Derive an expression for forces acting on a submerged body.

A body wholly immersed in a real fluid may be subjected to the following two kinds of forces due to the relative motion between the body and fluid. These are:

• Drag Force

• Lift Force

Drag Force:

The component of force in the direction of flow (free stream) acting on a submerged body which tends to resist the motion is called drag force.Drag force is equal to Thrust force at steady condition of aircraft(When aircraft travelling with constant speed).

Lift Force:

The component of force at right angles to the direction of flow is called lift force.Lift is the force on the aircraft which is equal to weight force of the aircraft in steady condition. Lift is in vertically upward direction of the aircraft. This force keeps aircraft in the air.

Differences between drag and lift force:

A body wholly immersed in a real fluid may be subjected to the following two kinds of forces due to the relative motion between the body and fluid. These are:

• Drag Force

• Lift Force

Drag Force:

The component of force in the direction of flow (free stream) acting on a submerged body which tends to resist the motion is called drag force.Drag force is equal to Thrust force at steady condition of aircraft(When aircraft travelling with constant speed).

Lift Force:

The component of force at right angles to the direction of flow is called lift force.Lift is the force on the aircraft which is equal to weight force of the aircraft in steady condition. Lift is in vertically upward direction of the aircraft. This force keeps aircraft in the air.

Differences between drag and lift force:

Consider an arbitrary shaped solid body placed in a real fluid, which is flowing with a uniform velocity `U` in the horizontal direction as shown in figure. Let us consider a small elemental area `dA` on the surface of the body. Now, the forces acting on the small surface `dA` are:

• Pressure force equal to `P*dA` acting perpendicular to the surface.

• Shear force equal to `tau_0*dA` acting along the tangential direction to the surface.

Let `theta` be the inclination of the tangent to the element with the direction of flow.

Drag Force:

The component of force due to P and `tau_x` along the direction of motion is known as drag force.

Drag force on the elemental area= force due to pressure in the direction of flow + flow due to shear stress in teh direction of fluid motion.

`dF_d=P*dA*sintheta + tau_0dA*costheta`

Total drag force is,

`F_D=intP*dA*sintheta+inttau_0*dA*costheta`

Lift Force:

The component of force perpendicular to the direction of flow is called lift force.

Lift force on the elemental area= force due to pressure in the direction perpendicular to the direction of motion + flow due to shear stress in the direction perpendicular to the direction of motion

`dF_L=tau_0*dA*sintheta - P*dA*costheta`

Now, the total lift is,

`F_L=inttau_0*dA*sintheta-intP*dA*costheta`

The drag and lift for a body moving in a fluid of density `rho` at uniform velocity `U` are calculated as:

`F_D=C_D*(rho*U^2)/2*A`

and `F_L=C_L*(rho*U^2)/2*A`

Where, `C_D=` coefficient of drag

`C_L=` coefficient of life

`A=` characteristic area

For calculating drag force `F_D`, the area A is taken as area projected on the plane perpendicular to the relative motion of fluid.

• For calculating the lift force `F_L`, the area A is taken as area projected on a plane at right angle to the direction of lift force.

• In case of air foil, the projection is conventionally taken on the plane of the chord i.e, the area of wings itself, independent of its inclination to the direction of the flow.

• area= span `l`* mean chord `c `