Harriet Dart’s hopes of breaking new ground at the US Open came to a ­bitter end in oppressive New York heat as she fell 7-6 (10), 6-1 in the second round to the 19th seed Marta Kostyuk of Ukraine.

Having only reached the third round of a grand slam on home soil at Wimbledon, Dart faced off against Kostyuk seeking her best result in New York. The 28-year-old has played some of the finest tennis in recent weeks, embedding herself inside the top 100 with a career-high ranking of 75. She started at Flushing Meadow with a dominant opening win against Chloé Paquet of France.

The conditions were suffocating from the beginning, the 34C heat feeling far worse both on and off the court. Both players struggled, with Kostyuk finding relief at the change of ends by placing bags of ice under each thigh and one on her head while the portable air conditioning machine blasted cold air in her face.

For much of the first set, Dart was clearly the better player against an erratic Kostyuk who was clearly battling to find her level. As the more solid and composed player in the exchanges, Dart took an early break and then held on to it until she served for the set at 5-3. She then served for the set again at 6-5. In the tie-break, the Briton then held five set points in total, including three in a row at 6-3.

But while Dart can hold her own from the baseline with many of the elite with her defence, ­consistency and her excellent ­ability to ­redirect pace off both sides, her serve is such a glaring weakness at the ­highest level. In the decisive moments of the opening set, her first serve ­percentage crumbled and her second serves floated over the net at around 65 miles per hour. After finally closing out the set, Kostyuk opened up her shoulders, struck the ball with freedom and marched into the third round.

After advancing into the third round when Alexander Shevchenko retired while two sets down, Frances Tiafoe said he prepared for the heat by bringing 20 shirts and three pairs of shoes on to the court: “Try to be overprepared,” he said. “I’m sweating crazy. Today, what was it, 95 degrees outside? Today was crazy.

“Yeah, I just try to bring as much as I can. I’ve got 20 shirts, I’ve got a bunch of shoes. I’m happy I’ve been able to keep everything with me, to be honest with you, because I lose everything. I’ve been doing pretty well with that so far, especially in the majors.”