 Sachin Deshmukh, working as an assistant professor in Vulture and Stop Technology. In today's session, we are going to learn about gradually varied flow and at the end of this session, you are able to explain the concept of gradually varied flow and you can able to derive the dynamic equation for gradually varied flow. Normally, it is called as a GVF equation. The definition of gradual varied flow is, it is a steady non-uniform flow in which the depth of flow or the elevation of water surface level varies gradually. That means, very slowly the characteristics are changing very slowly for long distance. The variation in the depth of flow may result due to change in longitudinal slope and frictional resistance of the channel. The gradual varied flow occurs near upstream side of the dam or we can say immediate downstream side of the sluice gate on the upstream side of the abrupt drop or fall we can say and in the channels which sudden changes in the longitudinal slope. What do you mean by total energy line and hydraulic gradient line? Total energy line is the line drawn with the addition of static potential and kinetic head and we can write that H total head is equal to Z that is static plus P by W that is potential plus kinetic that is V square upon 2G and hydraulic gradient line is drawn with addition of only static and potential head that is H is equal to Z plus P by W. Now, we will see what is a differential or dynamic equation for a gradual varied flow. Before starting of any, sorry for any equation, we must assume some realistic things that is channel is prismatic. The slope of the channel bottom is so small so that the depth measured in the vertical direction or normal to the channel pressure distribution is hydrostatic throughout the channel. The velocity distribution is practically fixed the head loss the head loss at any section is same for the uniform flow and the roughness coefficient of the channel bottom is independent of the flow. See this figure carefully this is a channel bottom and normally this slope is S0 it is called as S0 this blue line this is a water surface line and denoted as SW channel bottom is denoted as S0 it is SW and this top line is the energy grade line or energy gradient line this is denoted as SF particularly this SF I told you before also this is fluctuating somehow this is not constant. Now here this is section 1 this is section 2 this section is called as a controlled section this particular section is called as a controlled section where the characteristics are not going to be changed this is a datum line above this this is a datum head this is a pressure head and this is a velocity head or you can say kinetic head at section 1 similarly at section 2 this is a datum head this is a pressure head and this is a kinetic head or velocity head which is denoted as V square upon 2g consider the length between section 1 1 and section 2 2 of a gradual varied flow having a constant bottom slope S0 as shown in the figure A that is the lowest channel bottom slope that is the lowest one. Now here if you can see the figure again the total head at section 1 is Z1 plus Y1 plus V1 square upon 2g here at section 2 it is Z2 plus Y2 plus V2 square upon 2g alpha is a you can say you have to take we will take unity that is 1 that we will see but if you here you can measure head anywhere this is a addition of datum head plus pressure head plus your velocity head or kinetic head on the similar way when the total head or energy measured above the datum it is Z that is a datum head plus pressure head plus velocity head or kinetic head here it is again written Z is a distance of the channel bottom above the datum V is the average velocity Y is the depth measured normal to the channel S0 and alpha S0 is channel bottom you know it and alpha is the energy correction factor we have to introduce energy correction factor and we are taking that energy correction factor as 1. So the equation become H is equal to Z plus Y plus V square upon 2g this alpha is 1 assuming the channel bottom as an x axis so differentiating the above equation with respect to x that that is we are going ahead in the x axis differentiating dh by dx plus is equal to dz by dx plus dy by dx plus d by dx of V square upon 2g velocity we know that is discharge upon area so V square is q square upon a square substituting this value we will get the equation dh by dx is equal to dz by dx plus dy by dx plus dy dx of q square upon 2g s square we are we are getting the term dy by dx this is a very important term with respect to our concept of gradually varied flow this is nothing but the slope dy by dx and we are going to concentrate on this particular term see here dh by dx is a slope of energy line this is S0 and dz by dx is a slope of channel bottom it is Sf negative sign is given because when we travel in x axis direction the head is going to reduce so that it is written in minus it is given as a minus sign here I have written that is negative sign of the slope indicates that it is decreasing in the direction of the flow that means as we go forward the depth is going to reduce that is why minus Sf is equal to minus S0 plus dy by dx as it is plus d by dx of q square upon 2g s square this is the equation number 5 now we will see we will concentrate on this term that is q square upon 2g s square d by dx of q square upon 2g s square differentiate it we will get minus q square upon g a cube d by dy into dy by dx here see a simple figure of the channel this is a top width t this is a top width t and this is a y depth is y this is top width t and depth y here this particular term d by dy we are we are solving for this simplifying for this d by dy is nothing but dy dy of d into t here d is y so y into t dy get cancelled we will get the term t so d by dy is nothing but t in this term t please introduce in this equation we get we get d by dx of q square upon 2g s square is nothing but minus q square upon g a cube t bracket complete into dy by dx dy by dx is there substituting this value in the equation we get S0 which was there on the right hand side S0 which was there on the right hand side we are taking on the left hand side so S0 minus Sf is equal to dy by dx minus q square t upon g a cube dy by dx take dy by dx common S0 minus Sf is equal to because our aim is to find out what is a dy by dx that is why take dy by dx is common we will get 1 minus q square t upon g a cube or as I said dy by dx we are concentrating dy by dx is equal to S0 minus Sf upon 1 minus q square t upon g a cube this is the equation for a gradually varied flow now we can see here the terms q is there t is there g and a so again if you can simplify this again we can simplify this we will get instead of this q square t upon g a cube we will get the Froude's number so dy by dx if we can if we would like to derive in the sense of Froude's number we can introduce instead of q square t upon g a cube as a Froude's number square again we can modify this equation with respect to only depths where we can introduce Manning's formula as well as Chage's formula so that instead of these terms we will get the depths directly instead of these terms we will get depths directly and dy by dx this equation becomes again simpler if we can introduce Manning's formula and Chage's formula so that the critical depths normal depths and the field depths are introduced in the equation these are some reference books thank you.