Dimensions of Velocity Gradient

The velocity gradient is defined as the change in velocity per unit distance. The concept finds application in fluid mechanics, which is significant for studying fluid flow. It helps analyse important phenomena like viscosity, shear stress and fluid deformation. The concept is also useful for the understanding of flow behaviour and distribution of velocity within a fluid or fluid-like system.

Table of Contents

Velocity Gradient
Formula of Velocity Gradient
Dimensions of Velocity Gradient
Practice Problems
Frequently Asked Questions

Velocity Gradient

Velocity gradient can be explained by dividing the fluid into different layers. For instance, in an aquarium, the water remains in motion due to the movement of fishes. The upper layer of water remains static as the fishes are in the middle or bottom layer. Similarly, the layer of water around the fish where their fins are moving is in motion, while the water layer just below is moving with another velocity. It also resists the motion of the upper layer, which further resists the motion of fluid on its upper layer, and so on. This represents the velocity gradient where each layer is at different velocities.

Velocity gradient

Image: Velocity gradient

Formula of Velocity Gradient

The force applied on the fluid layers per unit is the velocity gradient and is represented as

formula , where T refers to Force, V is the velocity, and x is the perpendicular distance from the surface. dV indicates the change in velocity, and dx indicates the change in distance.

The proportional sign will be further replaced by the coefficient of viscosity, represented as formula . Hence, the complete formula is formula .

The equation can be used for Newtonian fluids or fluids with constant viscosity. It is not applicable for non-Newtonian fluids that do not have constant viscosity.

The S.I. unit of velocity gradient is per second.

Dimensions of Velocity Gradient

The dimensional formula is represented using the letters M, L and T, where M represents mass, L represents length and T represents time.

The formula of velocity gradient is

formula

Practice Problems

Q1. Which of these is Newtonian fluid?

a. Alcohol
b. Water
c. Toothpaste
d. Both a and b

Ans. d. Both a and b

The Newtonian fluids are alcohol, water, gasoline and motor oil.

Q2. What is the force required to move the metal plate of the area with a velocity of if it is resting on a thick oil layer of thickness. The dynamic viscosity or coefficient of viscosity of the oil is 1Nsm².

a. 0 N
b. 0.03 N
c. 0.06 N
d. 1

Ans. c. 0.06 N

formula

F = 0.06 N

Q3. What is the dimensional formula of velocity gradient?

formula

Frequently Asked Questions

Q1. What are the factors affecting velocity gradient?
Ans. The different factors affecting velocity gradient are temperature, pressure, suspended particles, size, shape and chemical nature of the molecules.

Q2. What is viscosity?
Ans. Viscosity refers to the shear force applied to the unit area to produce a unit velocity gradient.

Q3. How is velocity gradient measured experimentally?
Ans. The velocity gradient is measured using experimental techniques like Laser Doppler Velocimetry, Hot-wire anemometry and Particle image velocimetry.