TOTAL HYDROSTATIC FORCE ON CURVED SURFACES

We were discussing the basic definition and derivation of total pressure; centre of pressure and we have also secured the expression for total pressure and centre of pressure for inclined plane surface submerged in liquid in our previous posts. 

Today we will be interested here to see the total hydrostatic force on curved surfaces, in the subject of fluid mechanics, with the help of this post. 

Let us consider a curved surface AB sub-merged in a static liquid as displayed here in following figure.

Let us consider one small strip area dA at a depth of h from free surface of liquid. We have following data from above figure.

A = Total area of curved surface

ρ = Density of the liquid

g = Acceleration due to gravity 

Pressure intensity on small area dA = ρ g h 

Hydrostatic force on small area dA will be given by following formula as mentioned here.

dF= ρ g h x dA

Direction of this hydrostatic force will be normal to the curved surface and will vary from point to point. Therefore, in order to secure the value of total hydrostatic force we will not integrate the above equation. 

We will secure the value or expression for total hydrostatic force on curved surface by resolving the force dF in its two components or we can say that dF force will be resolved in X direction i.e. dFx and in Y direction i.e. dFy.

dFx = dF Sin θ = ρ g h x dA Sin θ

dFy = dF Cos θ = ρ g h x dA Cos θ

Total force in X- direction and in Y- direction will be given as mentioned here.

Let us analyze the above equation

FG will be dA Sin θ or vertical projection of area dA. Therefore, the expression for Fx will be total pressure force on the projected area of the curved surface on the vertical plane. 

Fx = Total pressure force on the projected area of the curved surface on the vertical plane 

EG will be dA Cos θ or horizontal projection of dA. Therefore, the expression for Fy will be the weight of the liquid contained between the curved surface extended up to free surface of liquid. 

Fy = Weight of the liquid contained between the curved surface extended up to free surface of liquid

We will discuss the basic concept of buoyancy, in the subject of fluid mechanics, in our next post. 

Do you have any suggestions? Please write in comment box. 

Reference:

Fluid mechanics, By R. K. Bansal

Image Courtesy: Google

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