An incompressible fluid flows steadily past a circular cylinder as shown in Fig. P3.8. The fluid

step 1 So in this question, we have been given a fluid. Right now, as per the statement of this questio

An incompressible fluid flows steadily past a circular...

Key Concepts

Pressure Gradient

The pressure gradient is a vector quantity that represents the spatial rate of change of pressure within the fluid. It plays a central role in the momentum conservation of fluid flow, as it drives acceleration or deceleration of the fluid particles. In the context of a given streamline, the pressure gradient can be linked directly to the derivative of the velocity and integrated to determine the pressure field along that path.

Bernoulli’s Equation

Bernoulli’s equation is a fundamental principle in fluid mechanics which states that along a streamline in steady, incompressible, and inviscid flow, the total mechanical energy (comprising pressure energy, kinetic energy, and potential energy) remains constant. This relation is used to connect the fluid's velocity and pressure, allowing for the determination of pressure changes due to variations in speed, such as those found near a stagnation point.

Stagnation Point

The stagnation point is defined as the point in the flow field where the fluid velocity becomes zero. At this point, the dynamic pressure is converted entirely into static pressure, leading to a local maximum in pressure. Understanding the stagnation point is crucial in applications such as aerodynamic profiling and in validating the predictions made by Bernoulli’s equation, which forecasts an increase in pressure corresponding to a decrease in velocity.

Incompressible Fluid Flow

In an incompressible fluid flow, the density of the fluid remains constant throughout the motion. This assumption simplifies the analysis by ensuring that the continuity equation reduces to a condition on the velocity field alone, without having to account for density variations. It is particularly valid for liquids and low-speed gas flows where compressibility effects are negligible.

Steady Flow

Steady flow implies that the fluid properties at any given point in space do not change with time. As a consequence, the velocity field and pressure distribution remain constant over time. This assumption is critical when applying conservation laws to predict the behavior of the fluid, as time derivatives in the governing equations vanish.

Transcript

00:01 So in this question, we have been given a fluid.

00:04 Right now, as per the statement of this question, there's fluid.

00:08 It's flowing steadily, right? and it is flowing steadily and the body through which it's flowing.

00:22 That is a circular cylinder.

00:29 Right.

00:29 Now it says that the velocity of this fluid is equal to v0 times 1 negative a square divided by x square, right? now over here this x, it is the radius of the cylinder, right? and this v0, it's the upstream velocity, right? now it says that there are three parts that are required to be solved right so in part a what we need to do is that we need to determine the pressure gradient right so let's solve part a now since we are given the function of the velocity over position right so we can start from the pressure gradient equation right so as per the gradient equation, right? we have the pressure energy negative gamma times sine theta negative, right? then we have the differential change of pressure along some elementary length, right? so that is equal to the product of the density times the velocity, and that is multiplied with the differential change in velocity.

02:04 Over some elementary length l.

02:07 Right.

02:08 So what we're going to do over here is that we are going to shift this, this term over here.

02:19 Right.

02:19 And before that, we are going to take this negative common over here, right, at the right hand side.

02:25 And we're going to deal with it separately, right? so when we'll shift it, we'll get a new equation.

02:31 So that would be the differential change in pressure.

02:35 Over the length that is our gradient um pressure gradient right that is equal to negative gamma times sine theta right and over here we have the um negative density times velocity times the differential change in velocity over length right so let's say that this is equation number one now before substituting the value of theta over here we have to express um this term that we have over here, right? so let us develop a relationship of this velocity with the upstream velocity, right? so for that, let's take a look over here, right? so we have that the differential change in velocity over some elementary length l.

03:27 It's equal to the upstream velocity times one negative.

03:34 The radius of the cylinder divided by x square right and then we have the derivative over this coordinate x right so over here we have v0 1 negative a squared divided by x square right now we have a given um expression for the velocity right so we are basically using this relation and we are taking its derivative uh with respect to um x right so now what we're going to do is that basically we are going to mathematically manipulate it right we are going to choose the basic mathematics techniques right so by mathematically manipulating it that is what i meant so we have v0 square bracket 1 negative a square divided by x square and as you can see over here this time it's remaining the same right and over here we have the other square bracket and inside the square bracket we have the supreme velocity parenthesis positive to a square divided by x cube right so that we are done over here with this um term is that we have taken its derivative right so we have gotten this result from the derivative right so what we need to do now is that we need to open up these square brackets and records basically means multiplication right so let's take a look that what we get from the this or from all of this work right so we have the square of v0 square bracket one negative a square divided by x square and over here we have 2a squared divided by x cube right now what we are going to do is that since we have this expression for the differential change in velocity over the elementary length l...

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