, 

T = η dV/dx, 

Here, η is called the coefficient of viscosity. 

This equation is applicable for the majority of the fluids that exist; these are called the Newtonian fluids. The fluids which do not follow this equation are called non-Newtonian fluids.

Dimension of the Velocity Gradient

The dimensional formula of velocity gradient is [M0 L0 T-1], where, 

• M = mass 
• L = length 
• T = time

Derivation.

Velocity gradient = velocity x distance-1 _____ (1)

The dimensional formula of velocity gradient is [M0 L0 T-1] _____ (2)

Dimensional formula of distance is [M0L1T0] _____ (3)

Placing (2) and (3) in (1) we get,

V = [M0 L1 T-1] × [M0 L1 T0]-1 = [M0 L0 T-1].

Factors Affecting the Viscosity of Fluids

Temperature influences viscosity to a great extent. In simple words, the viscosity of some fluids decreases with increasing temperature. 

Honey and sugar syrups become light and easy to flow when they are heated. Engine oils and other types of hydraulic fluids, on the other hand, thickens on cold days and affects the performance of vehicles and machines during the winter times. 

With the increment in temperature, the average speed of the molecules present in liquid increases and the average intermolecular forces also decreases. The relation between speed and temperature is nonlinear. It changes abruptly when the fluid changes the phase.

Normally viscosity is unaffected by pressure, but fluids under extreme pressure can experience an increment in viscosity as fluids are naturally incompressible. An increment in pressure does not really push the molecules closer to each other.

Fluids flow faster when they are hot, on the other hand, gases become thicker. Hence, the viscosity of gases increases as temperature increases and is proportional to the square root of the temperature. This happens because there is an increment in the frequency of the intermolecular collisions at an increased level of temperature.

The viscosity of fluids depends on the size, shape, and chemical nature of their molecules. The viscosity of the lyophilic colloid solution is high. The viscosity of water is also influenced by electrolytes. Small amounts of electrolytes decrease the viscosity, whereas large amounts of solids dissolved increase the viscosity.

Suspended particles in a solution or fluid increase the viscosity. The viscosity of blood is influenced by the emulsoid colloid system existing in plasma and also the number of suspended corpuscles. 

Conclusion

It is important to know what the velocity gradient is to study fluid mechanics. The velocity gradient is the difference in velocity between the different layers of fluid. It can be explained by the phenomenon of water flowing through a pipe. The layers of water in contact with the pipe walls are in a different motion than the layers in the centre.  

Apart from knowing what velocity gradient is, it is also helpful to study velocity gradient questions. They help understand the deeper aspects of the topic and solve sums.