Relative Equilibrium of Fluids: Examples, Problems, and Formulas!
Fig. 5.1 shows an open tank containing a liquid. When the tank moves horizontally with uniform acceleration, the surface of the liquid which was horizontal earlier will now be at some inclination θ with the horizontal. The level of liquid falls at the front of the tank and rises at the rear of the tank.
Now let us study the pressure variation on the liquid mass vertically and horizontally in the direction of motion.
Vertical Pressure Distribution:
Consider a vertical elemental cylinder of liquid of height h and area da the upper surface of the cylinder being at the liquid surface.
Since this vertical cylinder of liquid has no acceleration vertically the net vertical force on the cylinder is zero.
∴ pda – wdah = 0
∴ P = wh
This means, the pressure intensity varies vertically in a manner identical with the case of a liquid at rest.
Horizontal Pressure Distribution in the Direction of Accelerated Motion:
Let ax be the horizontal acceleration of the tank in the direction left to right. Let us consider an elemental horizontal cylinder of liquid of length L and area da. Let the left and the right ends of the cylinder of liquid be at depths h1 and h2 below the liquid surface.
Thus we find that the liquid surface has a uniform slope which depends entirely on the acceleration of the tank. The intensity of pressure on any plane parallel to the liquid surface will be uniform. If the tank moves with uniform velocity i.e., when there is no acceleration, tan θ = 0 and the liquid surface would remain horizontal.
Fig. 5.12 shows a tank containing a liquid accelerated vertically upwards. In this case the surface of the liquid will remain horizontal.
Consider a vertical elemental cylinder of liquid of height h and area da, the upper surface of the cylinder being at the liquid surface.
(a) Acceleration up the Slope:
Fig. 5.16 shows a tank moving up an inclined plane with an acceleration a. Let α be the inclination of the plane with the horizontal. Let θ be the inclination of the liquid surface with the horizontal.
The horizontal and vertical components of the acceleration are given by-
(b) Acceleration down the Slope:
Let the tank move down the slope at an acceleration a. See Fig. 5.17.
In this case proceeding similar to the previous case, the forces acting on a particle A on the liquid surface are the following:
(i) Weight of the particle = mg acting vertically downwards.
(ii) Pressure force F exerted by the fluid body on the particle A acting normal to the liquid surface.
To provide more stability to the tank cars the position of the wheels in relation to the tank is made as shown in Fig. 5.18. In this arrangement, when the tank car is at rest and depth of water at the two ends will be the same. This arrangement is very helpful particularly when the tank moves down the slope.
We know that any fluid mass having a motion along a curve possesses radial acceleration. The direction of the radial acceleration at any point in the path of the motion is towards the centre of motion.
This means any fluid mass in motion along a curve is under the action of a radially inward force. This force is the centripetal force. Such centripetal forces are produced by corresponding pressure changes. When a fluid continuously flows along a curved path about a fixed axis, the flow is called Vortex flow or whirling flow.
For the sake of simplicity let us assume that the fluid particles under discussion are in motion in horizontal plane (in order to avoid variation of potential energy). Let us consider an elemental cylindrical fluid element of area da and length ds, the axis of the cylinder being along the stream line. Let the pressure intensities at the ends of the cylinder be p and p + dp respectively.
Now let us consider an elemental cylindrical fluid element whose axis is in a radial direction, with respect to the stream line. Let da be the area of the cylinder and dr be its length. The ends of the cylinder are situated at radii r and r + dr respectively. Let the pressure intensities at these radii be p and p + dp respectively. Let v be the velocity along the stream line.
This is a vortex flow in which the fluid masses are made to move in a curved path under the action of an external agency. If the external agency acts continuously and constantly the entire fluid body will rotate with the constant angular velocity.
A forced vortex will occur when a cylindrical vessel containing a liquid is rotated about the vertical axis, with a uniform angular velocity. Let a vertical cylinder of radius R containing a liquid be rotated about its axis at an angular velocity ω.
This means the pressure and velocity heads increase together and equally, i.e.
Increase in pressure head = Increase in velocity head.
If the vessel is open at the upper end, and is large enough the liquid surface will take the profile shown in Fig. 5.30. Let C be the lowest point of the profile. Let P be any point on the profile. Let the coordinates of P be (x, y) with respect to C. Consider the points C and E. See Fig. 5.30.
Rise at End and Fall at Centre of the Liquid Surface:
Let MN be the liquid surface in the cylinder when the system is in absolute equilibrium.
When the cylinder is rotated, let ACB be the profile of the liquid surface.
Volumes of liquid above the level DE before and after rotation are equal.
Consider a closed cylinder of radius R and height H completely filled with a liquid. Let this cylinder be rotated (with its axis vertical) about its axis.
Free vortex is a vortex or circular flow in which no external energy is supplied. In this type of vortex flow the total energy head is the same at all points of the liquid.