# The mathematical meaning of a zero gradient pressure boundary condition in the Navier-Stokes equations

I would like to solve the Navier-Stokes equations for the unsteady problem of the flow around a circular cylinder. I would like to understand how to write mathematically the boundary condition for the pressure on the surface of the cylinder (essentially the circle). The domain of the problem is shown in the image below with the cylinder having a diameter D and centered in the origin.

I usually solve this problem using the CFD solver OpenFOAM, therefore, I will describe the boundary conditions for this problem which are common among OpenFOAM users. At the inlet the velocity $$u$$ is $$(1,0)$$ and zero gradient pressure is imposed. On the sides the so-called slip condition is imposed for velocity and zero gradient is used for pressure, while on the outlet we have zero gradient for velocity and zero pressure. In OpenFOAM, zero gradient pressure is often used for the obstacles in the domain such as the cylinder in this case and a zero velocity is assumed. Whenever a zero gradient condition is mentioned, what is meant is zero gradient in the direction perpendicular to the flow (for example on the outlet only the $$x$$-derivative is equal to zero for the velocity components $$U$$ and $$V$$ that is $$U_x = V_x = 0$$).

For the case of the cylinder boundary, is it correct to assume that the scalar product between the pressure gradient $$\nabla p = (p_x,p_y)$$ and the normal vector at the boundary $$\boldsymbol{n} = \frac{(x,y)}{0.5D}$$ has to be zero, in other words:

$$\nabla p \cdot \boldsymbol{n} = x p_x + y p_y = 0$$

Is that the correct formulation of the pressure boundary condition ? and is there another physical condition for the pressure at the cylinder boundary ?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
May 3, 2022 at 16:33
• Are you assuming that the circle is centered at the origin? May 3, 2022 at 17:25
• Separately, though: Why do you think that this is the "correct" boundary condition? What is the physical situation you are trying to model? May 3, 2022 at 17:25
• I have updated the question in order to give more details about the problem, @WolfgangBangerth yes the cylinder is centered at the origin, as for the boundary condition, I just used what is typically used in OpenFOAM, the point now is that I am trying to formulate the problem in another solver using OpenFOAM settings that's why I need to understand how each condition is written mathematically. Thank you. May 4, 2022 at 10:16
• The formula you show looks correct to me. Though I have no idea what a zero (normal) gradient pressure would actually mean. It's not a boundary condition I've ever encountered, and one cannot impose boundary conditions on the pressure to begin with in the Navier-Stokes equations. It may be the boundary condition chosen for the pressure Schur complement in split schemes, though. May 4, 2022 at 15:26