Dear Dr. Hayder, good question according to Bernoulli equation the pressure is inversely proportional to velocity but pay attention this only when we consider just the advection effects not with diffusion effects, here with diffusion effects if we are solving Navier-stocks equations for fully develop region and viscous flow in straight channel as in your case, we will see that the (pressure remain constant in Y-direction) while the velocity changed in this direction here which can be conclude that we can not always say that the pressure proportional to velocity especially with diffusion effects. regards

You can understand the link between pressure (actually pressure gradient) and velocity (actually velocity gradient) in a plane channel by the exact Poiseuille solution. The momentum equation reduces to dp/dx = 1/Re d2u/dy2= constant.

Dear Haydar,

Pressure is inversely proportional to velocity, according to Bernoulli equation :

P/rho + V^2/2 + gz = constant

With my best regards

Prof. Bachir ACHOUR

You may have confused with the actual value and the gradient of pressure. It is the pressure gradient term which is proportional to the velocity between two points.

Dear Dr. Hayder, good question according to Bernoulli equation the pressure is inversely proportional to velocity but pay attention this only when we consider just the advection effects not with diffusion effects, here with diffusion effects if we are solving Navier-stocks equations for fully develop region and viscous flow in straight channel as in your case, we will see that the (pressure remain constant in Y-direction) while the velocity changed in this direction here which can be conclude that we can not always say that the pressure proportional to velocity especially with diffusion effects. regards

Dear Bachir,

the original question was for the NS equations, thus you cannot use Bernoulli to express the relation between pressure and velocity. Of course, if you assume a vanishing viscosity, Re->+Inf and the pressure gradient in the channel vanishes, recovering Bernoulli.

Regards

Filippo

Dear Filippo Maria Denaro, Bachir Achour, Vikash Kumar Singh Chauhan and Mohammed saad Kamel,

I appreciate your comments.

Thank you very much.

Regards

Following Bernoulli’s law, an increase in velocity causes a significant decrease in the static pressure. Kindly see

https://www.quora.com/What-is-the-relationship-between-velocity-and-pressure

Dear Dr. Hayder

For any flow between two points the fluid energy is represented in terms of total pressure or stagnation pressure. For an adiabatic flow without friction the stagnation pressure is constant. Ans since the stagnation pressure is composed of static and dynamic components, then any increase in dynamic pressure (velocity) is accompanied by a decrease in static pressure. This would occur only when the flow cross sectional area changes. Therefore the relation in adiabatic invicid flow is always inversely proportional. If the channel is straight (cross sectional area is constant) then nothing will affect the pressure or velocity if no friction or heat transfer occur. In your case, I think, friction and heat transfer tend to change both static and dynamic pressure in a way that they change in proportional manner.