Consider the simple power-law model for a nonNewtonian fluid given by Eq. 2.16. Extend the analy

Consider the simple power-law model for a nonNewtonian fluid...

Key Concepts

Non-Newtonian Fluids

These are fluids whose stress?strain rate relationship departs from that of a Newtonian fluid, meaning the viscosity can change with the rate of shear. This category includes fluids that exhibit shear-thinning or shear-thickening behavior, and understanding them is critical in various engineering applications where traditional linear fluid models fail.

Power-Law Fluid Model

This model characterizes the behavior of non-Newtonian fluids through a simple mathematical relationship where the shear stress is proportional to a power of the shear rate. The model uses two parameters: a consistency index that scales the relationship and a flow behavior index that indicates whether a fluid is shear-thinning (n < 1) or shear-thickening (n > 1). It serves as a useful approximation for many real-world fluids in practical flow situations.

Fully Developed Laminar Flow

In fully developed laminar flow, the velocity profile does not change in the flow direction, implying that the effects of entry length have dissipated. This condition simplifies the analysis since the momentum balance reduces to a balance between pressure gradient and viscous forces, leading to a predictable and steady velocity distribution across the channel.

Channel Flow Between Parallel Plates

This flow configuration considers a fluid moving between two flat, stationary plates separated by a fixed distance. It is a common test case for studying viscous flows because of its geometric simplicity and serves as a fundamental model for understanding more complex flow behaviors in ducts, channels, and other confined spaces.

Velocity Profile Derivation

The derivation involves solving the momentum balance equations under the assumption of fully developed, pressure-driven flow. For power-law fluids, the constitutive equation replaces the Newtonian stress-strain rate relationship, leading to a non-linear ordinary differential equation whose solution yields the velocity profile. This process illustrates the interplay between the pressure gradient, rheological properties of the fluid, and geometric constraints.

Pressure-Driven Flow

In this context, the flow is induced by a pressure gradient along the channel. The pressure drop acts as the driving force that overcomes the resistance due to the fluid's viscosity and any non-Newtonian behavior, forming the basis for the velocity distribution within the channel.

Dimensionless Analysis and Profiling

This concept involves rewriting the physical variables in non-dimensional form to collapse different cases into a universal form. By expressing the velocity and spatial coordinates in dimensionless terms (such as u/U_max and y/h), one can effectively compare profiles for different fluid behavior indices and understand the influence of non-Newtonian parameters on the velocity distribution.

Transcript

00:01 We have a question okay we have problem number 18 in which we have to prove an equation which is being given to prove okay and we should assume that the flow is fully developed and laminar flow okay so let us first write the terms we will be using because it will very difficult to get those terms after we after we once we start solving the problem so let us start with f s x f s x is x component of sheer force component of shear force okay then f bx which is x component of body force then f bx which is x component of body force then f uh what is this we should write p x that is x component of pressure force pressure force okay and cv is controlled volume controlled volume and c s a is controlled surface controlled surface and v is volume and as you know small u is the velocity of the fully developed flow so these are the things you will be using extensively here okay now let us start with the question since it is given that the flow is fully developed and laminar so let us assume a section of the flow okay so let us start with the things that is let us start with all the forces that will be acting here so sheer force shear force that is it will act at the top and bottom of this that is he at this surface and at this surface of this flow so this is tau plus curly tau by curly y and d y and into w d x okay now this is the sheer force at the top and shear force at the bottom will be simply tau w d x let us write w like this okay here it will be this is um the pressure will be applied here so p w d y and here okay this is two x that is height of this or the width of this so here it will be uh the pressure force p plus curly p by curly x d x into wdy into wdy y into wdy y okay so now we are assuming this as controlled volume so let us start with the momentum equation which is which is given in the book momentum equation is given in the book and it is on 4 .18a equation number 4 .18a the momentum equation is f s x plus f bx that is net force equal to curly by curly t controlled volume okay you and this is the density row dv in that is change in volume plus controlled surface u row okay b dot d a so this is the controlled surface integrating over the controlled surface we should say okay now we have some exemptions over here so we have three assumptions let us start with assumption number one the flow is horizontal it is being given horizontal and steady flow steady flow and the flow is fully developed if we if you look at the question it is being given that the fully developed flow so we can easily make out that the it is horizontal flow is horizontal which means f bx should be equal to zero so let us cross it out this will be equal to zero and flow is steady which means this second term that is this term should be equal to zero that is because this is a steady flow okay so there will be no change in volume so that's why third is it is fully developed the flow is fully developed so if the flow is fully developed so this should be equal to zero okay we have f s x equal to zero so let us get to the idea get to the things what are the forces component of the sheer forces or forces along x x xx so let us start with the forces along x xxas okay so first is p w d y then minus p p plus curly p by curly x d x and w d y one force is remaining so this is tau that is sheer force plus curly tau by curly y and wd x and one more is tau d x this is tau wdx this is tau d x tau w d x should be equal to 0 because this is equal to 0 and one more thing to be noted that one more thing to be noted that we have assumed this direction as xx positive x and this direction as positive y so any force that we will be taking this direction should be negative now just open the brackets p w d y minus p w w w -d -y minus curly p by curly x d x into w -d -y plus tau w d x plus curly tis by curly y dy into w d x minus tau -w d x should be equal to zero okay these two will get cancelled out okay and what all will be cancelled out these two will get cancelled out okay this is curly x by curly y or curly p by curly x dx okay yes nothing will now get cancelled out so let us write curly p by or more clearly we should be writing w into curly p by curly x d x, d y equal to curly z oh sorry curly tau by curly y with negative sign and we should write w here just like left inside minus w d x d y now we can see that this negative and negative will get cancelled out w in w will get cancelled out dx ty dx ty will get cancelled out so from here we will be getting an equation that curly tau by curly y equal to curly p by curly x so this is the equation we are getting okay now let us write this equation number one we know that this shear force develops at the top of the at the top of the plate top of the plate means that is the upper side of the plate so we should write like this as a function of a function of y we should write okay or tau y x or we should say that simply tau function of y okay and the pressure pressure should be always written as the function of of x function of x okay now if this is a function of x and function of y so we should say that curly tau by curly y and curly p by curly u sorry curly x both should be the constant so in this case curly tau by curly y is constant and curly p by curly y should also be constant and since they are equal so we should be writing from equation number one that curly tau by oh sorry curly tau by curly y equal to curly p by here it is x curl x so curl tau will be equal to curl p by curl x d y okay now if we integrate with respect to y and with a certain limit so this could also be written as tau equal to y curl p by curl x now this is let us suppose equation number two now we have formula for tau x y tau x y sorry tau y x y x y x that k d u by d y rise raised to the power n and now we should say that equation number two tau y x and tau r one and the same thing that i have already written here so let us equate these two equations we will be getting after equating y curl p by curl x equal to k d u y d y raised to the power n okay this uh curly p by curly x could be written as minus y okay so okay this could be written as simply minus delta p by total length we should write l of the plate okay negative sign is due to the fact that we know delta p equal to always initial minus final so initial pressure or initial this curl p i'm talking about curl p is always initial minus final initial must be a little lesser than final that's why it's okay so this could be written as k d u by d y raised to the power n yeah okay so now we need to find the value of you so you should be writing this like d u by d y raised to the power n equal to minus y by k delta p by l which means d u by d y equal to minus y by k delta p by l raised to the power p by l raised to the power 1 by n okay and this should also be written as d u by d y equal to since we are to multiply we are to multiply like okay what is this good yes we have to integrate with respect to y because this is a d y so let us write it minus delta p by k l raised to the power 1 by n and y raised to the power 1 by n okay now let us write it d u equal to minus delta p by k l raised to the power 1 by n y raised to the power 1 by n so let us integrate it with respect to y but these all our constants so can be taken out so d u will be equal to minus delta p by k l raised to the power one by n and we have to integrate this so why raise to the power one by n plus one by one by n plus one and plus c which is the constant of integration so this is minus delta p by n and this will be y raised to the power one by n plus one by one plus n plus c okay i have made mistake this negative should be inside not upside okay so this is minus delta p by k l raised to the power one by n n y raised to the power one by n plus one or we should write n by n by 1 by n plus 1 is 1 plus n by n by n and divide by 1 plus n plus constant of integration c now we have a boundary condition that when y equal to h that is at the top of the plate at the upper surface of the plate that is here y equal to h you should be equal to 0 so let us plug in here so you should be equal to to 0 minus delta p by k l raised to the power 1 by n n by 1 plus n and this is y equal to h so h raised to the power 1 plus n plus c okay from here if we calculate we will be getting c equal to 1 by k l here should be delta p raised to the power 1 by n by 1 plus n and here 1 by n plus n h raised to the power 1 plus n by k so let us plug it back in original equation which is here we will be getting we will be getting u equal to u equal to delta p by k l raised to the power one by n by n by n plus one h raise to the power n plus one by n plus the c that is delta p by k l raised to the power one by n by n plus one h it is to the power one plus n by n okay so we should take this term as common and this term as common okay so certain terms which will be remaining okay is this good plus where is it yes i mean this is the negative sign sorry okay so delta p by k l raised to the power one by n will be taken as common n by n plus one h raised to the power n plus one by n will be taken as common it will be one minus one minus okay okay this is y yes okay yes this is why so i this was my mistake this should be equal to y and so this cannot be taken as common.

21:38 So these things can be taken as common.

21:41 So we will be left with h -d -r -raised -to -the -power 1 plus n by n minus y -raged to the power 1 plus n by n...

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