Fluid Dynamics: Calculating Force on Hemisphere in Uniform Flow

In summary, the problem involves calculating the force on each hemisphere of a sphere surrounded by a fluid with a uniform velocity. The pressure far from the sphere is assumed to be constant and the fluid is inviscid and irrotational. The equations used in the solution are the potential flow equation and the Bernoulli equation.
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jc09
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Homework Statement



A sphere of radius a is surrounded by a fluid, flowing with a uniform velocity U far from the sphere. Imagine the sphere divided into two hemispheres by a plane perpendicular to the direction of flow. Calculate the force on each hemisphere, assuming the pressure far from sphere is P. Hence show that the effect of the flow is to decrease the force on each hemisphere.

Assume fluid is both inviscid, of constant density and irrotational.


I am not sure how to start this problem at all. Would I use mapping of images or how would I start this off...
 
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Homework Equations The potential flow equation:$$\phi_x \frac{du}{dy}-\phi_y \frac{du}{dx}=0$$The Bernoulli equation:$$\frac{1}{2}\rho (u^2+v^2)+P=\text{constant} $$The Attempt at a Solution I am not sure how to start this problem at all. Would I use mapping of images or how would I start this off...
 

FAQ: Fluid Dynamics: Calculating Force on Hemisphere in Uniform Flow

How is force calculated in fluid dynamics?

The force on a hemisphere in uniform flow can be calculated by using the equation F = ρAV2, where ρ is the density of the fluid, A is the surface area of the hemisphere, and V is the velocity of the fluid.

What is meant by uniform flow in fluid dynamics?

Uniform flow refers to the condition in which the fluid has a constant velocity and direction over a given area. In other words, the flow does not vary in any way within the designated area.

How does the shape of the hemisphere affect the force calculation?

The shape of the hemisphere can affect the force calculation as it determines the surface area (A) in the equation. A larger surface area means a greater force acting on the hemisphere.

Can fluid density affect the calculation of force?

Yes, fluid density (ρ) plays a crucial role in the force calculation as it is a key component in the equation. A denser fluid will result in a higher force acting on the hemisphere.

What is the significance of calculating force on a hemisphere in fluid dynamics?

Calculating force on a hemisphere in fluid dynamics is important in understanding the overall dynamics and behavior of fluid flow. It can also help in designing and optimizing structures that are exposed to fluid forces, such as ships, airplanes, and hydroelectric dams.

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