*Answer the following (20 marks)*

### Explain frictional losses due to sudden expansion and contraction with necessary equations.

Head Loss Due to Sudden Contraction in Pipe

h

_{c}= 0.5 (V^{2}/2g)Head Loss Due to Sudden Expansion in Pipe

h

_{e}= (V_{1}- V_{2})^{2}/2### Write a short note on Creeping flow and Couette flow.

**Creeping flow.**Stokes flow also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. Re << 1. This is a typical situation in flows where the fluid velocities are very slow, and the viscosities are very large.

**Couette flow.**In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes shear stress on the fluid and induces flow.

### What is Reynolds's analogy?

Reynolds's analogy gives the interrelationship between fluid friction and newton’s law of viscosity.

The Reynolds analogy is given by,

N_{u}/R_{e}P_{r} = S_{t} = C_{f}/2

where

Nu = Local Nusselt number, Re = Local Reynolds number, St = Local Stanton number, Pr = Prandtl number, Cf = Skin friction coefficient

### Bernoulli’s theorem

The total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant.

### The physical significance of Reynolds and Froude's number

**Froude numbers**express a relationship between the free surface of a flow and the various waves and ruffles that form there, and bedforms at the sediment-water interface.

**Reynolds numbers**deal with the bulk characteristics of flow – whether it has a laminar or turbulent structure.

### What is the Mixing Length of Prandtl?

The mixing length is the distance that a fluid parcel will keep its original characteristics before dispersing them into the surrounding fluid. Here, the bar on the left side of the figure is the mixing length.

According to Prandtl, the mixing length l, is the distance between two layers in the transverse direction such that lumps of fluid particles from one layer could reach the other layer and the particles are mixed in the other layer in such a way that the momentum of the particles in the direction of x is same.

The relation between shear stress and mixing length is given by

### State the similarities and differences between Newton’s law of viscosity and Fourier’s law of heat conduction

Newton's law of viscosity describes that momentum flux in any fluid is proportional to the velocity gradients while Fourier's law describes that heat flux due to conduction is proportional to temperature gradients where velocity and temperature gradients are the driving force, respectively.

### Continuity equation

For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero.

AV = constant

### Define kinematic viscosity and eddy viscosity

The

**kinematic viscosity**formula is expressed as,ν = μ/ρ

Where μ= absolute or dynamic viscosity,

ρ = density

**Eddy viscosity**is the proportionality factor describing the turbulent transfer of energy as a result of moving eddies, giving rise to tangential stresses. It is also referred to as turbulent viscosity and doesn’t have any physical existence.

### How is the stream function defined, and why is it useful?

The stream function in a two-dimensional flow automatically satisfies the continuity equation. It is the scalar function of space and time.

The partial derivative of the stream function with respect to any direction gives the velocity component perpendicular to that direction. Hence it remains constant for a streamline.

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